From a8eef66f40115e31836cd6f69bda17b6bc10a7c6 Mon Sep 17 00:00:00 2001 From: RuiDongDR Date: Wed, 14 Jan 2026 22:57:58 -0500 Subject: [PATCH 1/3] revise from BF_21 to BF_12 --- Bayes_factor.ipynb | 36 +++++++++++++++------------ graphical_summary/slides.pptx | Bin 7344134 -> 7346638 bytes graphical_summary/slides/Slide1.png | Bin 292504 -> 293967 bytes graphical_summary/slides/Slide10.png | Bin 374928 -> 370744 bytes graphical_summary/slides/Slide11.png | Bin 358598 -> 355288 bytes graphical_summary/slides/Slide12.png | Bin 316621 -> 342310 bytes graphical_summary/slides/Slide13.png | Bin 384195 -> 382232 bytes graphical_summary/slides/Slide14.png | Bin 450650 -> 448448 bytes graphical_summary/slides/Slide15.png | Bin 413985 -> 412287 bytes graphical_summary/slides/Slide16.png | Bin 328682 -> 329376 bytes graphical_summary/slides/Slide17.png | Bin 336884 -> 332519 bytes graphical_summary/slides/Slide18.png | Bin 438330 -> 436428 bytes graphical_summary/slides/Slide19.png | Bin 552547 -> 548944 bytes graphical_summary/slides/Slide2.png | Bin 210801 -> 211506 bytes graphical_summary/slides/Slide20.png | Bin 526954 -> 527670 bytes graphical_summary/slides/Slide21.png | Bin 475433 -> 473441 bytes graphical_summary/slides/Slide22.png | Bin 462800 -> 463430 bytes graphical_summary/slides/Slide23.png | Bin 503217 -> 500471 bytes graphical_summary/slides/Slide24.png | Bin 387095 -> 389447 bytes graphical_summary/slides/Slide25.png | Bin 616993 -> 617219 bytes graphical_summary/slides/Slide26.png | Bin 413149 -> 413417 bytes graphical_summary/slides/Slide27.png | Bin 546486 -> 546304 bytes graphical_summary/slides/Slide28.png | Bin 360950 -> 360240 bytes graphical_summary/slides/Slide29.png | Bin 330760 -> 331258 bytes graphical_summary/slides/Slide3.png | Bin 186371 -> 186388 bytes graphical_summary/slides/Slide30.png | Bin 377590 -> 372577 bytes graphical_summary/slides/Slide31.png | Bin 612227 -> 610074 bytes graphical_summary/slides/Slide32.png | Bin 431688 -> 427732 bytes graphical_summary/slides/Slide4.png | Bin 465441 -> 465988 bytes graphical_summary/slides/Slide5.png | Bin 429107 -> 427129 bytes graphical_summary/slides/Slide6.png | Bin 270443 -> 272247 bytes graphical_summary/slides/Slide7.png | Bin 353180 -> 345329 bytes graphical_summary/slides/Slide8.png | Bin 399362 -> 400428 bytes graphical_summary/slides/Slide9.png | Bin 424015 -> 422525 bytes 34 files changed, 20 insertions(+), 16 deletions(-) diff --git a/Bayes_factor.ipynb b/Bayes_factor.ipynb index a710a7a..099e46d 100644 --- a/Bayes_factor.ipynb +++ b/Bayes_factor.ipynb @@ -36,7 +36,7 @@ "The Bayes factor quantifies the relative evidence for two competing model families $\\text{M}_1$ and $\\text{M}_2$ given observed data $\\text{D}$. It compares the marginal likelihoods by integrating over all possible parameter values weighted by their prior distributions:\n", "\n", "$$\n", - "\\text{BF}_{21} = \\frac{P(\\text{D}|\\text{M}_2)}{P(\\text{D}|\\text{M}_1)} = \\frac{\\int P(\\text{D}|\\theta_{2}, \\text{M}_{2}) P(\\theta_{2}|\\text{M}_{2})\\,d\\theta_{2}}{\\int P(\\text{D}|\\theta_{1},\\text{M}_{1}) P(\\theta_{1}|\\text{M}_{1})\\,d\\theta _{1}}\n", + "\\text{BF}_{12} = \\frac{P(\\text{D}\\mid \\text{M}_1)}{P(\\text{D}\\mid \\text{M}_2)} = \\frac{\\int P(\\text{D}\\mid \\theta_{1}, \\text{M}_{1}) \\, P(\\theta_{1}\\mid \\text{M}_{1}) \\, d\\theta_{1}}{\\int P(\\text{D}\\mid \\theta_{2}, \\text{M}_{2}) \\, P(\\theta_{2}\\mid \\text{M}_{2}) \\, d\\theta_{2}}\n", "$$\n", "\n", "where:\n", @@ -65,11 +65,13 @@ "## Interpretation Guidelines\n", "\n", "Following [Kass and Raftery (1995)](https://www.tandfonline.com/doi/abs/10.1080/01621459.1995.10476572), Bayes factors are typically interpreted as:\n", - "- $\\text{BF}_{21} < 1$: Evidence favors $\\text{M}_1$\n", - "- $1 < \\text{BF}_{21} < 3$: Weak evidence for $\\text{M}_2$\n", - "- $3 < \\text{BF}_{21} < 10$: Moderate evidence for $\\text{M}_2$\n", - "- $10 < \\text{BF}_{21} < 100$: Strong evidence for $\\text{M}_2$\n", - "- $\\text{BF}_{21} > 100$: Decisive evidence for $\\text{M}_2$" + "\n", + "| $\\text{BF}_{12}$ | Strength of evidence (for $\\text{M}_1$ over $\\text{M}_2$) |\n", + "|------------------|--------------------------------------------|\n", + "|1 to 3.2 | Not worth more than a bare mention |\n", + "|3.2 to 10 | Substantial |\n", + "|10 to 100 | Strong |\n", + "|> 100 | Decisive |" ] }, { @@ -119,7 +121,9 @@ "cell_type": "markdown", "id": "708774c1", "metadata": {}, - "source": "## Setup" + "source": [ + "## Setup" + ] }, { "cell_type": "code", @@ -246,7 +250,7 @@ "text": [ "Marginal Likelihood (Model Family 1, beta=0): 0.00192103 \n", "Marginal Likelihood (Model Family 2, beta~N(0,1)): 0.001008886 \n", - "Bayes Factor (BF_21): 0.5251796 \n", + "Bayes Factor (BF_12): 1.90411 \n", "Interpretation: Evidence favors Model Family 1 (no effect)\n" ] } @@ -285,18 +289,18 @@ "marginal_lik_model_family_1 <- likelihood(beta = 0, sd = true_sd, X = X, Y = Y) # Point mass at beta = 0\n", "marginal_lik_model_family_2 <- marginal_likelihood(X, Y, true_sd, prior_mean, prior_sd)\n", "\n", - "# Calculate Bayes Factor (BF_21 = Model 2 vs Model 1)\n", - "bf_21 <- marginal_lik_model_family_2 / marginal_lik_model_family_1\n", + "# Calculate Bayes Factor (BF_12 = Model 1 vs Model 2)\n", + "bf_12 <- marginal_lik_model_family_1 / marginal_lik_model_family_2\n", "\n", "cat(\"Marginal Likelihood (Model Family 1, beta=0):\", marginal_lik_model_family_1, \"\\n\")\n", "cat(\"Marginal Likelihood (Model Family 2, beta~N(0,1)):\", marginal_lik_model_family_2, \"\\n\")\n", - "cat(\"Bayes Factor (BF_21):\", bf_21, \"\\n\")\n", + "cat(\"Bayes Factor (BF_12):\", bf_12, \"\\n\")\n", "\n", "# Interpretation\n", - "if (bf_21 > 1) {\n", - " cat(\"Interpretation: Evidence favors Model Family 2 (some effect exists)\\n\")\n", - "} else {\n", + "if (bf_12 > 1) {\n", " cat(\"Interpretation: Evidence favors Model Family 1 (no effect)\\n\")\n", + "} else {\n", + " cat(\"Interpretation: Evidence favors Model Family 2 (some effect exists)\\n\")\n", "}" ] }, @@ -415,7 +419,7 @@ "id": "be1c0b37", "metadata": {}, "source": [ - "**Exercise**: Here the Bayes factor between the two models is ~0.52, which doesn't suggest very strong evidence. This is mainly because of the fact that there are only five samples in the data. Try to increase the number of samples and see how Bayes factor changes." + "**Exercise**: Here the Bayes factor between the two models is 1.90, which doesn't suggest very strong evidence. This is mainly because of the fact that there are only five samples in the data. Try to increase the number of samples and see how Bayes factor changes." ] }, { @@ -448,4 +452,4 @@ }, "nbformat": 4, "nbformat_minor": 5 -} \ No newline at end of file +} diff --git a/graphical_summary/slides.pptx b/graphical_summary/slides.pptx index 599971081eb457daced2308e44f9158e991b5f42..5819a84199f58f8b1cfcb88584592e562c2da96b 100644 GIT binary patch delta 204402 zcmZ^~LtrLM&@CL>b~3ST+nzkJZCg(!$;7s8+qP}nn%Mc@@4I(#H~*@;PWP%8r%#=# z3d!pOpK0s^4`>4A1;PoOvB$(>fq)#KrS98+Vgp$jJ?(6-^wymZB|UoFEAEkCYRxC? z;VoTF;;WsE2$jqpeXQ9NkFdagp%I}3fGS6Rbh-;+6U2!qZvWlq$j&<6pB&^b&4qY7=K0e2#B$a3@ zNr2bTGAWW&-I--qUw0=CKPMlbZSwDETiGe2{W1=kfVY41D!536D$4$u$0~5WdZxqF zWqUR(+!xsWnEY-@1V&mbF_I!#&lw+)b~m$(j0eoljgI^g3l;v-s1zcQ0!*P!`YlS6 ztzO#t;uQQLy8^4lTzBUCS_>ArX5Zyuq(Jv1{T_z<#!s<+;P>KjsA$B8aBWT!6d4Kp z78Zfj>hec}bxe^HYVM=|gP-m1S*o#6GI7&Pl1(zo`+93n%@@Kd>)ysr%xi9X(w$rn z^bXpMh0dshy0M9?f_Kg-NkkRxjR70#qI~+x9f!4|A!RpYuZ{zf(I$Rz76KLIO(1L2 z6>r9i&+Xati&WR_H5D+j@I)e$%YNsltOQDkTRU$uPXHmNmMH@DBlufpl}aG40{{7? 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