Merge pull request #21 from RuiDongDR/main

fix small typos

Author: gaow

Bayesian_model_comparison.ipynb

@@ -53,42 +53,13 @@
    "cell_type": "markdown",
    "id": "f5aa6a8c",
    "metadata": {},
-   "source": [
-    "# Technical Details\n",
-    "\n",
-    "## Bayes Factor\n",
-    "\n",
-    "Recall in [Lecture: Bayes factor](https://statfungen.github.io/statgen-primer/Bayes_factor.html), we learn that the **Bayes factor** quantifies the relative evidence data provides for competing models:\n",
-    "\n",
-    "$$\n",
-    "\\text{BF}_{21} = \\frac{L(\\text{M}_2|\\text{D})}{L(\\text{M}_1|\\text{D})} ={\\frac {\\int \\Pr(\\theta_{2}|M_{2})\\Pr(D|\\theta_{2},M_{2})\\,d\\theta _{2}}{\\int \\Pr(\\theta_{1}|M_{1})\\Pr(D|\\theta_{1},M_{1})\\,d\\theta _{1}}}\n",
-    "$$\n",
-    "\n",
-    "**Interpretation**: \n",
-    "- $\\text{BF}_{21} > 1$: Data favors $\\text{M}_2$ over $\\text{M}_1$\n",
-    "- $\\text{BF}_{21} < 1$: Data favors $\\text{M}_1$ over $\\text{M}_2$\n",
-    "- $\\text{BF}_{21} = 1$: Data uninformative for model choice\n"
-   ]
+   "source": "# Technical Details\n\n## Bayes Factor\n\nRecall in [Lecture: Bayes factor](https://statfungen.github.io/statgen-primer/Bayes_factor.html), we learn that the **Bayes factor** quantifies the relative evidence data provides for competing models:\n\n$$\n\\text{BF}_{12} = \\frac{L(\\text{M}_1|\\text{D})}{L(\\text{M}_2|\\text{D})} ={\\frac {\\int \\Pr(\\theta_{1}|M_{1})\\Pr(D|\\theta_{1},M_{1})\\,d\\theta _{1}}{\\int \\Pr(\\theta_{2}|M_{2})\\Pr(D|\\theta_{2},M_{2})\\,d\\theta _{2}}}\n$$\n\n**Interpretation**: \n- $\\text{BF}_{12} > 1$: Data favors $\\text{M}_1$ over $\\text{M}_2$\n- $\\text{BF}_{12} < 1$: Data favors $\\text{M}_2$ over $\\text{M}_1$\n- $\\text{BF}_{12} = 1$: Data uninformative for model choice\n"
   },
   {
    "cell_type": "markdown",
    "id": "649cf697",
    "metadata": {},
-   "source": [
-    "## Posterior Odds\n",
-    "\n",
-    "The Bayes factor updates prior beliefs about models:\n",
-    "\n",
-    "$$\n",
-    "\\frac{P(\\text{M}_2|D)}{P(\\text{M}_1|D)} = \\text{BF}_{21} \\times \\frac{P(\\text{M}_2)}{P(\\text{M}_1)}\n",
-    "$$\n",
-    "\n",
-    "This shows how conditioning on data transforms our uncertainty:\n",
-    "\n",
-    "$$\n",
-    "\\text{Posterior Uncertainty} = \\text{Evidence Update} \\times \\text{Prior Uncertainty}\n",
-    "$$"
-   ]
+   "source": "## Posterior Odds\n\nThe Bayes factor updates prior beliefs about models:\n\n$$\n\\frac{P(\\text{M}_1|D)}{P(\\text{M}_2|D)} = \\text{BF}_{12} \\times \\frac{P(\\text{M}_1)}{P(\\text{M}_2)}\n$$\n\nThis shows how conditioning on data transforms our uncertainty:\n\n$$\n\\text{Posterior Uncertainty} = \\text{Evidence Update} \\times \\text{Prior Uncertainty}\n$$"
   },
   {
    "cell_type": "markdown",

Bayesian_normal_mean_model.ipynb

@@ -22,7 +22,9 @@
    "cell_type": "markdown",
    "id": "bd95d8b1",
    "metadata": {},
-   "source": "![Fig](./graphical_summary/slides/Slide27.png)"
+   "source": [
+    "![Fig](./graphical_summary/slides/Slide27.png)"
+   ]
   },
   {
    "cell_type": "markdown",
@@ -115,7 +117,7 @@
    "source": [
     "## Posterior Distribution\n",
     "\n",
-    "Using Bayes' theorem, we update our uncertainty given the observed data:\n",
+    "Using Bayes' rule, we update our uncertainty given the observed data:\n",
     "\n",
     "$$\n",
     "p(\\beta|\\mathbf{Y}) \\propto p(\\mathbf{Y}|\\beta) \\cdot p(\\beta)\n",
@@ -150,7 +152,7 @@
     "where $w = \\frac{T_1/\\sigma^2}{T_1/\\sigma^2 + 1/\\sigma_0^2}$, $\\hat{\\beta}_{\\text{data}} = T_2/T_1$, and $T_1 = \\sum_{i=1}^N X_i^2$, $T_2 = \\sum_{i=1}^N X_i Y_i$.\n",
     "\n",
     "**Interpretation**:\n",
-    "- More data or lower noise → $w \\approx 1$ (trust data)\n",
+    "- More data or lower noise (small $\\sigma^2$) → $w \\approx 1$ (trust data)\n",
     "- Vague prior (large $\\sigma_0^2$) → $w \\approx 1$ (data dominates)\n",
     "- Strong prior (small $\\sigma_0^2$) → $w \\approx 0$ (prior dominates)"
    ]
@@ -619,4 +621,4 @@
  },
  "nbformat": 4,
  "nbformat_minor": 5
-}
+}

figures/Bayesian_hypothesis_testing.png

@@ -7,7 +7,7 @@
    "source": [
     "# $p$-value and Bayesian Hypothesis Testing\n",
     "\n",
-    "In hypothesis testing, frequentists generally use *p*-value to evaluate the evidence against the null hypothesis;*Bayesian hypothesis testing, however, directly computes the probability of any hypothesis given observed data."
+    "In hypothesis testing, frequentists generally use *p*-value to evaluate the evidence against the null hypothesis; Bayesian hypothesis testing, however, directly computes the probability of any hypothesis given observed data."
    ]
   },
   {
@@ -22,7 +22,9 @@
    "cell_type": "markdown",
    "id": "d470175e",
    "metadata": {},
-   "source": "![Fig](./graphical_summary/slides/Slide26.png)"
+   "source": [
+    "![Fig](./graphical_summary/slides/Slide26.png)"
+   ]
   },
   {
    "cell_type": "markdown",
@@ -1248,4 +1250,4 @@
  },
  "nbformat": 4,
  "nbformat_minor": 5
-}
+}