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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Bayesian Binomial Regression "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "This notebook is an introductory tutorial to Bayesian binomial regression with `RxInfer`."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "using RxInfer, ReactiveMP, Random, Plots, StableRNGs, LinearAlgebra, StatsPlots, LaTeXStrings"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Likelihood Specification\n",
+ "\n",
+ "For observations $y_i$ with predictors $\\mathbf{x}_i$, Binomial regression models the number of successes $y_i$ as a function of the predictors $\\mathbf{x}_i$ and the regression coefficients $\\boldsymbol{\\beta}$\n",
+ "\n",
+ "$$\\begin{equation}\n",
+ "y_i \\sim \\text{Binomial}(n_i, p_i)\\,,\n",
+ "\\end{equation}$$\n",
+ "\n",
+ "where:\n",
+ "\n",
+ "$y_i$ is the number of successes, $n_i$ is the number of trials, $p_i$ is the probability of success. The probability $p_i$ is linked to the predictors through the logistic function:\n",
+ "\n",
+ "$$\\begin{equation}\n",
+ "p_i = \\frac{1}{1 + e^{-\\mathbf{x}_i^T\\boldsymbol{\\beta}}}\n",
+ "\\end{equation}$$\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Prior Distributions\n",
+ "We specify priors for the regression coefficients:\n",
+ "\n",
+ "$$\\begin{equation}\n",
+ "\\boldsymbol{\\beta} \\sim \\mathcal{N}_{\\xi}(\\boldsymbol{\\xi}, \\boldsymbol{\\Lambda})\n",
+ "\\end{equation}$$\n",
+ "\n",
+ "as a Normal distribution in precision-weighted mean form.\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Model Specification\n",
+ "\n",
+ "The likelihood and the prior distributions form the probabilistic model\n",
+ "\n",
+ "$$p(y, x, \\beta, n) = p(\\beta) \\prod_{i=1}^N p(y_i \\mid x_i, \\beta, n_i),$$\n",
+ "\n",
+ "where the goal is to infer the posterior distributions $p(\\beta \\mid y, x, n)$. Due to logistic link function, the posterior distribution is not conjugate to the prior distribution. This means that we need to use a more complex inference algorithm to infer the posterior distribution. Before dwelling into the details of the inference algorithm, let's first generate some synthetic data to work with."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function generate_synthetic_binomial_data(\n",
+ " n_samples::Int,\n",
+ " true_beta::Vector{Float64};\n",
+ " seed::Int=42\n",
+ ")\n",
+ " n_features = length(true_beta)\n",
+ " rng = StableRNG(seed)\n",
+ " \n",
+ " X = randn(rng, n_samples, n_features)\n",
+ " \n",
+ " n_trials = rand(rng, 5:20, n_samples)\n",
+ " \n",
+ " logits = X * true_beta\n",
+ " probs = 1 ./ (1 .+ exp.(-logits))\n",
+ " \n",
+ " y = [rand(rng, Binomial(n_trials[i], probs[i])) for i in 1:n_samples]\n",
+ " \n",
+ " return X, y, n_trials\n",
+ "end\n",
+ "\n",
+ "\n",
+ "n_samples = 10000\n",
+ "true_beta = [-3.0 , 2.6]\n",
+ "\n",
+ "X, y, n_trials = generate_synthetic_binomial_data(n_samples, true_beta);\n",
+ "X = [collect(row) for row in eachrow(X)];"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We generate `X` as the design matrix and `y` as the number of successes and `n_trials` as the number of trials. Next task is to define the graphical model. RxInfer provides a `BinomialPolya` factor node that is a combination of a Binomial distribution and a PolyaGamma distribution introduced in [1]. The `BinomialPolya` factor node is used to model the likelihood of the binomial distribution. \n",
+ "\n",
+ "Due to non-conjugacy of the likelihood and the prior distribution, we need to use a more complex inference algorithm. RxInfer provides an Expectation Propagation (EP) [2] algorithm to infer the posterior distribution. Due to EP's approximation, we need to specify an inbound message for the regression coefficients while using the `BinomialPolya` factor node. This feature is implemented in the `dependencies` keyword argument during the creation of the `BinomialPolya` factor node. `ReactiveMP.jl` provides a `RequireMessageFunctionalDependencies` type that is used to specify the inbound message for the regression coefficients `β`. Refer to the ReactiveMP.jl documentation for more information."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "@model function binomial_model(prior_xi, prior_precision, n_trials, X, y) \n",
+ " β ~ MvNormalWeightedMeanPrecision(prior_xi, prior_precision)\n",
+ " for i in eachindex(y)\n",
+ " y[i] ~ BinomialPolya(X[i], n_trials[i], β) where {\n",
+ " dependencies = RequireMessageFunctionalDependencies(β = MvNormalWeightedMeanPrecision(prior_xi, prior_precision))\n",
+ " }\n",
+ " end\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ " This example uses the precision-weighted mean parametrization (`MvNormalWeightedMeanPrecision`) of the Gaussian distribution for efficiency reasons. While this is less conventional than the standard mean-covariance form, the example would work equally well with any parametrization. The choice of parametrization mainly affects computational efficiency and numerical stability, not the underlying model or results."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Havin specified the model, we can now utilize the `infer` function to infer the posterior distribution."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stderr",
+ "output_type": "stream",
+ "text": [
+ "\u001b[32mProgress: 100%|█████████████████████████████████████████| Time: 0:00:03\u001b[39m\u001b[K\n"
+ ]
+ },
+ {
+ "data": {
+ "text/plain": [
+ "Inference results:\n",
+ " Posteriors | available for (β)\n",
+ " Free Energy: | Real[21992.9, 16235.8, 13785.0, 12519.7, 11800.1, 11366.2, 11094.1, 10918.7, 10803.3, 10726.3 … 10558.2, 10558.2, 10558.2, 10558.1, 10558.1, 10558.1, 10558.1, 10558.1, 10558.1, 10558.1]\n"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "n_features = length(true_beta)\n",
+ "results = infer(\n",
+ " model = binomial_model(prior_xi = zeros(n_features), prior_precision = diageye(n_features),),\n",
+ " data = (X=X, y=y,n_trials=n_trials),\n",
+ " iterations = 50,\n",
+ " free_energy = true,\n",
+ " showprogress = true,\n",
+ " options = (\n",
+ " limit_stack_depth = 100, # to prevent stack-overflow errors\n",
+ " )\n",
+ ")\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We can now plot the free energy to see if the inference algorithm is converging."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "plot(results.free_energy)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Free energy is converging to a stable value, indicating that the inference algorithm is converging. Let's visualize the posterior distribution and how it compares to the true parameters."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "image/svg+xml": [
+ "\n",
+ "\n"
+ ],
+ "text/html": [
+ "\n",
+ "\n"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "m = mean(results.posteriors[:β][end])\n",
+ "Σ = cov(results.posteriors[:β][end])\n",
+ "\n",
+ "p = plot(xlims=(-3.1,-2.9), ylims=(2.5,2.7))\n",
+ "covellipse!(m, Σ, n_std=1, aspect_ratio=1, label=\"1σ Contours\", color=:green, fillalpha=0.2)\n",
+ "covellipse!(m, Σ, n_std=3, aspect_ratio=1, label=\"3σ Contours\", color=:blue, fillalpha=0.2)\n",
+ "scatter!([m[1]], [m[2]], label=\"Mean estimate\", color=:blue)\n",
+ "scatter!([true_beta[1]], [true_beta[2]], label=\"True Parameters\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# References\n",
+ "[1] Polson, N. G., Scott, J. G., & Windle, J. (2013). Bayesian inference for logistic models using Polya-Gamma latent variables. *Journal of the American Statistical Association*, 108(1), 136-146.\n",
+ "\n",
+ "[2] Minka, T. (2001). Expectation Propagation for approximate Bayesian inference. *Uncertainty in Artificial Intelligence*, 2, 362-369."
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 1.11.2",
+ "language": "julia",
+ "name": "julia-1.11"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.11.2"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/examples/Basic Examples/Bayesian Binomial Regression/Project.toml b/examples/Basic Examples/Bayesian Binomial Regression/Project.toml
new file mode 100644
index 0000000..0416ec2
--- /dev/null
+++ b/examples/Basic Examples/Bayesian Binomial Regression/Project.toml
@@ -0,0 +1,9 @@
+[deps]
+LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
+Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
+ReactiveMP = "a194aa59-28ba-4574-a09c-4a745416d6e3"
+RxInfer = "86711068-29c9-4ff7-b620-ae75d7495b3d"
+StableRNGs = "860ef19b-820b-49d6-a774-d7a799459cd3"
+StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
diff --git a/examples/Basic Examples/Bayesian Binomial Regression/meta.jl b/examples/Basic Examples/Bayesian Binomial Regression/meta.jl
new file mode 100644
index 0000000..50eb895
--- /dev/null
+++ b/examples/Basic Examples/Bayesian Binomial Regression/meta.jl
@@ -0,0 +1,15 @@
+meta = (
+ title = "Bayesian Binomial Regression",
+ description = """
+ An introductory tutorial to Bayesian binomial regression with RxInfer.
+ Learn how to model binary outcomes using logistic regression with proper Bayesian inference.
+ The example demonstrates the use of Expectation Propagation (EP) algorithm and Polya-Gamma augmentation.
+ """,
+ tags = [
+ "basic examples",
+ "regression",
+ "multivariate",
+ "expectation propagation",
+ "polya-gamma",
+ ]
+)
diff --git a/examples/Basic Examples/Bayesian Linear Regression Tutorial/Bayesian Linear Regression Tutorial.ipynb b/examples/Basic Examples/Bayesian Linear Regression/Bayesian Linear Regression.ipynb
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diff --git a/examples/Basic Examples/Bayesian Linear Regression Tutorial/Project.toml b/examples/Basic Examples/Bayesian Linear Regression/Project.toml
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rename from examples/Basic Examples/Bayesian Linear Regression Tutorial/Project.toml
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diff --git a/examples/Basic Examples/Bayesian Linear Regression Tutorial/hbr/osic_pulmonary_fibrosis.csv b/examples/Basic Examples/Bayesian Linear Regression/hbr/osic_pulmonary_fibrosis.csv
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rename from examples/Basic Examples/Bayesian Linear Regression Tutorial/hbr/osic_pulmonary_fibrosis.csv
rename to examples/Basic Examples/Bayesian Linear Regression/hbr/osic_pulmonary_fibrosis.csv
diff --git a/examples/Basic Examples/Bayesian Linear Regression Tutorial/meta.jl b/examples/Basic Examples/Bayesian Linear Regression/meta.jl
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--- a/examples/Basic Examples/Bayesian Linear Regression Tutorial/meta.jl
+++ b/examples/Basic Examples/Bayesian Linear Regression/meta.jl
@@ -1,5 +1,5 @@
return (
- title = "Bayesian Linear Regression Tutorial",
+ title = "Bayesian Linear Regression",
description = """
An extensive tutorial on Bayesian linear regression with RxInfer with a lot of examples, including multivariate and hierarchical linear regression.
""",