changr from BF_{21} to BF_{12} to be consistent

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
-}
+}

notations.ipynb

@@ -62,13 +62,13 @@
     "| $\\boldsymbol{\\Sigma}$ | Known covariance matrix | Residual covariance between height and weight |\n",
     "| **Prior Parameters** | | |\n",
     "| $\\beta_0$ | Prior mean (scalar) | Expected effect = 0 before seeing data |\n",
-    "| $\\boldsymbol{\\beta}_0$ | Prior mean vector | Expected effects = $\\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}$ before data |\n",
+    "| $\\boldsymbol{\\beta}_0$ | Prior mean vector | Expected effects = $\\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}$ before seeing data |\n",
     "| $\\sigma_0^2$ | Prior variance (scalar) | Prior uncertainty = 0.25 for effect size |\n",
     "| $\\boldsymbol{\\Sigma}_0$ | Prior covariance matrix | Prior uncertainty about joint effects |\n",
     "| **Posterior Parameters** | | |\n",
     "| $\\beta_1$ | Posterior mean (scalar) | Updated effect = 0.3 after seeing data |\n",
-    "| $\\boldsymbol{\\beta}_1$ | Posterior mean vector | Updated effects = $\\begin{pmatrix} 0.3 \\\\ 0.2 \\end{pmatrix}$ after data |\n",
-    "| $\\sigma_1^2$ | Posterior variance (scalar) | Reduced uncertainty = 0.1 after data |\n",
+    "| $\\boldsymbol{\\beta}_1$ | Posterior mean vector | Updated effects = $\\begin{pmatrix} 0.3 \\\\ 0.2 \\end{pmatrix}$ after seeing data |\n",
+    "| $\\sigma_1^2$ | Posterior variance (scalar) | Reduced uncertainty = 0.1 after seeing data |\n",
     "| $\\boldsymbol{\\Sigma}_1$ | Posterior covariance matrix | Reduced uncertainty about joint effects |\n",
     "| **Mixed Models** | | |\n",
     "| $\\mathbf{g}$ | Random effect vector in linear mixed models | Individual-specific genetic effects |\n",