{"id":26188,"date":"2026-07-03T13:53:20","date_gmt":"2026-07-03T13:53:20","guid":{"rendered":"https:\/\/umang.pk\/2026\/07\/03\/%d9%be%d8%b1%d9%88%da%88%da%a9%d9%b9-%da%a9%d8%a7-%d8%aa%d8%ac%d8%b1%d8%a8%db%81-%d8%a7%db%8c%d9%85-%d8%a7%db%8c%d8%b3-%d9%be%db%8c-%d8%a2%d8%b1-%d9%b9%db%8c-%da%a9%d8%a7-%d8%a7%d8%b3%d8%aa%d8%b9\/"},"modified":"2026-07-03T13:53:22","modified_gmt":"2026-07-03T13:53:22","slug":"%d9%be%d8%b1%d9%88%da%88%da%a9%d9%b9-%da%a9%d8%a7-%d8%aa%d8%ac%d8%b1%d8%a8%db%81-%d8%a7%db%8c%d9%85-%d8%a7%db%8c%d8%b3-%d9%be%db%8c-%d8%a2%d8%b1-%d9%b9%db%8c-%da%a9%d8%a7-%d8%a7%d8%b3%d8%aa%d8%b9","status":"publish","type":"post","link":"https:\/\/umang.pk\/en_us\/2026\/07\/03\/%d9%be%d8%b1%d9%88%da%88%da%a9%d9%b9-%da%a9%d8%a7-%d8%aa%d8%ac%d8%b1%d8%a8%db%81-%d8%a7%db%8c%d9%85-%d8%a7%db%8c%d8%b3-%d9%be%db%8c-%d8%a2%d8%b1-%d9%b9%db%8c-%da%a9%d8%a7-%d8%a7%d8%b3%d8%aa%d8%b9\/","title":{"rendered":"\u067e\u0631\u0648\u0688\u06a9\u0679 \u06a9\u0627 \u062a\u062c\u0631\u0628\u06c1: \u0627\u06cc\u0645 \u0627\u06cc\u0633 \u067e\u06cc \u0622\u0631 \u0679\u06cc \u06a9\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u067e\u06cc-\u06c1\u06cc\u06a9\u0646\u06af \u06a9\u06d2 \u0628\u063a\u06cc\u0631 \u0627\u0628\u062a\u062f\u0627\u0626\u06cc \u0631\u0648\u06a9\u0646\u0627 \u0627\u0648\u0631 \u0627\u0632\u06af\u0631 \u0645\u06cc\u06ba \u062a\u0631\u062a\u06cc\u0628 \u0648\u0627\u0631 \u062c\u0627\u0646\u0686"},"content":{"rendered":"\n<div id=\"\">\n<p>\u0622\u067e \u06a9\u06d2 AI \u067e\u0631\u0648\u0688\u06a9\u0679 \u06a9\u0627 \u062a\u062c\u0631\u0628\u06c1 30 \u062f\u0646 \u06a9\u06cc \u0645\u0646\u0635\u0648\u0628\u06c1 \u0628\u0646\u062f \u062f\u0648\u0691 \u06a9\u06d2 14 \u062f\u0646 \u06a9\u0648 \u0634\u0645\u0627\u0631\u06cc\u0627\u062a\u06cc \u0627\u06c1\u0645\u06cc\u062a \u062a\u06a9 \u067e\u06c1\u0646\u0686 \u06a9\u0631 \u06a9\u0627\u0631\u0622\u0645\u062f \u0627\u0633\u062a\u0646\u0628\u0627\u0637\u06cc \u0633\u0648\u0627\u0644\u0627\u062a \u06a9\u06cc \u067e\u06cc\u0645\u0627\u0626\u0634 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u06a9\u06cc\u0627 LLM \u067e\u0631 \u0645\u0628\u0646\u06cc \u062e\u0635\u0648\u0635\u06cc\u0627\u062a \u0646\u06d2 \u062d\u0642\u06cc\u0642\u062a \u0645\u06cc\u06ba \u0646\u062a\u0627\u0626\u062c \u06a9\u0648 \u0628\u06c1\u062a\u0631 \u0628\u0646\u0627\u06cc\u0627\u061f \u06a9\u0645\u0631\u06d2 \u0645\u06cc\u06ba \u0645\u0648\u062c\u0648\u062f \u06c1\u0631 \u067e\u0631\u0648\u0688\u06a9\u0679 \u0645\u06cc\u0646\u06cc\u062c\u0631 \u0688\u06cc\u0644\u06cc\u0648\u0631 \u06a9\u0631\u0646\u0627 \u0686\u0627\u06c1\u062a\u0627 \u06c1\u06d2\u06d4 \u0634\u0645\u0627\u0631\u06cc\u0627\u062a \u06a9\u0627 \u06a9\u06c1\u0646\u0627 \u06c1\u06d2 \u06a9\u06c1 30 \u062f\u0646 \u0627\u0646\u062a\u0638\u0627\u0631 \u06a9\u0631\u06cc\u06ba\u06d4 \u0628\u0635\u0648\u0631\u062a \u062f\u06cc\u06af\u0631\u060c p-value \u063a\u0644\u0637 \u06c1\u06d2\u06d4<\/p>\n<p>\u0628\u0631\u0627\u06c1 \u06a9\u0631\u0645 \u0627\u0646\u062a\u0638\u0627\u0631 \u06a9\u0631\u06cc\u06ba\u06d4 \u0627\u062b\u0631 30 \u062f\u0646 \u06a9\u06d2 \u0628\u0639\u062f \u0628\u06be\u06cc \u0648\u06c1\u06cc \u0631\u06c1\u062a\u0627 \u06c1\u06d2\u06d4 \u062a\u0627\u06c1\u0645\u060c 95% \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0627\u06cc\u06a9 \u0645\u0639\u0631\u0648\u0641 \u0641\u0646\u06a9\u0634\u0646 \u06a9\u0648 \u0627\u0646\u062c\u0627\u0645 \u062f\u06cc\u0646\u06d2 \u0645\u06cc\u06ba 16 \u062f\u0646 \u06af\u0632\u0627\u0631\u0646\u06d2 \u0633\u06d2 \u0627\u06af\u0644\u06d2 \u062a\u062c\u0631\u0628\u06d2 \u0645\u06cc\u06ba \u062a\u0627\u062e\u06cc\u0631 \u06c1\u0648\u0626\u06cc \u0627\u0648\u0631 \u06c1\u0645\u06cc\u06ba \u0645\u0648\u0642\u0639 \u0636\u0627\u0626\u0639 \u06c1\u0648\u0627\u06d4<\/p>\n<p>\u0627\u06af\u0631 \u0622\u067e \u06a9\u0644\u0627\u0633\u06a9 \u0641\u06a9\u0633\u0688 \u0633\u06cc\u0645\u067e\u0644 \u0679\u06cc\u0633\u0679 \u0686\u0644\u0627 \u0631\u06c1\u06d2 \u06c1\u06cc\u06ba\u060c \u062a\u0648 \u0634\u0645\u0627\u0631\u06cc\u0627\u062a \u062f\u0627\u0646 \u062a\u06a9\u0646\u06cc\u06a9\u06cc \u0637\u0648\u0631 \u067e\u0631 \u062f\u0631\u0633\u062a \u06c1\u06d2\u06d4 \u0645\u0639\u06cc\u0627\u0631\u06cc t-\u0679\u06cc\u0633\u0679 \u06a9\u06cc p-value \u0635\u0631\u0641 \u0627\u0633 \u0635\u0648\u0631\u062a \u0645\u06cc\u06ba \u062f\u0631\u0633\u062a \u06c1\u06d2 \u062c\u0628 \u0622\u067e \u0646\u0645\u0648\u0646\u06d2 \u06a9\u06d2 \u0633\u0627\u0626\u0632 \u06a9\u06cc \u067e\u06c1\u0644\u06d2 \u0633\u06d2 \u0648\u0636\u0627\u062d\u062a \u06a9\u0631\u06cc\u06ba \u0627\u0648\u0631 \u0646\u062a\u0627\u0626\u062c \u06a9\u0648 \u0627\u06cc\u06a9 \u0628\u0627\u0631 \u0686\u06cc\u06a9 \u06a9\u0631\u06cc\u06ba\u06d4 \u0627\u06af\u0631 \u0622\u067e \u067e\u06c1\u0644\u06d2 \u062f\u06cc\u06a9\u06be\u062a\u06d2 \u06c1\u06cc\u06ba \u0627\u0648\u0631 p <0.05 \u067e\u0631 \u0631\u06a9 \u062c\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u062a\u0648 \u063a\u0644\u0637 \u0645\u062b\u0628\u062a \u0634\u0631\u062d 30% \u062a\u06a9 \u062c\u0627\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<p>p-value \u06a9\u0648 \u0627\u06cc\u06a9 \u06c1\u06cc \u067e\u06c1\u0644\u06d2 \u0633\u06d2 \u06a9\u0645\u0679\u0688 \u0634\u06a9\u0644 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0688\u06cc\u0632\u0627\u0626\u0646 \u06a9\u06cc\u0627 \u06af\u06cc\u0627 \u06c1\u06d2\u06d4 \u06cc\u0639\u0646\u06cc\u060c \u06cc\u06c1 \u0645\u0642\u0631\u0631\u06c1 \u0627\u062e\u062a\u062a\u0627\u0645\u06cc \u0646\u0642\u0637\u0648\u06ba \u06a9\u06d2 \u0633\u0627\u062a\u06be \u062c\u0627\u0645\u062f \u062a\u062c\u0631\u0628\u0627\u062a \u06a9\u06d2 \u0644\u06cc\u06d2 \u0628\u0646\u0627\u06cc\u0627 \u06af\u06cc\u0627 \u06c1\u06d2\u06d4 \u0627\u0633\u06d2 \u0644\u0627\u0626\u06cc\u0648 \u0633\u0679\u0631\u06cc\u0645 \u0645\u06cc\u06ba \u0644\u0627\u06af\u0648 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u062c\u0633\u06d2 \u06a9\u0633\u06cc \u0628\u06be\u06cc \u0648\u0642\u062a \u062f\u06cc\u06a9\u06be\u0627 \u062c\u0627 \u0633\u06a9\u062a\u0627 \u06c1\u06d2 \u0631\u06cc\u0627\u0636\u06cc\u0627\u062a\u06cc \u0627\u0634\u06cc\u0627\u0621 \u06a9\u06d2 \u0628\u0627\u0644\u06a9\u0644 \u0645\u062e\u062a\u0644\u0641 \u0633\u06cc\u0679 \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u0648\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<p>\u062a\u0631\u062a\u06cc\u0628 \u0648\u0627\u0631 \u062c\u0627\u0646\u0686 \u0628\u0627\u0644\u06a9\u0644 \u0627\u0646 \u062d\u0627\u0644\u0627\u062a \u06a9\u06d2 \u0644\u06cc\u06d2 \u0688\u06cc\u0632\u0627\u0626\u0646 \u06a9\u06cc \u06af\u0626\u06cc \u06c1\u06d2\u06d4 \u0645\u06a9\u0633\u0688 \u0633\u06cc\u06a9\u0648\u06cc\u0646\u0634\u0644 \u067e\u0631\u0648\u0628\u06cc\u0628\u0644\u0679\u06cc \u0631\u06cc\u0634\u0648 \u0679\u06cc\u0633\u0679 (mSPRT) (Johari et al.) \u0627\u06cc\u06a9 \u0631\u06cc\u0627\u0636\u06cc\u0627\u062a\u06cc \u06c1\u0633\u062a\u06cc \u06a9\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u062c\u0633\u06d2 e-value \u06a9\u06c1\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2 \u062a\u0627\u06a9\u06c1 \u06c1\u0645\u06cc\u0634\u06c1 \u062f\u0631\u0633\u062a \u0627\u0646\u062f\u0627\u0632\u06c1 \u0644\u06af\u0627\u06cc\u0627 \u062c\u0627 \u0633\u06a9\u06d2\u06d4 \u0646\u062a\u0627\u0626\u062c \u06a9\u0648 \u0631\u0648\u0632\u0627\u0646\u06c1 \u0686\u06cc\u06a9 \u06a9\u06cc\u0627 \u062c\u0627 \u0633\u06a9\u062a\u0627 \u06c1\u06d2 \u0627\u0648\u0631 \u062b\u0628\u0648\u062a \u06a9\u0627\u0641\u06cc \u0645\u0636\u0628\u0648\u0637 \u06c1\u0648\u0646\u06d2 \u067e\u0631 \u0631\u0648\u06a9\u0627 \u062c\u0627 \u0633\u06a9\u062a\u0627 \u06c1\u06d2\u060c \u062c\u0633 \u0645\u06cc\u06ba \u063a\u0644\u0637 \u0645\u062b\u0628\u062a \u0634\u0631\u062d 5% \u0631\u06c1 \u062c\u0627\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<p>Netflix \u0646\u06d2 \u06c1\u0645\u06cc\u0634\u06c1 \u062f\u0631\u0633\u062a \u062a\u0631\u062a\u06cc\u0628 \u0648\u0627\u0631 \u062c\u0627\u0646\u0686 \u06a9\u06d2 \u0641\u0631\u06cc\u0645 \u0648\u0631\u06a9 (Lindon et al.) \u06a9\u06d2 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631\u06cc \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0648 \u062f\u0633\u062a\u0627\u0648\u06cc\u0632\u06cc \u0634\u06a9\u0644 \u062f\u06cc \u06c1\u06d2\u060c \u0627\u0648\u0631 \u0628\u0646\u06cc\u0627\u062f\u06cc \u062e\u06cc\u0627\u0644 \u062a\u0631\u062a\u06cc\u0628 \u0648\u0627\u0631 \u062a\u062c\u0632\u06cc\u06c1 \u067e\u0631 \u0648\u0627\u0644\u0688 \u06a9\u06d2 1945 \u06a9\u06d2 \u06a9\u0627\u0645 \u0627\u0648\u0631 \u0648\u06cc\u0644 \u06a9\u06cc 1939 \u06a9\u06cc \u0639\u062f\u0645 \u0645\u0633\u0627\u0648\u0627\u062a \u067e\u0631 \u0648\u0627\u067e\u0633 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u0633 \u0679\u06cc\u0648\u0679\u0648\u0631\u06cc\u0644 \u0645\u06cc\u06ba\u060c \u0622\u067e \u0648\u0627\u0636\u062d \u0637\u0648\u0631 \u067e\u0631 \u06a9\u0646\u06a9\u0634\u0646 \u0628\u0646\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0641\u0644\u0627\u0626\u06cc\u0679\u0688 \u0627\u06cc\u0631\u0631 \u0631\u06cc\u0679\u0633 \u06a9\u0648 \u0628\u0631\u0627\u06c1 \u0631\u0627\u0633\u062a \u062f\u06cc\u06a9\u06be\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2\u060c \u06c1\u0645 \u062c\u06be\u0627\u0646\u06a9\u0646\u06d2 \u06a9\u06d2 \u0645\u0633\u0626\u0644\u06d2 \u06a9\u06cc \u0646\u0642\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u060c Python \u0645\u06cc\u06ba \u0634\u0631\u0648\u0639 \u0633\u06d2 \u0627\u06cc\u06a9 \u0648\u0631\u06a9\u0646\u06af mSPRT \u06a9\u0648 \u0644\u0627\u06af\u0648 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u0627\u0633\u06d2 \u0645\u0634\u062a\u0631\u06a9\u06c1 \u0645\u0635\u0646\u0648\u0639\u06cc LLM \u067e\u0631\u0648\u0688\u06a9\u0679 \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u067e\u0631 \u0644\u0627\u06af\u0648 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u0627\u0648\u0631 \u062c\u0628 \u062a\u0631\u062a\u06cc\u0628 \u0648\u0627\u0631 \u0679\u06cc\u0633\u0679 \u0646\u0627\u06a9\u0627\u0645 \u06c1\u0648 \u062c\u0627\u062a\u06d2 \u06c1\u06cc\u06ba \u062a\u0648 \u0628\u0627\u0644\u06a9\u0644 \u0633\u0645\u062c\u06be\u06cc\u06ba\u06d4<\/p>\n<p><strong>\u0633\u0627\u062a\u06be\u06cc \u0644\u06cc\u067e \u0679\u0627\u067e:<\/strong> \u0627\u0633 \u062f\u0633\u062a\u0627\u0648\u06cc\u0632 \u06a9\u06d2 \u062a\u0645\u0627\u0645 \u06a9\u0648\u0688 \u0628\u0644\u0627\u06a9\u0633 \u06a9\u0648 \u0622\u062e\u0631 \u0633\u06d2 \u0622\u062e\u0631 \u062a\u06a9 \u0639\u0645\u0644 \u0645\u06cc\u06ba \u0644\u0627\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4 <code>msprt_demo.ipynb<\/code> \u06cc\u06c1 \u0633\u0627\u062a\u06be\u06cc \u0630\u062e\u06cc\u0631\u06c1 \u0645\u06cc\u06ba \u06c1\u06d2\u06d4<\/p>\n<h2 id=\"heading-table-of-contents\">\u0627\u0646\u0688\u06cc\u06a9\u0633<\/h2>\n<h2 id=\"heading-why-optional-stopping-breaks-classical-tests\">\u0645\u0646\u062a\u062e\u0628 \u0631\u06a9\u0646\u06d2 \u0633\u06d2 \u0645\u0648\u062c\u0648\u062f\u06c1 \u0679\u06cc\u0633\u0679 \u06a9\u06cc\u0648\u06ba \u0679\u0648\u0679 \u062c\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/h2>\n<p>\u0686\u0644\u062a\u06cc \u06c1\u0648\u0626\u06cc p-value \u06a9\u0648 \u062c\u06be\u0627\u0646\u06a9\u0646\u06d2 \u0633\u06d2 \u063a\u0644\u0637 \u0645\u062b\u0628\u062a \u0634\u0631\u062d 30% \u062a\u06a9 \u0628\u0691\u06be \u062c\u0627\u062a\u06cc \u06c1\u06d2\u06d4 \u06cc\u06c1 \u0648\u06c1 \u0646\u0645\u0628\u0631 \u06c1\u06d2 \u062c\u0648 \u0622\u067e \u06a9\u0648 \u062a\u0648\u0642\u0641 \u062f\u06cc\u062a\u0627 \u06c1\u06d2\u060c \u0627\u0648\u0631 \u0622\u0626\u06cc\u06d2 \u0627\u0633\u06d2 \u0630\u06cc\u0644 \u0645\u06cc\u06ba \u0645\u0631\u062d\u0644\u06c1 1 \u0645\u06cc\u06ba \u062f\u0648\u0628\u0627\u0631\u06c1 \u067e\u06cc\u0634 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<p>\u06a9\u0644\u0627\u0633\u06a9 \u0645\u0641\u0631\u0648\u0636\u06d2 \u06a9\u06d2 \u0679\u06cc\u0633\u0679 \u06a9\u06cc p-value \u0627\u06cc\u06a9 \u0645\u062e\u0635\u0648\u0635 \u0633\u0648\u0627\u0644 \u06a9\u0627 \u062c\u0648\u0627\u0628 \u062f\u06cc\u062a\u06cc \u06c1\u06d2\u06d4 \u0627\u06af\u0631 null \u062f\u0631\u0633\u062a \u06c1\u06d2 \u062a\u0648\u060c \u0627\u06af\u0631 \u0622\u067e \u067e\u06c1\u0644\u06d2 \u0633\u06d2 \u0648\u0639\u062f\u06c1 \u06a9\u0631\u062f\u06c1 \u0646\u0645\u0648\u0646\u06d2 \u06a9\u06d2 \u0633\u0627\u0626\u0632 \u06a9\u0648 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u062a\u062c\u0631\u0628\u06c1 \u06a9\u0648 \u0628\u0627\u0644\u06a9\u0644 \u0679\u06be\u06cc\u06a9 \u0645\u0646\u0635\u0648\u0628\u06c1 \u0628\u0646\u062f\u06cc \u06a9\u06d2 \u0645\u0637\u0627\u0628\u0642 \u0686\u0644\u0627\u062a\u06d2 \u06c1\u06cc\u06ba \u062a\u0648 \u0627\u0633 \u0637\u0631\u062d \u06a9\u06d2 \u0627\u0646\u062a\u06c1\u0627\u0626\u06cc \u0688\u06cc\u0679\u0627 \u06a9\u0648 \u062f\u06cc\u06a9\u06be\u0646\u06d2 \u06a9\u06cc \u06a9\u06cc\u0627 \u0645\u0634\u06a9\u0644\u0627\u062a \u06c1\u06cc\u06ba\u061f<\/p>\n<p>&quot;\u0628\u0627\u0644\u06a9\u0644 \u062c\u06cc\u0633\u0627 \u06a9\u06c1 \u0645\u0646\u0635\u0648\u0628\u06c1 \u0628\u0646\u062f\u06cc \u06a9\u06cc \u06af\u0626\u06cc \u06c1\u06d2&#8221; \u0634\u0642 \u0645\u0634\u06a9\u0644 \u06c1\u06d2\u06d4 \u0627\u06af\u0631 \u0622\u067e \u062f\u0646 5\u060c 10 \u0627\u0648\u0631 14 \u06a9\u0648 \u0646\u062a\u0627\u0626\u062c \u0686\u06cc\u06a9 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u0627\u0648\u0631 14 \u062f\u0646 \u06a9\u0648 \u0631\u0648\u06a9\u062a\u06d2 \u06c1\u06cc\u06ba \u06a9\u06cc\u0648\u0646\u06a9\u06c1 p <0.05\u060c \u062a\u0648 \u0622\u067e \u0646\u06d2 \u0645\u0646\u0635\u0648\u0628\u06c1 \u0628\u0646\u062f\u06cc \u06a9\u06d2 \u0645\u0637\u0627\u0628\u0642 \u062a\u062c\u0631\u0628\u06c1 \u0646\u06c1\u06cc\u06ba \u0686\u0644\u0627\u06cc\u0627\u06d4 \u06c1\u0645 \u0646\u06d2 14 \u0645\u062e\u062a\u0644\u0641 \u062a\u062c\u0631\u0628\u0627\u062a \u06a9\u06cc\u06d2\u060c \u06c1\u0631 \u062a\u062c\u0631\u0628\u06d2 \u06a9\u06d2 \u0646\u062a\u0627\u0626\u062c \u06a9\u0648 \u062f\u06cc\u06a9\u06be\u0627\u060c \u0627\u0648\u0631 \u062d\u062f \u0633\u06d2 \u062a\u062c\u0627\u0648\u0632 \u06a9\u0631\u0646\u06d2 \u0648\u0627\u0644\u0648\u06ba \u067e\u0631 \u0631\u06a9 \u06af\u0626\u06d2\u06d4 p-value \u0641\u0627\u0631\u0645\u0648\u0644\u06c1 \u06cc\u06c1 \u0646\u06c1\u06cc\u06ba \u062c\u0627\u0646\u062a\u0627\u06d4<\/p>\n<p>\u06cc\u06c1\u0627\u06ba \u0627\u0646\u062a\u0631\u062c\u0634\u062a\u06be\u0627\u0646 \u06c1\u06d2. \u06a9\u0627\u0644\u0639\u062f\u0645 \u0645\u0641\u0631\u0648\u0636\u06d2 \u06a9\u06d2 \u062a\u062d\u062a (\u06a9\u0648\u0626\u06cc \u0627\u062b\u0631 \u0646\u06c1\u06cc\u06ba)\u060c p-\u0648\u06cc\u0644\u06cc\u0648 0 \u0627\u0648\u0631 1 \u06a9\u06d2 \u062f\u0631\u0645\u06cc\u0627\u0646 \u062a\u0635\u0627\u062f\u0641\u06cc \u0637\u0648\u0631 \u067e\u0631 \u0627\u0686\u06be\u0627\u0644\u062a\u06cc \u06c1\u06d2\u06d4 \u06cc\u06c1 0.5 \u067e\u0631 \u0646\u06c1\u06cc\u06ba \u0631\u06c1\u062a\u06cc \u06c1\u06d2\u06d4 30 \u062f\u0646 \u06a9\u06cc \u062f\u0648\u0691 \u06a9\u06d2 \u062f\u0648\u0631\u0627\u0646\u060c null \u062a\u062c\u0631\u0628\u06c1 \u06a9\u0633\u06cc \u0648\u0642\u062a \u0632\u06cc\u0627\u062f\u06c1 \u0627\u0645\u06a9\u0627\u0646 \u06a9\u06d2 \u0633\u0627\u062a\u06be 0.05 \u0633\u06d2 \u0646\u06cc\u0686\u06d2 \u0622\u062c\u0627\u0626\u06d2 \u06af\u0627\u06d4 \u0627\u06af\u0631 \u0622\u067e \u06c1\u0631 \u0631\u0648\u0632 \u062f\u06cc\u06a9\u06be\u062a\u06d2 \u06c1\u06cc\u06ba \u0627\u0648\u0631 \u0627\u0633 \u0644\u0645\u062d\u06d2 \u06a9\u0648 \u0631\u0648\u06a9\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u062a\u06cc\u0627\u0631 \u06c1\u06cc\u06ba \u062c\u0628 \u0622\u067e p <0.05 \u062f\u06cc\u06a9\u06be\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u062a\u0648 \u0622\u067e \u06a9\u0648 \u062a\u0642\u0631\u06cc\u0628\u0627\u064b \u06c1\u0645\u06cc\u0634\u06c1 \u0646\u06cc\u0686\u06d2 \u06a9\u0627 \u0631\u062c\u062d\u0627\u0646 \u0645\u0644\u06d2 \u06af\u0627\u06d4 \u0622\u067e \u06a9\u0648 \u0641\u0627\u062a\u062d \u0642\u0631\u0627\u0631 \u062f\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4 \u0644\u06cc\u06a9\u0646 \u0627\u062b\u0631 \u062d\u0642\u06cc\u0642\u06cc \u0646\u06c1\u06cc\u06ba \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u0646\u06c1\u06cc\u06ba \u06a9\u0645 \u06a9\u062b\u0631\u062a \u0633\u06d2 \u062f\u06cc\u06a9\u06be\u0646\u06d2 \u0633\u06d2 \u0635\u0631\u0641 \u0627\u0646\u06c1\u06cc \u0645\u0633\u0627\u0626\u0644 \u0645\u06cc\u06ba \u062a\u0627\u062e\u06cc\u0631 \u06c1\u0648\u06af\u06cc\u06d4 \u0622\u067e \u06a9\u0648 \u0627\u0633\u06d2 \u0627\u06a9\u062b\u0631 \u0686\u06cc\u06a9 \u06a9\u0631\u0646\u0627 \u0686\u0627\u06c1\u0626\u06d2\u06d4 \u0622\u067e \u06a9\u0627 \u067e\u0631\u0648\u0688\u06a9\u0679 \u062a\u06cc\u0632\u06cc \u0633\u06d2 \u0622\u06af\u06d2 \u0628\u0691\u06be\u062a\u0627 \u06c1\u06d2\u060c \u0627\u0648\u0631 \u0636\u0631\u0648\u0631\u062a \u0633\u06d2 16 \u062f\u0646 \u0637\u0648\u06cc\u0644 \u062a\u062c\u0631\u0628\u06c1 \u0686\u0644\u0627\u0646\u06d2 \u0645\u06cc\u06ba \u062d\u0642\u06cc\u0642\u06cc \u0644\u0627\u06af\u062a\u060c \u062a\u0627\u062e\u06cc\u0631 \u0633\u06d2 \u0644\u0627\u0646\u0686 \u0627\u0648\u0631 \u0645\u0648\u0642\u0639 \u06a9\u06d2 \u0627\u062e\u0631\u0627\u062c\u0627\u062a \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0627\u0633 \u0633\u06d2 \u0642\u0637\u0639 \u0646\u0638\u0631 \u06a9\u06c1 \u0622\u067e \u06a9\u0628 \u0631\u0648\u06a9\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u0622\u067e \u06a9\u0648 \u062f\u0631\u0633\u062a \u0679\u06cc\u0633\u0679 \u06a9\u06d2 \u0627\u0639\u062f\u0627\u062f\u0648\u0634\u0645\u0627\u0631 \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u0648\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<h2 id=\"heading-what-a-sequential-test-actually-does\">\u062a\u0631\u062a\u06cc\u0628 \u0648\u0627\u0631 \u0679\u06cc\u0633\u0679 \u062f\u0631\u0627\u0635\u0644 \u06a9\u06cc\u0627 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/h2>\n<p>\u062a\u0631\u062a\u06cc\u0628 \u0648\u0627\u0631 \u0679\u06cc\u0633\u0679\u0648\u06ba \u06a9\u0648 \u0627\u0646\u062a\u062e\u0627\u0628\u06cc \u0637\u0648\u0631 \u067e\u0631 p-value \u06a9\u0648 \u0645\u062a\u0628\u0627\u062f\u0644 \u0627\u0639\u062f\u0627\u062f\u0648\u0634\u0645\u0627\u0631 \u0633\u06d2 \u062a\u0628\u062f\u06cc\u0644 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0688\u06cc\u0632\u0627\u0626\u0646 \u06a9\u06cc\u0627 \u06af\u06cc\u0627 \u06c1\u06d2 \u062c\u0633\u06d2 e-value \u06a9\u06c1\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<p>p-values \u200b\u200b\u06a9\u06d2 \u0628\u0631\u0639\u06a9\u0633\u060c e-values \u200b\u200b\u063a\u06cc\u0631 \u0645\u0646\u0641\u06cc \u06c1\u06cc\u06ba\u060c \u0627\u0648\u0631 \u0648\u0642\u062a \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0633\u0627\u062a\u06be e-values \u200b\u200b\u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06d2 \u062a\u0634\u06a9\u06cc\u0644 \u067e\u0627\u0646\u06d2 \u0648\u0627\u0644\u0627 \u0639\u0645\u0644 null \u06a9\u06d2 \u0646\u06cc\u0686\u06d2 \u0633\u067e\u0631\u0645\u0627\u0631\u0679\u0646\u06af\u06cc\u0644 \u067e\u0631\u0627\u067e\u0631\u0679\u06cc \u06a9\u0648 \u0645\u0637\u0645\u0626\u0646 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u062a\u0627\u0631\u06cc\u062e \u06a9\u06cc \u0628\u0646\u06cc\u0627\u062f \u067e\u0631\u060c \u0627\u06af\u0644\u06cc \u0645\u062a\u0648\u0642\u0639 \u0627\u06cc \u0648\u06cc\u0644\u06cc\u0648 \u0645\u0648\u062c\u0648\u062f\u06c1 \u0642\u062f\u0631 \u0633\u06d2 \u0628\u06c1\u062a\u0631\u06cc\u0646 \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u0633 \u0631\u0627\u0633\u062a\u06d2 \u06a9\u06cc \u0633\u0637\u062d \u067e\u0631 \u0633\u067e\u0631 \u0645\u0627\u0631\u0679\u0646\u06af\u06cc\u0644 \u06a9\u06cc \u062d\u0627\u0644\u062a \u0648\u06c1\u06cc \u06c1\u06d2 \u062c\u0648 \u0645\u0646\u062a\u062e\u0628 \u0631\u06a9\u0646\u06d2 \u06a9\u0648 \u0645\u062d\u0641\u0648\u0638 \u0628\u0646\u0627\u062a\u06cc \u06c1\u06d2\u06d4 \u06c1\u0631 \u0642\u062f\u0645 \u067e\u0631 \u0645\u0639\u0645\u0648\u0644\u06cc \u0627\u0648\u0633\u0637 \u06a9\u0648 1 \u0633\u06d2 \u0646\u06cc\u0686\u06d2 \u0631\u06a9\u06be\u0646\u0627 \u0636\u0631\u0648\u0631\u06cc \u06c1\u06d2\u060c \u0644\u06cc\u06a9\u0646 \u06a9\u0627\u0641\u06cc \u0646\u06c1\u06cc\u06ba \u06c1\u06d2\u06d4 \u0633\u067e\u0631\u0645\u0627\u0631\u0679\u0646\u06af\u06cc\u0644 \u06a9\u06cc \u062d\u0627\u0644\u062a \u0633\u062e\u062a\u06cc \u0633\u06d2 \u0645\u0636\u0628\u0648\u0637 \u06c1\u06d2\u060c \u062a\u0645\u0627\u0645 \u0631\u06a9\u0646\u06d2 \u06a9\u06d2 \u0627\u0648\u0642\u0627\u062a \u0645\u06cc\u06ba \u0628\u0627\u0624\u0646\u0688\u0631\u06cc \u06a9\u0648 \u06cc\u06a9\u0633\u0627\u06ba \u0631\u06a9\u06be\u0646\u0627\u06d4<\/p>\n<p>\u0627\u0633 \u06a9\u06cc \u0648\u062c\u06c1 \u06cc\u06c1 \u06c1\u06d2: \u0627\u06af\u0631 \u0627\u06cc-\u0648\u06cc\u0644\u06cc\u0648 \u06a9\u0627 \u0639\u0645\u0644 E \u06a9\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u0627\u06cc\u06a9 \u063a\u06cc\u0631 \u0645\u0646\u0641\u06cc \u0633\u067e\u0631 \u0645\u0627\u0631\u0679\u0646\u06af\u06cc\u0644 \u06c1\u06d2\u06d4[e_t] H0 \u0645\u06cc\u06ba \u2264 1 \u06a9\u0644\u0627\u0633\u06a9 \u0646\u062a\u06cc\u062c\u06c1 \u062f\u06cc\u062a\u0627 \u06c1\u06d2 \u062c\u0633\u06d2 \u0648\u06cc\u0644 \u06a9\u06cc \u0639\u062f\u0645 \u0645\u0633\u0627\u0648\u0627\u062a \u06a9\u06c1\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0627\u0633 \u0628\u0627\u062a \u06a9\u0627 \u0627\u0645\u06a9\u0627\u0646 \u06a9\u06c1 \u0639\u0645\u0644 \u06a9\u06cc \u0632\u06cc\u0627\u062f\u06c1 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0639\u0645\u0644 \u062f\u0631\u0622\u0645\u062f 1\/\u03b1 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u06c1\u0648 \u0632\u06cc\u0627\u062f\u06c1 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u03b1 \u06c1\u06d2\u06d4 \u03b1 = 0.05 \u0627\u0648\u0631 1\/\u03b1 = 20 \u06a9\u06cc \u0631\u06a9\u0646\u06d2 \u06a9\u06cc \u062d\u062f \u06a9\u06d2 \u0633\u0627\u062a\u06be\u060c null e-value \u06a9\u06d2 \u0639\u0645\u0644 \u06a9\u06d2 20 \u062a\u06a9 \u067e\u06c1\u0646\u0686\u0646\u06d2 \u06a9\u0627 \u0627\u0645\u06a9\u0627\u0646 \u0632\u06cc\u0627\u062f\u06c1 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 5% \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u0633 \u0642\u0633\u0645 \u06a9\u06cc \u063a\u0644\u0637\u06cc \u06a9\u06cc \u062d\u062f \u0628\u0631\u0642\u0631\u0627\u0631 \u0631\u06c1\u062a\u06cc \u06c1\u06d2 \u0627\u0633 \u0633\u06d2 \u0642\u0637\u0639 \u0646\u0638\u0631 \u06a9\u06c1 \u0622\u067e \u06a9\u0628 \u0631\u0648\u06a9\u062a\u06d2 \u06c1\u06cc\u06ba \u06cc\u0627 \u06a9\u062a\u0646\u06cc \u0628\u0627\u0631 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\u0686\u0627\u06c1\u06cc\u06d2\u06d4[e_{t+1} | e_1, &#8230;, e_t] e_t \u2264 H0 \u06a9\u06d2 \u062a\u062d\u062a\u06d4 \u06cc\u06c1\u0627\u06ba \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06c1\u0648\u0646\u06d2 \u0648\u0627\u0644\u06d2 \u0628\u06cc\u0679\u0627 \u0628\u0627\u0626\u0646\u0648\u0645\u06cc\u0626\u0644 \u0628\u06cc\u0632 \u0641\u06cc\u06a9\u0679\u0631 \u06a9\u06d2 \u0644\u06cc\u06d2\u060c \u06cc\u06c1 \u0627\u0633 \u0648\u0642\u062a \u062a\u06a9 \u0628\u0631\u0642\u0631\u0627\u0631 \u0631\u06c1\u062a\u0627 \u06c1\u06d2 \u062c\u0628 \u062a\u06a9 \u067e\u0631\u0627\u0626\u06cc\u0631 \u0645\u0646\u0627\u0633\u0628 \u06c1\u0648 (\u0628\u06cc\u0679\u0627(1\u060c1) \u06a9\u0648\u0627\u0644\u06cc\u0641\u0627\u0626\u06cc\u0688) \u0627\u0648\u0631 \u0645\u0634\u0627\u06c1\u062f\u0627\u062a \u06c1\u0631 \u0634\u0639\u0628\u06d2 \u0645\u06cc\u06ba \u0622\u0626\u06cc \u0622\u0626\u06cc \u0688\u06cc \u06c1\u06cc\u06ba\u06d4<\/p>\n<\/li>\n<li>\n<p><strong>\u062c\u06cc\u0648\u0646\u06af 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\u062f\u0631\u0645\u06cc\u0627\u0646\u06cc \u062a\u062c\u0631\u0628\u06d2 \u06a9\u0648 \u0645\u0646\u062a\u0642\u0644 \u06a9\u0631\u062a\u06cc \u06c1\u06d2\u060c \u062a\u0648 \u0627\u06cc-\u0648\u06cc\u0644\u06cc\u0648 \u0634\u0648\u0631 \u06a9\u0648 \u067e\u06a9\u0691 \u0644\u06cc\u062a\u06cc \u06c1\u06d2 \u062c\u0633\u06d2 \u062a\u062c\u0631\u0628\u06c1 \u0639\u0644\u0627\u062c \u06a9\u06d2 \u0627\u062b\u0631\u0627\u062a \u0633\u06d2 \u0627\u0644\u06af \u0646\u06c1\u06cc\u06ba \u06a9\u0631 \u0633\u06a9\u062a\u0627\u06d4<\/p>\n<\/li>\n<li>\n<p><strong>\u06c1\u0631 \u0628\u0627\u0632\u0648 \u06a9\u06d2 \u0627\u0646\u062f\u0631 \u0622\u0632\u0627\u062f \u0645\u0634\u0627\u06c1\u062f\u0627\u062a\u06d4<\/strong> \u06c1\u0631 \u0635\u0627\u0631\u0641 \u06a9\u06d2 \u0646\u062a\u0627\u0626\u062c \u062f\u0648\u0633\u0631\u06d2 \u0635\u0627\u0631\u0641\u06cc\u0646 \u06a9\u06d2 \u0646\u062a\u0627\u0626\u062c \u0633\u06d2 \u0622\u0632\u0627\u062f \u06c1\u0648\u0646\u06d2 \u0686\u0627\u06c1\u0626\u06cc\u06ba\u06d4 \u0646\u06cc\u0679 \u0648\u0631\u06a9 \u06a9\u06d2 \u0627\u062b\u0631\u0627\u062a\u060c \u0645\u0634\u062a\u0631\u06a9\u06c1 \u06a9\u0627\u0645 \u06a9\u06cc \u062c\u06af\u06c1\u06cc\u06ba\u060c \u06cc\u0627 \u0633\u0641\u0627\u0631\u0634\u06cc \u0646\u0638\u0627\u0645 \u06a9\u06d2 \u0627\u062b\u0631\u0627\u062a \u0627\u0633 \u06a9\u06cc \u062e\u0644\u0627\u0641 \u0648\u0631\u0632\u06cc \u06a9\u0631 \u0633\u06a9\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<\/li>\n<li>\n<p><strong>\u067e\u0631\u06cc \u0642\u06cc\u0627\u0633\u06d4<\/strong> \u0628\u06cc\u0679\u0627 (1\u060c1) \u067e\u06c1\u0644\u06d2 \u0627\u06cc\u06a9 \u0645\u0627\u0688\u0644\u0646\u06af \u0645\u0641\u0631\u0648\u0636\u06c1 \u06c1\u06d2\u06d4 \u0627\u06cc\u0645 \u0627\u06cc\u0633 \u067e\u06cc \u0622\u0631 \u0679\u06cc \u06a9\u06cc \u0637\u0627\u0642\u062a \u0627\u0633 \u0628\u0627\u062a \u067e\u0631 \u0645\u0646\u062d\u0635\u0631 \u06c1\u06d2 \u06a9\u06c1 \u0622\u06cc\u0627 \u067e\u0686\u06be\u0644\u06d2 \u0637\u0631\u06cc\u0642\u06d2 \u0635\u062d\u06cc\u062d \u0627\u062b\u0631 \u06a9\u06d2 \u0633\u0627\u0626\u0632 \u067e\u0631 \u0645\u0639\u0642\u0648\u0644 \u0645\u0627\u0633 \u0631\u06a9\u06be\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0646\u0627\u0642\u0635 \u0637\u0648\u0631 \u067e\u0631 \u0645\u062a\u0639\u06cc\u0646 \u06a9\u0631\u062f\u06c1 \u067e\u06cc\u0634 \u0628\u0646\u062f\u06cc \u0642\u0633\u0645 I \u06a9\u06cc \u063a\u0644\u0637\u06cc \u06a9\u06cc \u0636\u0645\u0627\u0646\u062a\u0648\u06ba \u06a9\u0648 \u0646\u06c1\u06cc\u06ba \u062a\u0648\u0691\u062a\u06cc \u06c1\u06d2\u060c \u0644\u06cc\u06a9\u0646 \u06cc\u06c1 \u062d\u062f \u06a9\u0648 \u0639\u0628\u0648\u0631 \u06a9\u06cc\u06d2 \u0628\u063a\u06cc\u0631 \u0646\u0645\u0648\u0646\u06d2 \u06a9\u06d2 \u0628\u062c\u0679 \u06a9\u0648 \u062e\u062a\u0645 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2\u060c \u0627\u06cc \u0642\u062f\u0631\u0648\u06ba \u06a9\u0648 \u0628\u06c1\u062a \u0622\u06c1\u0633\u062a\u06c1 \u0622\u06c1\u0633\u062a\u06c1 \u0628\u0691\u06be\u0646\u06d2 \u06a9\u0627 \u0633\u0628\u0628 \u0628\u0646 \u0633\u06a9\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<\/li>\n<\/ol>\n<h2 id=\"heading-prerequisites\">\u0634\u0631\u0627\u0626\u0637<\/h2>\n<ul>\n<li>\n<p>Python 3.11+<\/p>\n<\/li>\n<li>\n<p>\u067e\u0627\u0646\u0688\u0627\u0633 2.x (<code>pip install pandas<\/code>)<\/p>\n<\/li>\n<li>\n<p>NumPy 1.26+ (<code>pip install numpy<\/code>)<\/p>\n<\/li>\n<li>\n<p>\u0633\u0627\u0626\u0641\u06cc 1.12+ (<code>pip install scipy<\/code>)<\/p>\n<\/li>\n<li>\n<p>matplotlib 3.8+ (<code>pip install matplotlib<\/code>)<\/p>\n<\/li>\n<\/ul>\n<p>\u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u062d\u0627\u0635\u0644 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2\u060c \u0633\u0627\u062a\u06be\u06cc \u0631\u06cc\u067e\u0648\u0632\u0679\u0631\u06cc \u06a9\u0648 \u06a9\u0644\u0648\u0646 \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<pre><code class=\"language-bash\">git clone https:\/\/github.com\/RudrenduPaul\/product-experimentation-causal-inference-genai-llm.git\ncd product-experimentation-causal-inference-genai-llm\npython data\/generate_data.py --seed 42 --n-users 50000 --out data\/synthetic_llm_logs.csv\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u06cc\u06c1 \u0627\u0633 \u0633\u06cc\u0631\u06cc\u0632 \u06a9\u06d2 \u0644\u06cc\u06d2 \u062a\u0645\u0627\u0645 13 \u0633\u0627\u062a\u06be\u06cc \u0646\u0648\u0679 \u0628\u06a9\u0633 \u067e\u0631 \u0645\u0634\u062a\u0645\u0644 \u0631\u06cc\u067e\u0648\u0632\u0679\u0631\u06cc \u06a9\u0648 \u06a9\u0644\u0648\u0646 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u0627\u0648\u0631 \u0627\u06cc\u06a9 \u0645\u0634\u062a\u0631\u06a9\u06c1 50,000 \u0635\u0627\u0631\u0641 \u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u0628\u0646\u0627\u062a\u0627 \u06c1\u06d2\u060c \u062c\u0633 \u0645\u06cc\u06ba \u0630\u062e\u06cc\u0631\u06c1 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2: <code>data\/synthetic_llm_logs.csv<\/code>. \u0633\u06cc\u0631\u06cc\u0632 \u06a9\u06d2 \u062a\u0645\u0627\u0645 \u0645\u0636\u0627\u0645\u06cc\u0646 \u0627\u06cc\u06a9 \u06c1\u06cc CSV \u06a9\u06d2 \u062e\u0644\u0627\u0641 \u0686\u0644\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u0622\u067e \u06a9\u0648 \u0637\u0631\u06cc\u0642\u0648\u06ba \u06a9\u0627 \u0628\u0631\u0627\u06c1 \u0631\u0627\u0633\u062a \u0645\u0648\u0627\u0632\u0646\u06c1 \u06a9\u0631\u0646\u06d2 \u06a9\u06cc \u0627\u062c\u0627\u0632\u062a \u062f\u06cc\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0688\u06cc\u0679\u0627 \u062c\u0646\u0631\u06cc\u0679\u0631 \u0644\u06c1\u0631 1 \u06a9\u06d2 \u0635\u0627\u0631\u0641\u06cc\u0646 \u06a9\u06d2 \u0644\u06cc\u06d2 \u06a9\u0627\u0645\u0648\u06ba \u06a9\u06cc \u062a\u06a9\u0645\u06cc\u0644 \u067e\u0631 +5% \u067e\u0648\u0627\u0626\u0646\u0679 \u06a9\u0627\u0632\u0644 \u0627\u062b\u0631 \u06a9\u0627 \u0627\u0637\u0644\u0627\u0642 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<h2 id=\"heading-setting-up-the-working-example\">\u0648\u0631\u06a9\u0646\u06af \u0645\u062b\u0627\u0644 \u0633\u06cc\u0679 \u0627\u067e<\/h2>\n<p>\u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 50,000 \u0635\u0627\u0631\u0641\u06cc\u0646 \u06a9\u06d2 \u0633\u0627\u062a\u06be SaaS AI \u0627\u0645\u062f\u0627\u062f\u06cc \u067e\u0631\u0648\u0688\u06a9\u0679 \u06a9\u06cc \u0646\u0642\u0644 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u06a9\u06c1 <code>task_completed<\/code> \u06a9\u0627\u0644\u0645 \u0631\u06cc\u06a9\u0627\u0631\u0688 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u06a9\u06c1 \u0622\u06cc\u0627 AI \u0646\u06d2 \u0635\u0627\u0631\u0641 \u06a9\u0627 \u06a9\u0627\u0645 \u06a9\u0627\u0645\u06cc\u0627\u0628\u06cc \u0633\u06d2 \u0645\u06a9\u0645\u0644 \u06a9\u06cc\u0627 (1) \u06cc\u0627 \u0646\u06c1\u06cc\u06ba (0)\u06d4 \u06a9\u06c1 <code>wave<\/code> \u06a9\u0627\u0644\u0645 \u0635\u0627\u0631\u0641\u06cc\u0646 \u06a9\u0648 \u06af\u0631\u0648\u067e\u0633 \u06a9\u0648 \u062a\u0641\u0648\u06cc\u0636 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 Wave 1 \u0646\u0626\u06cc AI \u062e\u0635\u0648\u0635\u06cc\u0627\u062a \u062d\u0627\u0635\u0644 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u0627\u0648\u0631 Wave 2 \u06c1\u0648\u0644\u0688 \u0622\u0624\u0679 \u06a9\u0646\u0679\u0631\u0648\u0644 \u06c1\u06d2\u06d4<\/p>\n<p><em>\u0634\u06a9\u0644 1: \u062a\u0635\u0648\u0631\u0627\u062a\u06cc \u0627\u06cc \u0642\u062f\u0631 \u06a9\u06cc \u0631\u0641\u062a\u0627\u0631\u06d4 \u0646\u06cc\u0644\u0627 \u0631\u0627\u0633\u062a\u06c1 (\u062d\u0642\u06cc\u0642\u06cc \u0627\u062b\u0631) \u0627\u0679\u06be\u062a\u0627 \u06c1\u06d2 \u0627\u0648\u0631 \u0633\u0628\u0632 \u0646\u0642\u0637\u0648\u06ba \u0648\u0627\u0644\u06cc \u0644\u06a9\u06cc\u0631 \u067e\u0631 \u0631\u06a9\u0646\u06d2 \u06a9\u06cc \u062f\u06c1\u0644\u06cc\u0632 \u06a9\u0648 \u0639\u0628\u0648\u0631 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u062c\u0627\u0645\u0646\u06cc \u0631\u0646\u06af \u06a9\u0627 \u0631\u0627\u0633\u062a\u06c1 (\u06a9\u0645\u0632\u0648\u0631 \u0627\u062b\u0631) \u0628\u0691\u06be\u062a\u0627 \u06c1\u06d2 \u0644\u06cc\u06a9\u0646 30 \u062f\u0646\u0648\u06ba \u06a9\u06d2 \u0627\u0646\u062f\u0631 \u0646\u06c1\u06cc\u06ba \u06a9\u0627\u0679\u062a\u0627 \u06c1\u06d2\u06d4 \u0633\u0631\u0645\u0626\u06cc \u0631\u0627\u0633\u062a\u06c1 (\u0646\u0627\u0644) \u06c1\u0648\u0627 \u0645\u062c\u0645\u0648\u0639\u06cc \u0637\u0648\u0631 \u067e\u0631 1 \u06a9\u06d2 \u0642\u0631\u06cc\u0628 \u06c1\u06d2\u06d4 \u0633\u0631\u062e \u0688\u06cc\u0634\u0688 \u0644\u0627\u0626\u0646 1\/\u03b1 = 20 \u067e\u0631 \u0633\u0679\u06cc\u0634\u0646\u0631\u06cc \u0628\u0627\u0624\u0646\u0688\u0631\u06cc \u06c1\u06d2\u06d4 \u0627\u0633 \u06a9\u0627 \u0630\u06cc\u0644 \u0645\u06cc\u06ba \u0634\u06a9\u0644 2 \u0633\u06d2 \u0645\u0648\u0627\u0632\u0646\u06c1 \u06a9\u0631\u06cc\u06ba\u060c \u062c\u0648 \u0627\u06cc\u06a9 \u062d\u0642\u06cc\u0642\u06cc \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0627\u0635\u0644 \u0627\u06cc-\u0648\u06cc\u0644\u06cc\u0648 \u06a9\u06cc \u0631\u0641\u062a\u0627\u0631 \u06a9\u0648 \u0638\u0627\u06c1\u0631 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/em><\/p>\n<pre><code class=\"language-python\">import pandas as pd\nimport numpy as np\n\ndf = pd.read_csv(\"data\/synthetic_llm_logs.csv\")\n\ntreated = df[df[\"wave\"] == 1][\"task_completed\"].values\ncontrol = df[df[\"wave\"] == 2][\"task_completed\"].values\n\nprint(f\"Treated: n={len(treated):,}, mean={treated.mean():.4f}\")\nprint(f\"Control: n={len(control):,}, mean={control.mean():.4f}\")\nprint(f\"Observed lift: {treated.mean() - control.mean():.4f}\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">Treated: n=24,937, mean=0.6202\nControl: n=25,063, mean=0.5718\nObserved lift: 0.0485\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> 50,000 \u0642\u0637\u0627\u0631\u0648\u06ba \u06a9\u0627 \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u0644\u0648\u0688 \u06a9\u0631\u06cc\u06ba \u0627\u0648\u0631 \u0627\u0633\u06d2 \u0644\u06c1\u0631\u0648\u06ba \u06a9\u06d2 \u0630\u0631\u06cc\u0639\u06d2 \u062a\u0642\u0633\u06cc\u0645 \u06a9\u0631\u06cc\u06ba\u06d4 Wave 1 \u06a9\u06d2 \u067e\u0627\u0633 24,937 \u067e\u0631\u0648\u0633\u06cc\u0633 \u0634\u062f\u06c1 \u0635\u0627\u0631\u0641\u06cc\u0646 \u062a\u06be\u06d2\u060c \u062c\u0633 \u06a9\u06cc \u0679\u0627\u0633\u06a9 \u0645\u06a9\u0645\u0644 \u06a9\u0631\u0646\u06d2 \u06a9\u06cc \u0634\u0631\u062d 62.0% \u062a\u06be\u06cc\u06d4 \u0648\u06cc\u0648 2 \u0645\u06cc\u06ba 25,063 \u06a9\u0646\u0679\u0631\u0648\u0644 \u0635\u0627\u0631\u0641\u06cc\u0646 \u06c1\u06cc\u06ba\u06d4 <em>\u06a9\u0627\u0645 \u06a9\u06cc \u062a\u06a9\u0645\u06cc\u0644 \u06a9\u06cc \u0634\u0631\u062d 57.2%<\/em>. \u0645\u0634\u0627\u06c1\u062f\u06c1 \u06a9\u06cc\u0627 \u06af\u06cc\u0627 4.85 \u0641\u06cc\u0635\u062f \u067e\u0648\u0627\u0626\u0646\u0679 \u0627\u0636\u0627\u0641\u06c1 \u0688\u06cc\u0679\u0627 \u062c\u0646\u0631\u06cc\u0679\u0631 \u0645\u06cc\u06ba 5 \u067e\u06cc \u067e\u06cc \u06af\u0631\u0627\u0624\u0646\u0688 \u0633\u0686\u0627\u0626\u06cc \u06a9\u06d2 \u0642\u0631\u06cc\u0628 \u06c1\u06d2\u060c \u0646\u0645\u0648\u0646\u06d2 \u0644\u06cc\u0646\u06d2 \u06a9\u06d2 \u0634\u0648\u0631 \u06a9\u06cc \u0648\u062c\u06c1 \u0633\u06d2 \u0686\u06be\u0648\u0679\u06d2 \u0641\u0631\u0642 \u06a9\u06d2 \u0633\u0627\u062a\u06be\u06d4 \u06cc\u06c1 \u0627\u0646\u062a\u0638\u0627\u0645 \u0627\u06cc\u06a9 \u0648\u0642\u062a \u0645\u06cc\u06ba \u0627\u06cc\u06a9 \u0645\u0634\u0627\u06c1\u062f\u06d2 \u06a9\u06cc \u062a\u0631\u062a\u06cc\u0628 \u0648\u0627\u0631 \u062c\u0627\u0646\u0686 \u0641\u0631\u0627\u06c1\u0645 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u060c \u062c\u06cc\u0633\u0627 \u06a9\u06c1 \u0630\u06cc\u0644 \u06a9\u06d2 \u0645\u0631\u0627\u062d\u0644 \u0645\u06cc\u06ba \u0628\u06cc\u0627\u0646 \u06a9\u06cc\u0627 \u06af\u06cc\u0627 \u06c1\u06d2\u06d4<\/p>\n<h2 id=\"heading-step-1-simulate-the-peeking-problem\">\u0645\u0631\u062d\u0644\u06c1 1: \u062c\u06be\u0627\u0646\u06a9\u0646\u06d2 \u06a9\u06d2 \u0645\u0633\u0626\u0644\u06d2 \u06a9\u06cc \u062a\u0642\u0644\u06cc\u062f \u06a9\u0631\u06cc\u06ba\u06d4<\/h2>\n<p>\u062c\u06be\u0627\u0646\u06a9\u0646\u06d2 \u06a9\u0627 \u0645\u0633\u0626\u0644\u06c1 \u062d\u0642\u06cc\u0642\u06cc \u0627\u0648\u0631 \u0642\u0627\u0628\u0644 \u067e\u06cc\u0645\u0627\u0626\u0634 \u06c1\u06d2\u06d4 30 \u062f\u0646\u0648\u06ba \u06a9\u06d2 \u0644\u06cc\u06d2 \u0631\u0648\u0632\u0627\u0646\u06c1 \u06a9\u06cc \u0646\u06af\u0631\u0627\u0646\u06cc \u063a\u0644\u0637 \u0645\u062b\u0628\u062a \u0634\u0631\u062d \u06a9\u0648 4.2% \u0633\u06d2 30.2% \u062a\u06a9 \u0628\u0691\u06be\u0627 \u062f\u06cc\u062a\u06cc \u06c1\u06d2\u06d4 \u0627\u0633 \u06a9\u06cc \u062a\u0635\u062f\u06cc\u0642 \u0646\u06cc\u0686\u06d2 \u062f\u06cc\u06d2 \u06af\u0626\u06d2 \u062a\u062e\u0631\u0648\u067e\u0646 \u0645\u06cc\u06ba \u06c1\u0648\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<p>\u06cc\u06c1 \u062a\u062e\u0631\u0648\u067e\u0646 1,000 \u06a9\u0627\u0644\u0639\u062f\u0645 \u062a\u062c\u0631\u0628\u0627\u062a \u0686\u0644\u0627\u062a\u0627 \u06c1\u06d2 (\u062c\u06c1\u0627\u06ba \u0639\u0644\u0627\u062c \u06a9\u0627 \u06a9\u0648\u0626\u06cc \u0627\u062b\u0631 \u0646\u06c1\u06cc\u06ba \u06c1\u0648\u062a\u0627 \u06c1\u06d2) \u0627\u0648\u0631 \u06c1\u0631 \u0631\u0648\u0632 \u06cc\u06c1 \u062f\u06cc\u06a9\u06be\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0686\u06cc\u06a9 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u06a9\u06c1 \u0686\u0644 \u0631\u06c1\u06cc p-value 0.05 \u0633\u06d2 \u06a9\u0645 \u06c1\u06d2 \u06cc\u0627 \u0646\u06c1\u06cc\u06ba\u06d4 \u06cc\u06c1 \u0645\u0646\u0638\u0631 30 \u062f\u0646 \u06a9\u06d2 \u062a\u062c\u0631\u0628\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 60 \u0635\u0627\u0631\u0641\u06cc\u0646 \u0641\u06cc \u0628\u0627\u0632\u0648 \u0641\u06cc \u062f\u0646 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u0641\u06cc \u0628\u0627\u0632\u0648 \u0645\u0634\u0627\u06c1\u062f\u0627\u062a \u06a9\u06cc \u06a9\u0644 \u062a\u0639\u062f\u0627\u062f 1,800 \u06c1\u06d2\u060c \u062c\u0648 \u062f\u0631\u0645\u06cc\u0627\u0646\u06d2 \u0633\u0627\u0626\u0632 \u06a9\u06d2 SaaS \u067e\u0631\u0648\u0688\u06a9\u0679 \u06a9\u06d2 \u0644\u06cc\u06d2 \u062d\u0642\u06cc\u0642\u062a \u067e\u0633\u0646\u062f\u0627\u0646\u06c1 \u0633\u0627\u0626\u0632 \u06c1\u06d2\u06d4<\/p>\n<pre><code class=\"language-python\">from scipy import stats\nimport numpy as np\n\nnp.random.seed(42)\n\nN_SIMS = 1000\nN_DAYS = 30\nUSERS_PER_ARM_PER_DAY = 60\nNULL_RATE = 0.60\n\nfalse_positives_peeking = 0\nfalse_positives_single_look = 0\n\nfor _ in range(N_SIMS):\n    control_outcomes = []\n    treated_outcomes = []\n    stopped_early = False\n\n    for day in range(N_DAYS):\n        control_outcomes.extend(np.random.binomial(1, NULL_RATE, USERS_PER_ARM_PER_DAY))\n        treated_outcomes.extend(np.random.binomial(1, NULL_RATE, USERS_PER_ARM_PER_DAY))\n\n        # The peeking problem: checking the test every single day\n        if len(control_outcomes) >= 10:\n            _, p = stats.ttest_ind(treated_outcomes, control_outcomes)\n            if p < 0.05 and not stopped_early:\n                false_positives_peeking += 1\n                stopped_early = True\n\n    # The fixed-sample approach: checking only once at the very end\n    _, p_final = stats.ttest_ind(treated_outcomes, control_outcomes)\n    if p_final < 0.05:\n        false_positives_single_look += 1\n\nprint(f\"False positive rate (peeking daily):  {false_positives_peeking \/ N_SIMS:.1%}\")\nprint(f\"False positive rate (single look):    {false_positives_single_look \/ N_SIMS:.1%}\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">False positive rate (peeking daily):  30.2%\nFalse positive rate (single look):    4.2%\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u06c1\u0631 \u062a\u062e\u0631\u0648\u067e\u0646 \u0627\u06cc\u06a9 \u06c1\u06cc 60% \u062a\u06a9\u0645\u06cc\u0644 \u06a9\u06cc \u0634\u0631\u062d \u067e\u0631 \u062a\u06cc\u0627\u0631 \u06a9\u0631\u062f\u06c1 \u062f\u0648\u0646\u0648\u06ba \u0628\u0627\u0632\u0648\u0624\u06ba \u06a9\u06d2 \u0633\u0627\u062a\u06be \u06a9\u0627\u0644\u0639\u062f\u0645 \u0688\u06cc\u0679\u0627 \u062a\u06cc\u0627\u0631 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u060c \u0644\u06c1\u0630\u0627 \u067e\u062a\u06c1 \u0686\u0644\u0627 \u0627\u062b\u0631 \u062e\u0627\u0644\u0635 \u0634\u0648\u0631 \u06c1\u06d2\u06d4 \u0627\u0646\u062f\u0631\u0648\u0646\u06cc \u0644\u0648\u067e \u06c1\u0631 \u0631\u0648\u0632 60 \u0645\u0634\u0627\u06c1\u062f\u0627\u062a \u0641\u06cc \u0628\u0627\u0632\u0648 \u06a9\u0627 \u0627\u0636\u0627\u0641\u06c1 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u0627\u0648\u0631 \u0627\u0633 \u062f\u0646 \u06a9\u06d2 \u0644\u06cc\u06d2 \u062c\u0645\u0639 \u06a9\u0631\u062f\u06c1 \u0688\u06cc\u0679\u0627 \u067e\u0631 \u0679\u06cc \u0679\u06cc\u0633\u0679 \u0686\u0644\u0627\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u062c\u0628 p-value \u067e\u06c1\u0644\u06d2 0.05 \u0633\u06d2 \u0646\u06cc\u0686\u06d2 \u0622\u062c\u0627\u062a\u06cc \u06c1\u06d2\u060c \u0633\u0645\u0648\u0644\u06cc\u0634\u0646 \u062c\u06be\u0648\u0679\u06d2 \u0645\u062b\u0628\u062a \u06a9\u0648 \u062c\u06be\u0646\u0688\u0627 \u0644\u06af\u0627\u062a\u0627 \u06c1\u06d2 \u0627\u0648\u0631 \u0631\u06a9 \u062c\u0627\u062a\u0627 \u06c1\u06d2 (\u0627\u06cc\u0633\u06cc \u0679\u06cc\u0645 \u06a9\u06cc \u0646\u0642\u0644 \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u062c\u0648 \u0627\u06c1\u0645\u06cc\u062a \u06a9\u0627 \u067e\u062a\u06c1 \u0644\u06af\u0627\u0646\u06d2 \u067e\u0631 \u0631\u06cc\u0644\u06cc\u0632 \u06a9\u0631\u062a\u06cc \u06c1\u06d2)\u06d4<\/p>\n<p>\u062f\u0646 30 \u067e\u0631 \u0627\u06cc\u06a9 \u06c1\u06cc \u0686\u06cc\u06a9 \u0627\u06cc\u06a9 \u0627\u06cc\u0645\u0627\u0646\u062f\u0627\u0631 \u0645\u0642\u0631\u0631\u06c1 \u0646\u0645\u0648\u0646\u06c1 \u0679\u06cc\u0633\u0679 \u06c1\u06d2\u06d4 \u0627\u06cc\u06a9 \u0641\u0648\u0631\u06cc \u0646\u0638\u0631 4.2% \u06a9\u06cc \u063a\u0644\u0637 \u0645\u062b\u0628\u062a \u0634\u0631\u062d \u06a9\u0648 \u0638\u0627\u06c1\u0631 \u06a9\u0631\u062a\u06cc \u06c1\u06d2\u060c \u062c\u0648 \u06a9\u06c1 \u0628\u0631\u0627\u0626\u06d2 \u0646\u0627\u0645 \u06a9\u06d2 \u0642\u0631\u06cc\u0628 \u06c1\u06d2\u06d4 \u0631\u0648\u0632\u0627\u0646\u06c1 \u062c\u06be\u0627\u0646\u06a9\u0646\u06d2 \u06a9\u06cc \u0634\u0631\u062d 30.2% \u06c1\u06d2\u06d4 \u0627\u0633 \u06a9\u0627 \u0645\u0637\u0644\u0628 \u06c1\u06d2 \u06a9\u06c1 4 \u0645\u06cc\u06ba \u0633\u06d2 1 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \"\u0627\u06c1\u0645\" \u062a\u062c\u0631\u0628\u0627\u062a \u0634\u0648\u0631 \u06a9\u0627 \u067e\u062a\u06c1 \u0644\u06af\u0627 \u0631\u06c1\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<h2 id=\"heading-step-2-implement-the-msprt-e-value\">\u0645\u0631\u062d\u0644\u06c1 2: mSPRT \u0627\u06cc-\u0648\u06cc\u0644\u06cc\u0648 \u0646\u0627\u0641\u0630 \u06a9\u0631\u06cc\u06ba\u06d4<\/h2>\n<p>mSPRT \u06c1\u0631 \u0628\u0627\u0631 \u0642\u062f\u0645 \u067e\u0631 Bayes \u0641\u06cc\u06a9\u0679\u0631 \u06a9\u0627 \u062d\u0633\u0627\u0628 \u0644\u06af\u0627\u062a\u0627 \u06c1\u06d2\u06d4 \u06cc\u0639\u0646\u06cc\u060c null \u06a9\u06d2 \u0645\u0642\u0627\u0628\u0644\u06d2 \u0645\u06cc\u06ba \u0645\u062a\u0628\u0627\u062f\u0644 \u0645\u0631\u06a9\u0628 \u06a9\u06d2 \u062a\u062d\u062a \u06a9\u062a\u0646\u0627 \u0632\u06cc\u0627\u062f\u06c1 \u0688\u06cc\u0679\u0627 \u062f\u06cc\u06a9\u06be\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u061f \u06c1\u0631 \u0634\u0639\u0628\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u062a\u06a9\u0645\u06cc\u0644 \u06a9\u06cc \u0634\u0631\u062d \u0633\u06d2 \u067e\u06c1\u0644\u06d2 beta(1,1) \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0628\u0627\u0626\u0646\u0631\u06cc \u0646\u062a\u0627\u0626\u062c \u06a9\u06d2 \u0644\u06cc\u06d2\u060c \u0686\u0644 \u0631\u06c1\u0627 Bayes \u0641\u06cc\u06a9\u0679\u0631 \u0644\u0627\u06af-\u0628\u06cc\u0679\u0627 \u0641\u0646\u06a9\u0634\u0646 \u06a9\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u0627\u06cc\u06a9 \u0628\u0646\u062f \u0634\u06a9\u0644 \u0631\u06a9\u06be\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<pre><code class=\"language-python\">from scipy.special import betaln\n\ndef compute_evalue_running(outcomes_treated, outcomes_control,\n                           alpha_prior=1.0, beta_prior=1.0):\n    \"\"\"\n    Compute the running mSPRT e-value for two Bernoulli arms.\n\n    Parameters\n    ----------\n    outcomes_treated : array-like of 0\/1\n    outcomes_control : array-like of 0\/1\n    alpha_prior, beta_prior : Beta prior hyperparameters (default: uniform)\n\n    Returns\n    -------\n    e_values : np.ndarray of shape (n,), one e-value per observation\n    \"\"\"\n    outcomes_treated = np.asarray(outcomes_treated, dtype=float)\n    outcomes_control = np.asarray(outcomes_control, dtype=float)\n    n = min(len(outcomes_treated), len(outcomes_control))\n\n    cum_t = np.cumsum(outcomes_treated[:n])\n    cum_c = np.cumsum(outcomes_control[:n])\n    t_arr = np.arange(1, n + 1, dtype=float)\n\n    # Alternative hypothesis: each arm has its own independent Beta prior on completion rate\n    log_ml_t = (betaln(alpha_prior + cum_t, beta_prior + t_arr - cum_t)\n                - betaln(alpha_prior, beta_prior))\n    log_ml_c = (betaln(alpha_prior + cum_c, beta_prior + t_arr - cum_c)\n                - betaln(alpha_prior, beta_prior))\n\n    # Null hypothesis: both arms share a single pooled Beta prior on the common rate\n    pooled_successes = cum_t + cum_c\n    pooled_n = 2 * t_arr\n    log_ml_h0 = (betaln(alpha_prior + pooled_successes,\n                        beta_prior + pooled_n - pooled_successes)\n                 - betaln(alpha_prior, beta_prior))\n\n    # Log Bayes factor is the difference in log marginal likelihoods\n    log_bf = log_ml_t + log_ml_c - log_ml_h0\n\n    return np.exp(log_bf)\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u06cc\u06c1 \u0641\u0646\u06a9\u0634\u0646 0\/1 \u0646\u062a\u0627\u0626\u062c \u06a9\u06cc \u062f\u0648 \u0635\u0641\u06cc\u06ba \u0644\u06cc\u062a\u0627 \u06c1\u06d2 \u062c\u0648 \u062a\u0627\u0631\u06cc\u062e \u06a9\u06cc \u062a\u0631\u062a\u06cc\u0628 \u0645\u06cc\u06ba \u0622\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u06c1\u0631 \u0628\u0627\u0631 \u0642\u062f\u0645 t \u06a9\u06d2 \u0644\u06cc\u06d2\u060c \u06c1\u0645 \u06c1\u0631 \u0648\u0631\u0698\u0646 \u06a9\u06d2 \u0644\u06cc\u06d2 \u06a9\u0627\u0645\u06cc\u0627\u0628\u06cc\u0648\u06ba \u0627\u0648\u0631 \u06a9\u0648\u0634\u0634\u0648\u06ba \u06a9\u06cc \u0645\u062c\u0645\u0648\u0639\u06cc \u062a\u0639\u062f\u0627\u062f \u06a9\u0627 \u062d\u0633\u0627\u0628 \u0644\u06af\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<p><code>betaln<\/code>    \u0628\u06cc\u0679\u0627 \u0641\u0646\u06a9\u0634\u0646 \u06a9\u0627 \u0644\u0648\u06af\u0627\u0631\u062a\u06be\u0645 \u0641\u0631\u0627\u06c1\u0645 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u060c \u062c\u0648 \u06a9\u06c1 \u0628\u06cc\u0679\u0627 \u0628\u0627\u0626\u0646\u0648\u0645\u06cc\u0644 \u0645\u0639\u0645\u0648\u0644\u06cc \u0627\u0645\u06a9\u0627\u0646 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0646\u0627\u0631\u0645\u0644\u0627\u0626\u0632\u06cc\u0634\u0646 \u0645\u0633\u062a\u0642\u0644 \u06c1\u06d2\u06d4 H1 \u06c1\u0631 \u0628\u0627\u0632\u0648 \u06a9\u06d2 \u062a\u0646\u0627\u0633\u0628 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0622\u0632\u0627\u062f \u0628\u06cc\u0679\u0627 \u067e\u0631\u06cc\u0631\u0632 \u06a9\u0648 \u0634\u0627\u0645\u0644 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 H0 \u067e\u06c1\u0644\u06d2 \u06a9\u06d2 \u0627\u06cc\u06a9 \u062d\u0635\u06c1 \u06a9\u06d2 \u062a\u0646\u0627\u0633\u0628 \u067e\u0631 \u0636\u0645 \u06c1\u0648\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0644\u0627\u06af Bayes \u0639\u0646\u0635\u0631 \u0641\u0631\u0642 \u06c1\u06d2. \u0627\u0633 \u06a9\u06cc \u0648\u0636\u0627\u062d\u062a \u06a9\u0631\u0646\u0627 \u0627\u06cc \u0648\u06cc\u0644\u06cc\u0648 \u062f\u06cc\u062a\u0627 \u06c1\u06d2\u06d4 \u0627\u06af\u0631 \u0639\u0644\u0627\u062c \u06a9\u0627 \u062d\u0642\u06cc\u0642\u06cc \u0627\u062b\u0631 \u06c1\u0648 \u062a\u0648 \u0648\u0642\u062a \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0633\u0627\u062a\u06be \u0627\u06cc \u0648\u06cc\u0644\u06cc\u0648 \u0628\u0691\u06be \u062c\u0627\u062a\u06cc \u06c1\u06d2\u06d4 \u06cc\u06c1 \u0627\u06cc\u06a9 \u0633\u067e\u0631 \u0645\u0627\u0631\u0679\u0646\u06af\u06cc\u0644 \u06c1\u06d2 \u062c\u0648 \u0628\u063a\u06cc\u0631 \u06a9\u0633\u06cc \u0627\u062b\u0631 \u06a9\u06d2 1 \u06a9\u06d2 \u0627\u0631\u062f \u06af\u0631\u062f \u0627\u0686\u06be\u0627\u0644\u062a\u0627 \u06c1\u06d2 \u0627\u0648\u0631 H0 \u0633\u06d2 \u0646\u06cc\u0686\u06d2 \u063a\u06cc\u0631 \u0645\u0646\u0641\u06cc \u06c1\u06d2\u06d4<\/p>\n<p>null \u0688\u06cc\u0679\u0627 \u067e\u0631 \u0641\u0648\u0631\u06cc \u0633\u0646\u062c\u06cc\u062f\u06af\u06cc \u06a9\u06cc \u062c\u0627\u0646\u0686 \u0645\u062a\u0648\u0642\u0639 \u0631\u0648\u06cc\u06d2 \u06a9\u06cc \u062a\u0635\u062f\u06cc\u0642 \u06a9\u0631\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<pre><code class=\"language-python\">np.random.seed(0)\nnull_t = np.random.binomial(1, 0.60, 500)\nnull_c = np.random.binomial(1, 0.60, 500)\nev_null = compute_evalue_running(null_t, null_c)\nprint(f\"E-value at end under null (should be near 1): {ev_null[-1]:.3f}\")\nprint(f\"Max e-value under null: {ev_null.max():.3f}\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">E-value at end under null (should be near 1): 0.078\nMax e-value under null: 2.188\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> null \u06a9\u06d2 \u062a\u062d\u062a\u060c \u062d\u062a\u0645\u06cc e-value 1 \u06a9\u06d2 \u0642\u0631\u06cc\u0628 \u062e\u062a\u0645 \u06c1\u0648 \u062c\u0627\u062a\u06cc \u06c1\u06d2 (\u06cc\u06c1\u0627\u06ba \u0646\u0645\u0648\u0646\u06d2 \u0644\u06cc\u0646\u06d2 \u06a9\u06d2 \u062a\u063a\u06cc\u0631 \u06a9\u06cc \u0648\u062c\u06c1 \u0633\u06d2 0.078 \u06c1\u06d2)\u060c \u0627\u0648\u0631 500 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0645\u0634\u0627\u06c1\u062f\u0627\u062a \u06a9\u06cc \u0633\u0628 \u0633\u06d2 \u0628\u0691\u06cc \u062a\u0639\u062f\u0627\u062f 20 \u06a9\u06d2 \u0631\u06a9\u0646\u06d2 \u06a9\u06cc \u062d\u062f \u0633\u06d2 \u0628\u0627\u0644\u06a9\u0644 \u0646\u06cc\u0686\u06d2 \u0631\u06c1\u062a\u06cc \u06c1\u06d2\u06d4 Ville \u06a9\u06cc \u0639\u062f\u0645 \u0645\u0633\u0627\u0648\u0627\u062a \u06a9\u06d2 \u0645\u0637\u0627\u0628\u0642\u060c \u0627\u0633 \u0628\u0627\u062a \u06a9\u0627 \u0627\u0645\u06a9\u0627\u0646 \u06c1\u06d2 \u06a9\u06c1 \u0627\u06cc\u06a9 \u062f\u0631\u0633\u062a null e-value \u0633\u0628 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 20 \u0641\u06cc\u0635\u062f \u062a\u06a9 \u067e\u06c1\u0646\u0686 \u062c\u0627\u0626\u06d2 \u06af\u0627\u060c \u062c\u0633 \u06a9\u06d2 \u0639\u0645\u0644 \u06a9\u06d2 \u0633\u0627\u062a\u06be 5 \u0641\u06cc\u0635\u062f \u062a\u06a9 \u067e\u06c1\u0646\u0686 \u062c\u0627\u0626\u06d2 \u06af\u06cc\u06d4 5% \u0642\u0633\u0645 I \u063a\u0644\u0637\u06cc \u06a9\u06cc \u0634\u0631\u062d\u06d4 \u0627\u0633 \u0648\u0627\u062d\u062f 500 \u0645\u0634\u0627\u06c1\u062f\u06d2 \u06a9\u06cc \u062f\u0648\u0691 \u0645\u06cc\u06ba\u060c \u0632\u06cc\u0627\u062f\u06c1 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 2.188 \u06c1\u06d2\u060c \u062c\u0648 \u0645\u062a\u0648\u0642\u0639 \u0631\u0648\u06cc\u06c1 \u06c1\u06d2\u06d4<\/p>\n<h2 id=\"heading-step-3-apply-msprt-to-the-real-dataset\">\u0645\u0631\u062d\u0644\u06c1 3: \u0627\u0635\u0644\u06cc \u0688\u06cc\u0679\u0627 \u0633\u06cc\u0679 \u067e\u0631 mSPRT \u0644\u06af\u0627\u0626\u06cc\u06ba\u06d4<\/h2>\n<p>\u0627\u0628 \u0679\u06cc\u0633\u0679 \u06a9\u0648 \u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627 \u067e\u0631 \u0644\u06af\u0627\u0626\u06cc\u06ba \u062c\u06c1\u0627\u06ba \u0639\u0644\u0627\u062c \u06a9\u06d2 \u062d\u0642\u06cc\u0642\u06cc \u0627\u062b\u0631\u0627\u062a \u0645\u0648\u062c\u0648\u062f \u06c1\u0648\u06ba\u06d4 \u06c1\u0631 \u062f\u0646 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0686\u0644\u0646\u06d2 \u0648\u0627\u0644\u06cc \u0627\u06cc \u0648\u06cc\u0644\u06cc\u0648 \u06a9\u0627 \u062d\u0633\u0627\u0628 \u0644\u06af\u0627\u0626\u06cc\u06ba \u0627\u0648\u0631 \u067e\u06c1\u0644\u0627 \u062f\u0646 \u062a\u0644\u0627\u0634 \u06a9\u0631\u06cc\u06ba \u062c\u0628 \u06cc\u06c1 \u0631\u06a9\u0646\u06d2 \u06a9\u06cc \u062d\u062f \u0633\u06d2 \u062a\u062c\u0627\u0648\u0632 \u06a9\u0631 \u062c\u0627\u0626\u06d2\u06d4<\/p>\n<pre><code class=\"language-python\">import matplotlib.pyplot as plt\n\nnp.random.seed(42)\ntreated_shuffled = treated.copy()\ncontrol_shuffled = control.copy()\nnp.random.shuffle(treated_shuffled)\nnp.random.shuffle(control_shuffled)\n\nUSERS_PER_ARM_PER_DAY = 60\nN_DAYS_RUN = 30\nn_per_arm = USERS_PER_ARM_PER_DAY * N_DAYS_RUN  # 1,800\n\ntreated_seq = treated_shuffled[:n_per_arm]\ncontrol_seq = control_shuffled[:n_per_arm]\n\ne_values = compute_evalue_running(treated_seq, control_seq)\n\nALPHA = 0.05\nTHRESHOLD = 1 \/ ALPHA  # = 20\n\ndays = np.arange(1, len(e_values) + 1) \/ USERS_PER_ARM_PER_DAY\ncross_indices = np.where(e_values >= THRESHOLD)[0]\nif len(cross_indices) > 0:\n    stopping_day = days[cross_indices[0]]\n    print(f\"mSPRT stopping day: {stopping_day:.1f}\")\n    print(f\"E-value at stopping: {e_values[cross_indices[0]]:.1f}\")\nelse:\n    stopping_day = None\n    print(\"mSPRT did not cross threshold in this window\")\n\nprint(f\"Final e-value on day {N_DAYS_RUN}: {e_values[-1]:.2f}\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">mSPRT stopping day: 25.9\nE-value at stopping: 20.9\nFinal e-value on day 30: 75.64\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u06c1\u0645 \u0639\u0644\u0627\u062c \u0627\u0648\u0631 \u06a9\u0646\u0679\u0631\u0648\u0644 \u06a9\u06d2 \u0627\u0646\u062a\u0638\u0627\u0645\u0627\u062a \u0645\u06cc\u06ba \u0631\u062f\u0648\u0628\u062f\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u062a\u0627\u06a9\u06c1 \u0635\u0627\u0631\u0641\u06cc\u0646 \u06a9\u06cc \u0631\u0648\u0632\u0627\u0646\u06c1 \u06a9\u06cc \u0628\u06d2 \u062a\u0631\u062a\u06cc\u0628 \u0622\u0645\u062f \u06a9\u0627 \u0627\u0646\u062f\u0627\u0632\u06c1 \u0644\u06af\u0627\u06cc\u0627 \u062c\u0627 \u0633\u06a9\u06d2 (\u062d\u0642\u06cc\u0642\u06cc \u062a\u062c\u0631\u0628\u0627\u062a \u0645\u06cc\u06ba\u060c \u06c1\u0645 \u0635\u0627\u0631\u0641\u06cc\u0646 \u06a9\u0648 \u06a9\u0633\u06cc \u062e\u0627\u0635 \u062a\u0631\u062a\u06cc\u0628 \u0645\u06cc\u06ba \u0641\u0631\u0627\u06c1\u0645 \u0646\u06c1\u06cc\u06ba \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba) \u0627\u0648\u0631 \u067e\u06be\u0631 \u067e\u06c1\u0644\u06d2 1,800 \u0627\u0641\u0631\u0627\u062f \u06a9\u0648 \u0641\u06cc \u0628\u0627\u0632\u0648 \u0645\u0646\u062a\u062e\u0628 \u06a9\u0631\u06cc\u06ba\u06d4 <code>compute_evalue_running<\/code> \u0627\u06cc\u06a9 \u0648\u0642\u062a \u0645\u06cc\u06ba \u0627\u06cc\u06a9 \u0645\u0634\u0627\u06c1\u062f\u06c1\u06d4 \u0627\u06cc \u0642\u062f\u0631 25.9 \u062f\u0646\u0648\u06ba \u0645\u06cc\u06ba 20 \u06a9\u06cc \u062d\u062f \u0633\u06d2 \u062a\u062c\u0627\u0648\u0632 \u06a9\u0631 \u062c\u0627\u062a\u06cc \u06c1\u06d2\u06d4 \u0627\u0633 \u06a9\u0627 \u0645\u0637\u0644\u0628 \u06cc\u06c1 \u06c1\u06d2 \u06a9\u06c1 \u062a\u062c\u0631\u0628\u06d2 \u06a9\u0648 4 \u062f\u0646 \u067e\u06c1\u0644\u06d2 \u0628\u0644\u0627\u06cc\u0627 \u062c\u0627 \u0633\u06a9\u062a\u0627 \u06c1\u06d2\u060c \u0645\u06a9\u0645\u0644 \u0637\u0648\u0631 \u067e\u0631 \u062f\u0631\u0633\u062a \u0627\u0646\u062f\u0627\u0632\u06c1 \u06a9\u0648 \u06cc\u0642\u06cc\u0646\u06cc \u0628\u0646\u0627 \u06a9\u0631\u06d4 30 \u0648\u06cc\u06ba \u062f\u0646\u060c \u0627\u06cc-\u0648\u06cc\u0644\u06cc\u0648 75.64 \u062a\u06a9 \u067e\u06c1\u0646\u0686 \u06af\u0626\u06cc\u060c \u062d\u062f \u0633\u06d2 \u0628\u06c1\u062a \u0627\u0648\u067e\u0631\u06d4<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/umang.pk\/wp-content\/uploads\/2026\/07\/1783086800_798_\u067e\u0631\u0648\u0688\u06a9\u0679-\u06a9\u0627-\u062a\u062c\u0631\u0628\u06c1-\u0627\u06cc\u0645-\u0627\u06cc\u0633-\u067e\u06cc-\u0622\u0631-\u0679\u06cc-\u06a9\u0627-\u0627\u0633\u062a\u0639\u0645\u0627\u0644.png\" alt=\"82ae7b10-c598-4597-80e6-375fa76b209d\" style=\"display:block;margin:0 auto\" width=\"1486\" height=\"947\" loading=\"lazy\" title=\"\"><\/p>\n<p><em>\u0634\u06a9\u0644 2: \u062d\u0642\u06cc\u0642\u06cc 50,000 \u0635\u0627\u0631\u0641 \u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u06a9\u06d2 \u0644\u06cc\u06d2 \u062d\u0642\u06cc\u0642\u06cc mSPRT \u0627\u06cc-\u0648\u06cc\u0644\u06cc\u0648 \u06a9\u06cc \u0631\u0641\u062a\u0627\u0631 (\u0648\u06cc\u0648 1 \u0679\u0631\u06cc\u0679\u0645\u0646\u0679 \u0628\u0645\u0642\u0627\u0628\u0644\u06c1 \u0648\u06cc\u0648 2 \u06a9\u0646\u0679\u0631\u0648\u0644)\u06d4 \u0646\u06cc\u0644\u06cc \u0644\u06a9\u06cc\u0631 \u0627\u06cc \u0648\u06cc\u0644\u06cc\u0648 \u06c1\u06d2 \u062c\u0648 \u0644\u0648\u06af\u0627\u0631\u06cc\u062a\u06be\u0645\u06a9 \u067e\u06cc\u0645\u0627\u0646\u06d2 \u067e\u0631 \u0686\u0644\u0627\u0626\u06cc \u062c\u0627\u062a\u06cc \u06c1\u06d2\u06d4 \u0633\u0631\u062e \u0688\u06cc\u0634\u0688 \u0644\u0627\u0626\u0646 1\/\u03b1 = 20 \u06a9\u06cc \u0631\u06a9\u0646\u06d2 \u06a9\u06cc \u062d\u062f \u06c1\u06d2\u06d4<\/em><\/p>\n<p><em>\u0633\u0628\u0632 \u0646\u0642\u0637\u06d2 \u0648\u0627\u0644\u06cc \u0639\u0645\u0648\u062f\u06cc \u0644\u0627\u0626\u0646 25.9 \u062f\u0646 \u06a9\u0648 \u0646\u0634\u0627\u0646 \u0632\u062f \u06a9\u0631\u062a\u06cc \u06c1\u06d2 \u062c\u0628 \u0627\u06cc \u0648\u06cc\u0644\u06cc\u0648 \u067e\u06c1\u0644\u06d2 \u062d\u062f \u0633\u06d2 \u062a\u062c\u0627\u0648\u0632 \u06a9\u0631 \u062c\u0627\u062a\u06cc \u06c1\u06d2\u06d4 \u0646\u06cc\u0686\u06d2 \u0648\u0627\u0644\u0627 \u067e\u06cc\u0646\u0644 \u0641\u06cc \u0628\u0627\u0632\u0648 \u067e\u0631 \u0645\u062c\u0645\u0648\u0639\u06cc \u06a9\u0627\u0645 \u06a9\u06cc \u062a\u06a9\u0645\u06cc\u0644 \u06a9\u0627 \u0641\u06cc\u0635\u062f \u062f\u06a9\u06be\u0627\u062a\u0627 \u06c1\u06d2\u060c \u062c\u0648 \u0688\u06cc\u0679\u0627 \u06a9\u06d2 \u062c\u0645\u0639 \u06c1\u0648\u0646\u06d2 \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0633\u0627\u062a\u06be \u0628\u062f\u0644 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4 \u0634\u06a9\u0644 1 \u0645\u06cc\u06ba \u0627\u0633\u06a9\u06cc\u0645\u06cc\u0679\u06a9 \u06a9\u06d2 \u0628\u0631\u0639\u06a9\u0633\u060c \u06cc\u06c1 \u0645\u0634\u062a\u0631\u06a9\u06c1 \u0688\u06cc\u0679\u0627 \u0633\u06cc\u0679 \u0633\u06d2 \u062d\u0642\u06cc\u0642\u06cc \u0688\u06cc\u0679\u0627 \u06c1\u06d2\u060c \u0627\u0648\u0631 \u0627\u0633 \u0645\u06cc\u06ba \u062d\u0642\u06cc\u0642\u06cc 4.85 \u067e\u06cc \u067e\u06cc \u0627\u0636\u0627\u0641\u06c1 \u06c1\u06d2\u06d4<\/em><\/p>\n<h2 id=\"heading-step-4-compare-power-against-a-fixed-sample-test\">\u0645\u0631\u062d\u0644\u06c1 4: \u0641\u06a9\u0633\u0688 \u0646\u0645\u0648\u0646\u06c1 \u0679\u06cc\u0633\u0679 \u06a9\u06d2 \u0633\u0627\u062a\u06be \u0637\u0627\u0642\u062a \u06a9\u0627 \u0645\u0648\u0627\u0632\u0646\u06c1 \u06a9\u0631\u06cc\u06ba\u06d4<\/h2>\n<p>mSPRT \u06a9\u06d2 \u062d\u0642\u06cc\u0642\u06cc \u0627\u062e\u0631\u0627\u062c\u0627\u062a \u06c1\u06cc\u06ba\u06d4 \u0627\u062b\u0631 \u0686\u0627\u0644\u0648 \u06c1\u0648\u0646\u06d2 \u06a9\u06d2 \u0628\u0639\u062f\u060c \u0622\u067e \u0627\u0633\u06d2 \u0627\u0633 \u06a9\u06d2 \u0645\u0642\u0631\u0631\u06c1 \u0627\u062e\u062a\u062a\u0627\u0645\u06cc \u0648\u0642\u062a \u0633\u06d2 \u067e\u06c1\u0644\u06d2 \u0631\u0648\u06a9 \u0633\u06a9\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0637\u0627\u0642\u062a \u06a9\u0627 \u062c\u0631\u0645\u0627\u0646\u06c1 \u0627\u06c1\u0645 \u06c1\u06d2 \u0627\u06af\u0631 \u0627\u062b\u0631 \u067e\u06c1\u0644\u06d2 \u06a9\u06cc \u062a\u0648\u0642\u0639 \u0633\u06d2 \u06a9\u0645 \u06c1\u06d2 \u06cc\u0627 \u0627\u06af\u0631 \u0622\u067e \u0646\u0645\u0648\u0646\u06d2 \u06a9\u06d2 \u0645\u0639\u0645\u0648\u0644\u06cc \u0633\u0627\u0626\u0632 \u06a9\u06d2 \u0633\u0627\u062a\u06be \u06a9\u0627\u0645 \u06a9\u0631 \u0631\u06c1\u06d2 \u06c1\u06cc\u06ba\u06d4 \u06cc\u06c1 \u0646\u0642\u0644\u06cc \u0627\u06cc\u0645\u0627\u0646\u062f\u0627\u0631\u06cc \u06a9\u06d2 \u0633\u0627\u062a\u06be \u062a\u062c\u0627\u0631\u062a \u06a9\u06cc \u0645\u0642\u062f\u0627\u0631 \u06a9\u0648 \u062f\u0631\u0633\u062a \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<pre><code class=\"language-python\">from scipy.stats import ttest_ind\n\nnp.random.seed(42)\n\nN_SIMS = 1000\nTRUE_EFFECT = 0.05\nBASE_RATE = 0.60\nN_PER_ARM = 1800          # 30 days x 60 users\/arm\/day\nDAILY_BATCH = 60\nTHRESHOLD = 20\n\nmsprt_stopping_days = []\nmsprt_detected = 0\nttest_detected = 0\n\nfor sim in range(N_SIMS):\n    t_obs = np.random.binomial(1, BASE_RATE + TRUE_EFFECT, N_PER_ARM)\n    c_obs = np.random.binomial(1, BASE_RATE, N_PER_ARM)\n\n    e_vals = compute_evalue_running(t_obs, c_obs)\n    days = np.arange(1, N_PER_ARM + 1) \/ DAILY_BATCH\n    crosses = np.where(e_vals >= THRESHOLD)[0]\n    if len(crosses) > 0:\n        msprt_detected += 1\n        msprt_stopping_days.append(days[crosses[0]])\n    else:\n        msprt_stopping_days.append(30.0)\n\n    _, p = ttest_ind(t_obs, c_obs)\n    if p < 0.05:\n        ttest_detected += 1\n\nmsprt_power = msprt_detected \/ N_SIMS\nttest_power = ttest_detected \/ N_SIMS\nmedian_stop = np.median(msprt_stopping_days)\npct_stopped_early = np.mean(np.array(msprt_stopping_days) < 30.0)\n\nprint(f\"mSPRT power:               {msprt_power:.1%}\")\nprint(f\"Fixed-sample t-test power: {ttest_power:.1%}\")\nprint(f\"Median mSPRT stop day:     {median_stop:.1f} \/ 30\")\nprint(f\"Fraction stopping early:   {pct_stopped_early:.1%}\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">mSPRT power:               49.3%\nFixed-sample t-test power: 88.7%\nMedian mSPRT stop day:     30.0 \/ 30\nFraction stopping early:   49.3%\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u06c1\u0645 \u0627\u06cc\u06a9 \u062d\u0642\u06cc\u0642\u06cc 5pp \u0644\u0641\u0679 \u06a9\u06d2 \u0633\u0627\u062a\u06be 1,000 \u0646\u0642\u0644\u06cc\u06ba \u0686\u0644\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 mSPRT \u06a9\u06d2 \u0644\u06cc\u06d2\u060c \u0645\u062c\u0645\u0648\u0639\u06cc \u0627\u06cc \u0648\u06cc\u0644\u06cc\u0648 \u06a9\u0627 \u062d\u0633\u0627\u0628 \u0644\u06af\u0627\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2 \u0627\u0648\u0631 20 \u06a9\u0627 \u067e\u06c1\u0644\u0627 \u0627\u0646\u0679\u0631\u0633\u06cc\u06a9\u0634\u0646 \u0631\u06cc\u06a9\u0627\u0631\u0688 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0645\u0642\u0631\u0631\u06c1 \u0646\u0645\u0648\u0646\u06d2 \u06a9\u06cc \u062c\u0627\u0646\u0686 \u06a9\u06d2 \u0644\u06cc\u06d2\u060c \u06c1\u0645 30 \u062f\u0646 \u06a9\u06d2 \u0622\u062e\u0631 \u0645\u06cc\u06ba \u0627\u06cc\u06a9 \u067e\u0631 \u0627\u06cc\u06a9 \u0646\u0638\u0631 \u0688\u0627\u0644\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0646\u062a\u0627\u0626\u062c \u0627\u06cc\u06a9 \u0628\u0627\u0645\u0639\u0646\u06cc \u0637\u0627\u0642\u062a \u06a9\u06d2 \u0641\u0631\u0642 \u06a9\u0648 \u0638\u0627\u06c1\u0631 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u0627\u06cc\u0645 \u0627\u06cc\u0633 \u067e\u06cc \u0622\u0631 \u0679\u06cc \u0646\u06d2 49.3 \u0641\u06cc\u0635\u062f \u0679\u0631\u0627\u0626\u0644\u0632 \u0645\u06cc\u06ba \u0627\u062b\u0631 \u06a9\u0627 \u067e\u062a\u06c1 \u0644\u06af\u0627\u06cc\u0627\u060c \u062c\u0628 \u06a9\u06c1 \u0641\u06a9\u0633\u0688 \u0633\u06cc\u0645\u067e\u0644 \u0679\u06cc\u0633\u0679 \u0646\u06d2 88.7 \u0641\u06cc\u0635\u062f \u0645\u06cc\u06ba \u0627\u062b\u0631 \u067e\u0627\u06cc\u0627\u06d4 5 \u067e\u06cc \u067e\u06cc \u0644\u0641\u0679 \u0627\u0648\u0631 1,800 \u0645\u0634\u0627\u06c1\u062f\u0627\u062a \u0641\u06cc \u0628\u0627\u0632\u0648 \u06a9\u06d2 \u0633\u0627\u062a\u06be\u060c mSPRT \u06a9\u0648 \u0627\u06cc\u06a9 \u0645\u0642\u0631\u0631\u06c1 \u0646\u0645\u0648\u0646\u06c1 \u0679\u06cc\u0633\u0679 \u06a9\u06cc \u0637\u0627\u0642\u062a \u0633\u06d2 \u0645\u0644\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u062a\u0642\u0631\u06cc\u0628\u0627\u064b \u062f\u0648 \u06af\u0646\u0627 \u0632\u06cc\u0627\u062f\u06c1 \u0645\u0634\u0627\u06c1\u062f\u0627\u062a \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u0648\u062a\u06cc \u06c1\u06d2\u06d4<\/p>\n<p>\u06cc\u06c1 \u0627\u06cc\u06a9 \u0648\u0627\u0631\u0646\u0679\u06cc \u06a9\u06cc \u0642\u06cc\u0645\u062a \u06c1\u06d2 \u062c\u0648 \u06c1\u0645\u06cc\u0634\u06c1 \u0644\u0627\u06af\u0648 \u06c1\u0648\u062a\u06cc \u06c1\u06d2\u06d4 \u062c\u0628 \u0622\u067e \u0631\u0648\u0632\u0627\u0646\u06c1 \u0686\u06cc\u06a9 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u062a\u0648 \u0622\u067e \u06a9\u0648 \u062c\u0648 \u0645\u0644\u062a\u0627 \u06c1\u06d2 \u0648\u06c1 \u06c1\u06d2 \u0679\u0627\u0626\u067e I \u0627\u06cc\u0631\u0631 \u06a9\u0646\u0679\u0631\u0648\u0644\u06d4 \u0631\u0648\u0632\u0627\u0646\u06c1 \u062c\u06be\u0627\u0646\u06a9\u0646\u06d2 \u06a9\u06d2 \u0627\u06cc\u06a9 \u0645\u0642\u0631\u0631\u06c1 \u0646\u0645\u0648\u0646\u06d2 \u06a9\u06cc \u062c\u0627\u0646\u0686 \u06a9\u0631\u0646\u06d2 \u0633\u06d2 \u0634\u0631\u062d 30.2% \u062a\u06a9 \u0628\u0691\u06be \u062c\u0627\u062a\u06cc \u06c1\u06d2\u06d4 mSPRT 5% \u067e\u0631 \u0631\u06c1\u062a\u0627 \u06c1\u06d2 \u0642\u0637\u0639 \u0646\u0638\u0631 \u0627\u0633 \u06a9\u06d2 \u06a9\u06c1 \u06cc\u06c1 \u06a9\u0628 \u0631\u06a9 \u062c\u0627\u0626\u06d2\u06d4<\/p>\n<p>\u0635\u062d\u06cc\u062d \u0627\u0646\u062a\u062e\u0627\u0628 \u06a9\u0627 \u0627\u0646\u062d\u0635\u0627\u0631 \u0627\u0633 \u0628\u0627\u062a \u067e\u0631 \u06c1\u06d2 \u06a9\u06c1 \u0622\u067e \u06a9\u06cc \u0679\u06cc\u0645 \u06a9\u0648 \u06a9\u06cc\u0627 \u0632\u06cc\u0627\u062f\u06c1 \u062e\u0631\u0686 \u0622\u062a\u0627 \u06c1\u06d2\u06d4 \u06cc\u0627 \u062a\u0648 \u062a\u062c\u0631\u0628\u06c1 \u0632\u06cc\u0627\u062f\u06c1 \u0686\u0644\u062a\u0627 \u06c1\u06d2 \u06cc\u0627 \u06cc\u06c1 \u063a\u0644\u0637 \u0645\u062b\u0628\u062a \u067e\u06cc\u0634 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u0632\u06cc\u0627\u062f\u06c1 \u062a\u0631 \u0679\u06cc\u0645\u06cc\u06ba \u0627\u067e\u0646\u06cc \u0637\u0627\u0642\u062a \u06a9\u06d2 \u0627\u062e\u0631\u0627\u062c\u0627\u062a \u06a9\u0648 \u06a9\u0645 \u0646\u06c1\u06cc\u06ba \u0633\u0645\u062c\u06be\u062a\u06cc \u06c1\u06cc\u06ba \u062c\u0628 \u062a\u06a9 \u06a9\u06c1 \u0648\u06c1 \u062e\u0648\u062f \u0627\u0646 \u0646\u0642\u0627\u0644\u06cc \u06a9\u0648 \u0646\u06c1\u06cc\u06ba \u0686\u0644\u0627\u062a\u06d2\u06d4<\/p>\n<h2 id=\"heading-validate-against-ground-truth\">\u0632\u0645\u06cc\u0646\u06cc \u0633\u0686\u0627\u0626\u06cc \u06a9\u06cc \u062a\u0635\u062f\u06cc\u0642<\/h2>\n<p>\u0645\u0635\u0646\u0648\u0639\u06cc \u0688\u06cc\u0679\u0627\u0633\u06cc\u0679 \u0645\u06cc\u06ba \u0627\u06cc\u06a9 \u0645\u0639\u0631\u0648\u0641 5pp \u0627\u0636\u0627\u0641\u06c1 \u0634\u0627\u0645\u0644 \u06c1\u06d2\u060c \u062c\u0648 \u06c1\u0645\u06cc\u06ba \u0627\u0633 \u0628\u0627\u062a \u06a9\u06cc \u062a\u0635\u062f\u06cc\u0642 \u06a9\u0631\u0646\u06d2 \u06a9\u06cc \u0627\u062c\u0627\u0632\u062a \u062f\u06cc\u062a\u0627 \u06c1\u06d2 \u06a9\u06c1 30 \u062f\u0646 \u06a9\u06cc \u0645\u062f\u062a \u06a9\u06d2 \u0628\u0639\u062f \u0645\u0632\u06cc\u062f \u0688\u06cc\u0679\u0627 \u062f\u0633\u062a\u06cc\u0627\u0628 \u06c1\u0648\u0646\u06d2 \u067e\u0631 mSPRT \u0627\u062b\u0631 \u06a9\u06cc \u0635\u062d\u06cc\u062d \u0634\u0646\u0627\u062e\u062a \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<pre><code class=\"language-python\">np.random.seed(0)\nt_full = treated_shuffled\nc_full = control_shuffled[:len(t_full)]\n\ne_full = compute_evalue_running(t_full, c_full)\ndays_full = np.arange(1, len(e_full) + 1) \/ USERS_PER_ARM_PER_DAY\n\ncross_full = np.where(e_full >= THRESHOLD)[0]\nif len(cross_full) > 0:\n    print(f\"mSPRT correctly detected the effect.\")\n    print(f\"Could have stopped on day {days_full[cross_full[0]]:.1f}\")\n    print(f\"True effect in data: {treated.mean() - control.mean():.4f}\")\n    print(f\"E-value at stopping point: {e_full[cross_full[0]]:.1f}\")\nelse:\n    print(\"mSPRT did not cross threshold with this data slice.\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">mSPRT correctly detected the effect.\nCould have stopped on day 27.1\nTrue effect in data: 0.0485\nE-value at stopping point: 22.2\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u062c\u0628 \u0627\u06cc\u0645 \u0627\u06cc\u0633 \u067e\u06cc \u0622\u0631 \u0679\u06cc \u06a9\u0648 \u067e\u0648\u0631\u06cc \u0634\u0641\u0644\u0688 \u0627\u0631\u06cc (24,937 \u0679\u0631\u06cc\u0679\u0645\u0646\u0679\u0633\u060c 25,063 \u06a9\u0646\u0679\u0631\u0648\u0644\u0632) \u067e\u0631 \u0686\u0644\u0627\u062a\u06d2 \u06c1\u06cc\u06ba \u062a\u0648 \u0627\u06cc \u0648\u06cc\u0644\u06cc\u0648 27.1 \u062f\u0646\u0648\u06ba \u0645\u06cc\u06ba \u062d\u062f \u0633\u06d2 \u062a\u062c\u0627\u0648\u0632 \u06a9\u0631 \u062c\u0627\u062a\u06cc \u06c1\u06d2\u06d4 \u0627\u0639\u062f\u0627\u062f \u0648 \u0634\u0645\u0627\u0631 \u0645\u06cc\u06ba \u0627\u0635\u0644 \u0648\u062c\u06c1 \u0627\u062b\u0631\u060c 4.85pp\u060c \u062c\u0646\u0631\u06cc\u0679\u0631 \u06a9\u06cc 5pp \u06a9\u06cc \u062d\u0642\u06cc\u0642\u06cc \u0642\u062f\u0631 \u06a9\u06d2 \u0642\u0631\u06cc\u0628 \u06c1\u06d2 \u0627\u0648\u0631 \u0627\u0633 \u06a9\u0627 \u0635\u062d\u06cc\u062d \u0637\u0648\u0631 \u067e\u0631 \u067e\u062a\u06c1 \u0686\u0644\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>30 \u062f\u0646\u0648\u06ba \u06a9\u06d2 \u0644\u06cc\u06d2 \u0688\u06cc\u0632\u0627\u0626\u0646 \u06a9\u06cc\u06d2 \u06af\u0626\u06d2 \u0641\u06a9\u0633\u0688 \u0646\u0645\u0648\u0646\u06d2 \u06a9\u06d2 \u0679\u06cc\u0633\u0679 30 \u062f\u0646 \u062a\u06a9 \u0628\u0631\u0642\u0631\u0627\u0631 \u0631\u06a9\u06be\u06d2 \u062c\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u06cc\u06c1\u0627\u06ba \u062a\u06a9 \u06a9\u06c1 \u062c\u0628 \u062b\u0628\u0648\u062a \u067e\u06c1\u0644\u06d2 \u06c1\u06cc \u062c\u0645\u0639 \u06c1\u0648 \u0686\u06a9\u06d2 \u06c1\u0648\u06ba\u06d4 60 \u06cc\u0648\u0632\u0631 \u0641\u06cc \u0628\u0627\u0632\u0648 \u0641\u06cc \u062f\u0646 \u06a9\u06d2 \u0633\u0627\u062a\u06be\u060c mSPRT \u06c1\u0645\u06cc\u06ba 27.1 \u062f\u0646\u0648\u06ba \u0645\u06cc\u06ba \u0631\u06cc\u0644\u06cc\u0632 \u06a9\u0631\u0646\u06d2 \u06a9\u06cc \u0627\u062c\u0627\u0632\u062a \u062f\u06cc\u062a\u0627 \u06c1\u06d2\u060c \u0627\u0633 \u062e\u0635\u0648\u0635\u06cc\u062a \u067e\u0631 \u06c1\u0645\u06cc\u06ba \u062a\u0642\u0631\u06cc\u0628\u0627\u064b 3 \u062f\u0646 \u06a9\u06cc \u0628\u0686\u062a \u06c1\u0648\u062a\u06cc \u06c1\u06d2 \u062c\u0648 \u06c1\u0645\u06cc\u0634\u06c1 \u0631\u06cc\u0644\u06cc\u0632 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0645\u0642\u0631\u0631 \u062a\u06be\u06cc\u06d4<\/p>\n<h2 id=\"heading-step-5-bootstrap-confidence-intervals\">\u0645\u0631\u062d\u0644\u06c1 5: \u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u0627 \u0648\u0642\u0641\u06c1<\/h2>\n<p>\u0627\u0633\u0679\u0627\u067e \u0688\u06cc\u0679 \u0622\u067e \u06a9\u0648 \u0628\u062a\u0627\u062a\u06cc \u06c1\u06d2 \u06a9\u06c1 \u062a\u062c\u0631\u0628\u06c1 \u06a9\u0628 \u06a9\u0627\u0644 \u06a9\u0631\u0646\u0627 \u06c1\u06d2\u060c \u0644\u06cc\u06a9\u0646 \u06cc\u06c1 \u0622\u067e \u06a9\u0648 \u06cc\u06c1 \u0646\u06c1\u06cc\u06ba \u0628\u062a\u0627\u062a\u06cc \u06a9\u06c1 \u0627\u062b\u0631 \u06a9\u062a\u0646\u0627 \u0628\u0691\u0627 \u06c1\u0648\u06af\u0627 \u06cc\u0627 \u0627\u0633 \u06a9\u0627 \u0627\u0646\u062f\u0627\u0632\u06c1 \u06a9\u062a\u0646\u0627 \u062f\u0631\u0633\u062a \u06c1\u0648\u06af\u0627\u06d4 \u0628\u0648\u0679\u0633\u0679\u0631\u06cc\u067e \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u06d2 \u0648\u0642\u0641\u06d2 \u062f\u0648\u0646\u0648\u06ba \u0641\u0631\u0627\u06c1\u0645 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<pre><code class=\"language-python\">rng = np.random.default_rng(7)\npoint_est = treated.mean() - control.mean()\n\nboot_diffs = np.array([\n    rng.choice(treated, size=len(treated), replace=True).mean() -\n    rng.choice(control, size=len(control), replace=True).mean()\n    for _ in range(500)\n])\n\nlower = float(np.percentile(boot_diffs, 2.5))\nupper = float(np.percentile(boot_diffs, 97.5))\n\nprint(f\"Point estimate (treated - control): {point_est:.4f} ({point_est*100:.2f}pp)\")\nprint(f\"95% bootstrap CI: [{lower:.4f}, {upper:.4f}]  \"\n      f\"([{lower*100:.2f}pp, {upper*100:.2f}pp])\")\nprint(f\"Ground-truth 5pp is {'inside' if lower <= 0.05 <= upper else 'outside'} the CI.\")\n<\/code><\/pre>\n<p><strong>\u0645\u062a\u0648\u0642\u0639 \u067e\u06cc\u062f\u0627\u0648\u0627\u0631:<\/strong><\/p>\n<pre><code class=\"language-text\">Point estimate (treated - control): 0.0485 (4.85pp)\n95% bootstrap CI: [0.0407, 0.0581]  ([4.07pp, 5.81pp])\nGround-truth 5pp is inside the CI.\n<\/code><\/pre>\n<p><strong>\u0645\u0648\u062c\u0648\u062f\u06c1 \u0635\u0648\u0631\u062a\u062d\u0627\u0644 \u06a9\u0686\u06be \u06cc\u0648\u06ba \u06c1\u06d2:<\/strong> \u06c1\u0645 \u0639\u0644\u0627\u062c \u0634\u062f\u06c1 \u0627\u0648\u0631 \u06a9\u0646\u0679\u0631\u0648\u0644 \u0635\u0641\u0648\u06ba \u06a9\u0648 500 \u0628\u0627\u0631 \u0622\u0632\u0627\u062f\u0627\u0646\u06c1 \u0637\u0648\u0631 \u067e\u0631 \u062a\u0628\u062f\u06cc\u0644 \u06a9\u0631\u06a9\u06d2 \u062f\u0648\u0628\u0627\u0631\u06c1 \u0646\u0645\u0648\u0646\u06c1 \u0628\u0646\u0627\u062a\u06d2 \u06c1\u06cc\u06ba \u0627\u0648\u0631 \u06c1\u0631 \u0628\u0627\u0631 \u0630\u0631\u0627\u0626\u0639 \u0645\u06cc\u06ba \u0641\u0631\u0642 \u06a9\u0627 \u062d\u0633\u0627\u0628 \u0644\u06af\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 500 \u0641\u0631\u0642\u0648\u06ba \u06a9\u0627 2.5 \u0648\u0627\u06ba \u0627\u0648\u0631 97.5 \u0641\u06cc\u0635\u062f \u0627\u0639\u062a\u0645\u0627\u062f \u06a9\u06d2 \u0648\u0642\u0641\u06d2 \u0628\u0646\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 CI 4.07pp \u0633\u06d2 5.81pp \u062a\u06a9 \u0686\u0644\u062a\u0627 \u06c1\u06d2\u060c \u0628\u0634\u0645\u0648\u0644 \u062d\u0642\u06cc\u0642\u06cc 5pp \u0627\u0648\u0631 0 \u06a9\u0648 \u0686\u06be\u0648\u0691 \u06a9\u0631 \u06cc\u06c1 \u06cc\u0642\u06cc\u0646\u06cc \u0628\u0646\u0627\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u06a9\u06c1 \u0627\u062b\u0631 \u062d\u0642\u06cc\u0642\u06cc \u06c1\u06d2\u06d4 \u0627\u0633 \u0628\u0627\u062a \u067e\u0631 \u063a\u0648\u0631 \u06a9\u0631\u062a\u06d2 \u06c1\u0648\u0626\u06d2 \u06a9\u06c1 \u0641\u06cc \u0633\u06cc\u06af\u0645\u0646\u0679 \u0645\u06cc\u06ba 25,000 \u0635\u0627\u0631\u0641\u06cc\u0646 \u06c1\u06cc\u06ba\u060c \u0648\u0642\u0641\u06c1 \u06a9\u0627\u0631\u06cc \u06a9\u0627\u0641\u06cc \u062a\u0646\u06af \u06c1\u06d2\u060c \u0627\u06cc\u06a9 \u06c1\u06cc \u0642\u062f\u0645 \u0645\u06cc\u06ba \"\u06a9\u06cc\u0627 \u0627\u0633 \u0646\u06d2 \u06a9\u0627\u0645 \u06a9\u06cc\u0627\" (\u06c1\u0627\u06ba) \u0627\u0648\u0631 \"\u06a9\u062a\u0646\u0627\" (4.07 \u0627\u0648\u0631 5.81 \u0641\u06cc\u0635\u062f \u067e\u0648\u0627\u0626\u0646\u0679\u0633 \u06a9\u06d2 \u062f\u0631\u0645\u06cc\u0627\u0646) \u06a9\u0627 \u062c\u0648\u0627\u0628 \u062f\u0648\u0646\u0648\u06ba \u0641\u0631\u0627\u06c1\u0645 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<h2 id=\"heading-when-msprt-fails\">\u062c\u0628 mSPRT \u0646\u0627\u06a9\u0627\u0645 \u06c1\u0648\u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4<\/h2>\n<p>\u062a\u0631\u062a\u06cc\u0628 \u0648\u0627\u0631 \u062c\u0627\u0646\u0686 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0627\u0628 \u0628\u06be\u06cc \u062a\u062c\u0631\u0628\u0627\u062a\u06cc \u0633\u062e\u062a\u06cc \u06a9\u06cc \u0636\u0631\u0648\u0631\u062a \u06c1\u06d2\u06d4 \u0686\u0627\u0631 \u062d\u0627\u0644\u0627\u062a \u0636\u0645\u0627\u0646\u062a \u06a9\u0648 \u062a\u0648\u0691 \u062f\u06cc\u062a\u06d2 \u06c1\u06cc\u06ba \u06cc\u0627 \u0637\u0631\u06cc\u0642\u06c1 \u06a9\u0627\u0631 \u06a9\u0648 \u0639\u0645\u0644\u06cc \u0637\u0648\u0631 \u067e\u0631 \u0628\u06cc\u06a9\u0627\u0631 \u06a9\u0631 \u062f\u06cc\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4<\/p>\n<h3 id=\"heading-badly-misspecified-prior\">\u067e\u06c1\u0644\u06d2 \u063a\u0644\u0637 \u0637\u0631\u06cc\u0642\u06d2 \u0633\u06d2 \u0628\u06cc\u0627\u0646 \u06a9\u06cc\u0627 \u06af\u06cc\u0627 \u062a\u06be\u0627\u06d4<\/h3>\n<p>\u0627\u06cc\u0645 \u0627\u06cc\u0633 \u067e\u06cc \u0622\u0631 \u0679\u06cc \u06c1\u0631 \u0642\u062f\u0645 \u06a9\u06cc \u062a\u06a9\u0645\u06cc\u0644 \u06a9\u06cc \u0634\u0631\u062d \u06a9\u06d2 \u0644\u06cc\u06d2 \u0627\u06cc\u06a9 \u067e\u06cc\u0634\u06af\u06cc \u0628\u06cc\u0679\u0627 (1,1) \u0641\u0631\u0636 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u060c \u06cc\u0639\u0646\u06cc \u0645\u0627\u0688\u0644\u0646\u06af \u06a9\u06d2 \u0627\u0646\u062a\u062e\u0627\u0628 \u062c\u0646 \u06a9\u06d2 \u0646\u062a\u06cc\u062c\u06d2 \u0645\u06cc\u06ba \u062d\u0642\u06cc\u0642\u06cc \u0646\u062a\u0627\u0626\u062c \u0628\u0631\u0622\u0645\u062f \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba\u06d4 \u06cc\u06c1 \u067e\u06cc\u0634\u06af\u06cc \u062a\u0635\u0631\u06cc\u062d \u06a9\u06d2 \u0645\u0641\u0631\u0648\u0636\u06d2 \u06a9\u06cc \u062e\u0644\u0627\u0641 \u0648\u0631\u0632\u06cc \u06a9\u0631\u062a\u0627 \u06c1\u06d2 \u0627\u06af\u0631 \u0627\u0635\u0644 \u0627\u062b\u0631 \u067e\u06c1\u0644\u06d2 \u06a9\u06cc \u0637\u0631\u0641 \u0633\u06d2 \u0645\u062a\u0648\u0642\u0639 \u062d\u062f \u0633\u06d2 \u0628\u06c1\u062a \u0628\u0627\u06c1\u0631 \u0622\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u06cc\u06a9\u0633\u0627\u06ba \u0628\u06cc\u0679\u0627 (1,1) \u067e\u06c1\u0644\u06d2 3-10pp \u06a9\u06cc \u062d\u062f \u0645\u06cc\u06ba \u062f\u0631\u0645\u06cc\u0627\u0646\u06d2 \u0627\u062b\u0631\u0627\u062a \u06a9\u06d2 \u0644\u06cc\u06d2 \u062a\u0642\u0631\u06cc\u0628\u0627\u064b 60% \u06a9\u06cc \u0628\u0646\u06cc\u0627\u062f\u06cc \u0634\u0631\u062d \u067e\u0631 \u0645\u0639\u0642\u0648\u0644 \u062d\u062f \u062a\u06a9 \u0627\u0686\u06be\u06cc \u06a9\u0627\u0631\u06a9\u0631\u062f\u06af\u06cc \u06a9\u0627 \u0645\u0638\u0627\u06c1\u0631\u06c1 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 \u0627\u06af\u0631 \u0627\u0635\u0644 \u0627\u062b\u0631 0.3pp \u0627\u0636\u0627\u0641\u06c1 \u06c1\u06d2 (\u0627\u06cc\u06a9 \u0645\u0639\u0645\u0648\u0644\u06cc AI \u062e\u0635\u0648\u0635\u06cc\u062a \u0645\u06cc\u06ba \u062a\u0628\u062f\u06cc\u0644\u06cc \u06a9\u06d2 \u0644\u06cc\u06d2 \u0627\u06cc\u06a9 \u062d\u0642\u06cc\u0642\u062a \u067e\u0633\u0646\u062f\u0627\u0646\u06c1 \u0646\u062a\u06cc\u062c\u06c1)\u060c \u062a\u0648 \u0627\u06cc \u0648\u06cc\u0644\u06cc\u0648 \u0628\u06c1\u062a \u0622\u06c1\u0633\u062a\u06c1 \u0622\u06c1\u0633\u062a\u06c1 \u0628\u0691\u06be\u062a\u0627 \u06c1\u06d2\u06d4 \u0646\u0645\u0648\u0646\u06c1 \u0628\u062c\u0679 \u062d\u062f \u0633\u06d2 \u0628\u0691\u06be\u0646\u06d2 \u0633\u06d2 \u067e\u06c1\u0644\u06d2 \u06c1\u06cc \u062e\u062a\u0645 \u06c1\u0648 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<p>\u0627\u067e\u0646\u06cc \u0644\u063a\u062a \u06a9\u0648 \u0627\u067e\u0646\u06d2 \u067e\u0631\u0648\u0688\u06a9\u0679 \u06a9\u06d2 \u062a\u0627\u0631\u06cc\u062e\u06cc A\/B \u0679\u06cc\u0633\u0679\u0646\u06af \u0688\u06cc\u0679\u0627 \u06a9\u06d2 \u062e\u0644\u0627\u0641 \u06a9\u06cc\u0644\u06cc\u0628\u0631\u06cc\u0679 \u06a9\u0631\u06cc\u06ba\u06d4 \u06c1\u0645 \u0628\u06cc \u0679\u0627 \u06c1\u0627\u0626\u067e\u0631 \u067e\u06cc\u0631\u0627\u0645\u06cc\u0679\u0631 \u06a9\u0648 \u062a\u0627\u0631\u06cc\u062e\u06cc \u0627\u062b\u0631 \u06a9\u06d2 \u0633\u0627\u0626\u0632 \u06a9\u06cc \u062a\u0642\u0633\u06cc\u0645 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0641\u0679 \u06a9\u0631\u0646\u06d2 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0633\u06d2 \u0632\u06cc\u0627\u062f\u06c1 \u0627\u0645\u06a9\u0627\u0646\u0627\u062a \u06a9\u0627 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u062a\u06d2 \u06c1\u06cc\u06ba \u0627\u0648\u0631 \u0627\u0633 \u0628\u0627\u062a \u06a9\u0648 \u06cc\u0642\u06cc\u0646\u06cc \u0628\u0646\u0627\u062a\u06d2 \u06c1\u06cc\u06ba \u06a9\u06c1 \u0646\u062a\u06cc\u062c\u06c1 \u0645\u06cc\u06ba \u067e\u06cc\u0634\u06af\u06cc \u06a9\u0645 \u0627\u0632 \u06a9\u0645 \u0642\u0627\u0628\u0644 \u0634\u0646\u0627\u062e\u062a \u0627\u062b\u0631 \u06a9\u06d2 \u0642\u0631\u06cc\u0628 \u0628\u0627\u0645\u0639\u0646\u06cc \u0645\u0627\u0633 \u0631\u06a9\u06be\u062a\u0627 \u06c1\u06d2\u06d4<\/p>\n<h3 id=\"heading-non-stationary-outcomes\">\u063a\u06cc\u0631 \u0645\u0639\u0645\u0648\u0644\u06cc \u0646\u062a\u0627\u0626\u062c<\/h3>\n<p>\u06af\u0627\u0631\u0646\u0679\u06cc \u06a9\u0627 \u062a\u0642\u0627\u0636\u06c1 \u06c1\u06d2 \u06a9\u06c1 \u0627\u06cc-\u0648\u06cc\u0644\u06cc\u0648 \u06a9\u0627 \u0639\u0645\u0644 \u06a9\u0627\u0644\u0639\u062f\u0645 \u06a9\u06d2 \u062a\u062d\u062a \u0627\u06cc\u06a9 \u063a\u06cc\u0631 \u0645\u0646\u0641\u06cc \u0633\u067e\u0631 \u0645\u0627\u0631\u0679\u0646\u06af\u06cc\u0644 \u06c1\u0648\u060c \u062c\u0633 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0636\u0631\u0648\u0631\u06cc \u06c1\u06d2 \u06a9\u06c1 \u0688\u06cc\u0679\u0627 \u062c\u0646\u0631\u06cc\u0634\u0646 \u06a9\u0627 \u0639\u0645\u0644 \u0637\u06d2 \u06a9\u06cc\u0627 \u062c\u0627\u0626\u06d2\u06d4 \u0627\u06af\u0631 AI \u0645\u0627\u0688\u0644 \u06a9\u0648 \u062a\u062c\u0631\u0628\u06d2 \u06a9\u06d2 \u0648\u0633\u0637 \u0645\u06cc\u06ba \u0627\u067e \u0688\u06cc\u0679 \u06a9\u06cc\u0627 \u062c\u0627\u062a\u0627 \u06c1\u06d2\u060c \u0627\u06af\u0631 \u0635\u0627\u0631\u0641 \u06a9\u06cc \u0622\u0628\u0627\u062f\u06cc \u062d\u0631\u06a9\u062a \u06a9\u0631\u062a\u06cc \u06c1\u06d2 (\u062c\u06cc\u0633\u06d2 \u0645\u0627\u0631\u06a9\u06cc\u0679\u0646\u06af \u06a9\u06cc \u0645\u06c1\u0645 \u062c\u0633 \u06a9\u06d2 \u0646\u062a\u06cc\u062c\u06d2 \u0645\u06cc\u06ba 12 \u062a\u0627\u0631\u06cc\u062e \u06a9\u0648 \u0645\u062e\u062a\u0644\u0641 \u06af\u0631\u0648\u06c1 \u06c1\u0648\u062a\u06d2 \u06c1\u06cc\u06ba)\u060c \u06cc\u0627 \u0627\u06af\u0631 \u06a9\u0627\u0645 \u06a9\u06cc \u062f\u0634\u0648\u0627\u0631\u06cc \u067e\u0631 \u06c1\u0641\u062a\u06d2 \u06a9\u06d2 \u062f\u0646 \u06a9\u0627 \u0627\u062b\u0631 \u06c1\u0648\u062a\u0627 \u06c1\u06d2\u060c \u062a\u0648 \u0627\u06cc-\u0648\u06cc\u0644\u06cc\u0648 \u0645\u0627\u062d\u0648\u0644\u06cc\u0627\u062a\u06cc \u0634\u0648\u0631 \u06a9\u0648 \u062c\u0630\u0628 \u06a9\u0631\u06d2 \u06af\u06cc \u062c\u0633\u06d2 \u062a\u062c\u0631\u0628\u06c1 \u067e\u0631\u0648\u0633\u06cc\u0633\u0646\u06af \u0627\u062b\u0631\u0627\u062a \u0633\u06d2 \u0627\u0644\u06af \u0646\u06c1\u06cc\u06ba \u06a9\u0631 \u0633\u06a9\u062a\u0627\u06d4<\/p>\n<p>\u06c1\u0648\u0644\u0688 \u0622\u0624\u0679 A\/A \u062a\u062c\u0631\u0628\u06c1 \u0645\u06cc\u06ba \u0627\u06cc-\u0648\u06cc\u0644\u06cc\u0648 \u06a9\u06d2 \u0646\u0641\u0627\u0630 \u06a9\u0648 \u0686\u0644\u0627 \u06a9\u0631 \u0628\u06d2 \u0636\u0627\u0628\u0637\u06af\u06cc\u0648\u06ba \u06a9\u06cc \u062a\u0634\u062e\u06cc\u0635 \u06a9\u0631\u06cc\u06ba\u06d4 \u0627\u06af\u0631 null e-value \u06a9\u0627 \u0639\u0645\u0644 \u0627\u0648\u067e\u0631 \u06a9\u06cc \u0637\u0631\u0641 \u0628\u0691\u06be \u0631\u06c1\u0627 \u06c1\u06d2 \u062c\u0628 \u0627\u0633\u06d2 1 \u06a9\u06d2 \u0642\u0631\u06cc\u0628 \u06c1\u0648\u0646\u0627 \u0686\u0627\u06c1\u06cc\u06d2\u060c \u062a\u0648 \u0645\u0627\u062d\u0648\u0644 \u0627\u062a\u0646\u0627 \u0645\u0633\u062a\u062d\u06a9\u0645 \u0646\u06c1\u06cc\u06ba \u06c1\u06d2 \u06a9\u06c1 \u0637\u0631\u06cc\u0642\u06c1 \u0642\u0627\u0628\u0644 \u0627\u0639\u062a\u0645\u0627\u062f \u06c1\u0648\u06d4<\/p>\n<h3 id=\"heading-multiple-metrics-without-multiplicity-correction\">\u0636\u0631\u0628 \u06a9\u06cc \u0627\u0635\u0644\u0627\u062d \u06a9\u06d2 \u0628\u063a\u06cc\u0631 \u0645\u062a\u0639\u062f\u062f \u0645\u06cc\u0679\u0631\u06a9\u0633<\/h3>\n<p>mSPRT \u0627\u06cc\u06a9 \u06c1\u06cc \u0645\u0648\u0627\u0632\u0646\u06c1 \u06a9\u06d2 \u0644\u06cc\u06d2 \u0642\u0633\u0645 I \u06a9\u06cc \u063a\u0644\u0637\u06cc \u06a9\u0648 \u06a9\u0646\u0679\u0631\u0648\u0644 \u06a9\u0631\u062a\u0627 \u06c1\u06d2\u06d4 20 \u0645\u06cc\u0679\u0631\u06a9\u0633 \u06a9\u06cc \u062c\u0627\u0646\u0686 \u062e\u0648\u062f \u0637\u0631\u06cc\u0642\u06c1 \u06a9\u0627\u0631 \u06a9\u0648 \u0646\u0627\u06a9\u0627\u0645 \u0646\u06c1\u06cc\u06ba \u06a9\u0631\u062a\u06cc \u06c1\u06d2\u060c \u0644\u06c1\u0630\u0627 \u06c1\u0631 \u0627\u0646\u0641\u0631\u0627\u062f\u06cc \u0627\u06cc \u0648\u06cc\u0644\u06cc\u0648 \u062f\u0631\u0633\u062a \u0631\u06c1\u062a\u06cc \u06c1\u06d2\u06d4 \u062c\u0648 \u0686\u06cc\u0632 \u0646\u0627\u06a9\u0627\u0645 \u06c1\u0648\u062a\u06cc \u06c1\u06d2 \u0648\u06c1 \u062e\u0627\u0646\u062f\u0627\u0646 \u06a9\u06d2 \u0644\u062d\u0627\u0638 \u0633\u06d2 \u063a\u0644\u0637\u06cc \u06a9\u06cc \u0634\u0631\u062d \u06c1\u06d2\u06d4 \u0627\u06af\u0631 \u0622\u067e \u0628\u06cc\u06a9 \u0648\u0642\u062a 20 \u0627\u0634\u0627\u0631\u06cc\u0648\u06ba \u067e\u0631 \u0627\u06cc\u0645 \u0627\u06cc\u0633 \u067e\u06cc \u0622\u0631 \u0679\u06cc \u0686\u0644\u0627\u062a\u06d2 \u06c1\u06cc\u06ba \u0627\u0648\u0631 \u062c\u0628 \u06a9\u0648\u0626\u06cc \u0628\u06be\u06cc 20 \u0633\u06d2 \u062a\u062c\u0627\u0648\u0632 \u06a9\u0631 \u062c\u0627\u062a\u0627 \u06c1\u06d2 \u062a\u0648 \u0631\u06a9 \u062c\u0627\u062a\u06d2 \u06c1\u06cc\u06ba\u060c \u06a9\u0645 \u0627\u0632 \u06a9\u0645 \u0627\u06cc\u06a9 \u063a\u0644\u0637 \u0645\u062b\u0628\u062a \u06c1\u0648\u0646\u06d2 \u06a9\u0627 \u0627\u0645\u06a9\u0627\u0646 5% \u0633\u06d2 \u0628\u06c1\u062a \u0632\u06cc\u0627\u062f\u06c1 \u06c1\u0648\u06af\u0627\u06d4<\/p>\n<p>\u03b1=0.05 \u067e\u0631 m=20 \u0645\u06cc\u0679\u0631\u06a9\u0633 \u06a9\u06d2 \u0644\u06cc\u06d2\u060c \u062d\u062f \u06a9\u0648 1\/(\u03b1\/m) = 400 \u062a\u06a9 \u0628\u0691\u06be\u0627 \u06a9\u0631 \u0628\u0648\u0646\u0641\u0631\u0648\u0646\u06cc \u062a\u0635\u062d\u06cc\u062d \u06a9\u0627 \u0627\u0637\u0644\u0627\u0642 \u06a9\u0631\u06cc\u06ba \u06cc\u0627 \u062a\u062c\u0631\u0628\u06d2 \u06a9\u06d2 \u0627\u062e\u062a\u062a\u0627\u0645 \u067e\u0631 \u062d\u062a\u0645\u06cc \u0627\u06cc-\u0648\u06cc\u0644\u06cc\u0648 \u067e\u0631 \u0628\u06cc\u0646\u062c\u0645\u06cc\u0646\u06cc-\u06c1\u0686\u0628\u0631\u06af \u0637\u0631\u06cc\u0642\u06c1 \u06a9\u0627\u0631 \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u06a9\u0631\u06cc\u06ba\u06d4<\/p>\n<p>\u0636\u0631\u0628 \u06a9\u0627 \u0645\u0633\u0626\u0644\u06c1 \u0648\u06c1\u06cc \u06c1\u06d2 \u062c\u0648 \u0641\u06a9\u0633\u0688 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