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Merge pull request #22 from RuiDongDR/main
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‎Bayesian_mixture_model.ipynb‎

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"cell_type": "markdown",
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"id": "0fea5f8a",
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"metadata": {},
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"source": "![Fig](./graphical_summary/slides/Slide30.png)"
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"source": [
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"![Fig](./graphical_summary/slides/Slide30.png)"
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]
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"cell_type": "markdown",
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"In reality, genetic variants follow different biological mechanisms that we cannot condition on:\n",
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"\n",
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"- Most variants have null effects on both traits\n",
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"- Some affect only Height (growth pathways)\n",
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"- Others affect only Weight (metabolism) \n",
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"- Some affect only height (growth pathways)\n",
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"- Others affect only weight (metabolism) \n",
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"- A few affect both traits with varying correlations\n",
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"\n",
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"Since we cannot condition on which mechanism generated each variant, we observe only a heterogeneous scatter of effects with no clear pattern -- a mixture of all models!\n",
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"We generate data for multiple genetic variants from the same list of models as in [Lecture: Bayesian model comparison](https://statfungen.github.io/statgen-primer/Bayesian_model_comparison.html):\n",
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"\n",
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"- $\\text{M}_0$ (No effect): $\\boldsymbol{\\beta} = (0, 0)$\n",
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"- $\\text{M}_1$ (Height Only): $\\beta_2 = 0$\n",
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"- $\\text{M}_2$ (Weight Only): $\\beta_1 = 0$\n",
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"- $\\text{M}_1$ (Height Only): $\\beta_1 \\neq 0, \\beta_2 = 0$\n",
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"- $\\text{M}_2$ (Weight Only): $\\beta_1 = 0, \\beta_2 \\neq 0$\n",
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"- $\\text{M}_3$ (Perfect Correlation): $\\text{cor}(\\beta_1, \\beta_2) = 1$\n",
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"- $\\text{M}_4$ (Weak Correlation): $\\text{cor}(\\beta_1, \\beta_2) = 0.1$\n",
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"- $\\text{M}_5$ (Medium Correlation): $\\text{cor}(\\beta_1, \\beta_2) = 0.5$\n",
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},
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"nbformat": 4,
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‎Bayesian_model_comparison.ipynb‎

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"cell_type": "markdown",
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"id": "6988ab81",
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"metadata": {},
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"source": "![Fig](./graphical_summary/slides/Slide29.png)"
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"source": [
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"![Fig](./graphical_summary/slides/Slide29.png)"
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]
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"cell_type": "markdown",
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"cell_type": "markdown",
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"id": "f5aa6a8c",
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"metadata": {},
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"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"
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"source": [
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"# Technical Details\n",
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"\n",
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"## Bayes Factor\n",
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"\n",
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"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",
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"\n",
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"$$\n",
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"\\text{BF}_{12} = \\frac{L(\\text{M}_1|\\text{D})}{L(\\text{M}_2|\\text{D})} ={\\frac {\\int \\Pr(\\theta_{1}|M_{1})\\Pr(D|\\theta_{1},M_{1})\\,d\\theta _{1}}{\\int \\Pr(\\theta_{2}|M_{2})\\Pr(D|\\theta_{2},M_{2})\\,d\\theta _{2}}}\n",
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"$$\n",
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"\n",
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"**Interpretation**: \n",
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"- $\\text{BF}_{12} > 1$: Data favors $\\text{M}_1$ over $\\text{M}_2$\n",
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"- $\\text{BF}_{12} < 1$: Data favors $\\text{M}_2$ over $\\text{M}_1$\n",
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"- $\\text{BF}_{12} = 1$: Data uninformative for model choice\n"
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]
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"cell_type": "markdown",
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"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$$\n\\text{Posterior Uncertainty} = \\text{Evidence Update} \\times \\text{Prior Uncertainty}\n$$"
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"source": [
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"## Posterior Odds\n",
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"\n",
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"The Bayes factor updates prior beliefs about models:\n",
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"\n",
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"$$\n",
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"\\frac{P(\\text{M}_1|D)}{P(\\text{M}_2|D)} = \\text{BF}_{12} \\times \\frac{P(\\text{M}_1)}{P(\\text{M}_2)}\n",
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"$$\n",
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"\n",
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"This shows how conditioning on data transforms our uncertainty:\n",
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"\n",
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"$$\n",
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"\\text{Posterior Uncertainty} = \\text{Evidence Update} \\times \\text{Prior Uncertainty}\n",
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"$$"
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]
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"In practice, this variant could affect height and weight $\\boldsymbol{\\beta} = (\\beta_1, \\beta_2)$ in multiple ways:\n",
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"\n",
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"- $\\text{M}_0$ (No effect): $\\boldsymbol{\\beta} = (0, 0)$\n",
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"- $\\text{M}_1$ (Height Only): $\\beta_2 = 0$\n",
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"- $\\text{M}_2$ (Weight Only): $\\beta_1 = 0$\n",
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"- $\\text{M}_1$ (Height Only): $\\beta_1 \\neq 0, \\beta_2 = 0$\n",
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"- $\\text{M}_2$ (Weight Only): $\\beta_1 = 0, \\beta_2 \\neq 0$\n",
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"- $\\text{M}_3$ (Perfect Correlation): $\\text{cor}(\\beta_1, \\beta_2) = 1$\n",
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"- $\\text{M}_4$ (Weak Correlation): $\\text{cor}(\\beta_1, \\beta_2) = 0.1$\n",
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"- $\\text{M}_5$ (Medium Correlation): $\\text{cor}(\\beta_1, \\beta_2) = 0.5$\n",
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"nbformat": 4,
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‎Bayesian_multivariate_normal_mean_model.ipynb‎

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"cell_type": "markdown",
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"source": "![Fig](./graphical_summary/slides/Slide28.png)"
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"source": [
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"![Fig](./graphical_summary/slides/Slide28.png)"
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]
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"### Posterior as Weighted Combination\n",
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"$$\n",
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"\\boldsymbol{\\beta}_1 = \\boldsymbol{W} \\cdot \\frac{\\mathbf{T}_2}{T_1} + (\\boldsymbol{I} - \\boldsymbol{W}) \\cdot \\boldsymbol{\\beta}_0\n",
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"\\boldsymbol{\\beta}_1 = \\boldsymbol{W} \\cdot \\frac{\\mathbf{T}_2}{\\mathbf{T}_1} + (\\boldsymbol{I} - \\boldsymbol{W}) \\cdot \\boldsymbol{\\beta}_0\n",
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"$$\n",
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"\n",
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"where $\\boldsymbol{W}$ balances prior beliefs and observed data based on their relative information content.\n"
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}

‎graphical_summary/slides.pptx‎

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‎graphical_summary/slides/Slide33.png‎

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‎hidden_Markov_model.ipynb‎

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"source": "![Fig](./graphical_summary/slides/Slide33.png)"
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"source": [
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"![Fig](./graphical_summary/slides/Slide33.png)"
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]
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"cell_type": "markdown",
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"metadata": {},
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"** Emission Probability** $P(h_m | Y_m = k)$\n",
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"**Emission Probability** $P(h_m | Y_m = k)$\n",
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"Given ancestral haplotype $k$, the observed allele follows a Bernoulli distribution:\n",
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‎p_value_and_Bayesian_hypothesis_testing.ipynb‎

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