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| import numpy as np | |
| import jax.numpy as jnp | |
| import matplotlib.pyplot as plt | |
| import numpyro | |
| import numpyro.distributions as dist | |
| from numpyro.infer import MCMC, NUTS | |
| from sklearn.datasets import make_regression | |
| from jax import random | |
| import streamlit as st | |
| # Define the model | |
| def linear_regression(X, y, alpha_prior, beta_prior, sigma_prior): | |
| alpha = numpyro.sample('alpha', alpha_prior) | |
| beta = numpyro.sample('beta', beta_prior) | |
| sigma = numpyro.sample('sigma', sigma_prior) | |
| mean = alpha + beta * X | |
| numpyro.sample('obs', dist.Normal(mean, sigma), obs=y) | |
| def run_linear_regression(X, y, alpha_prior, beta_prior, sigma_prior): | |
| # Run MCMC | |
| rng_key = random.PRNGKey(0) | |
| nuts_kernel = NUTS(linear_regression) | |
| mcmc = MCMC(nuts_kernel, num_warmup=50, num_samples=1000) | |
| mcmc.run(rng_key, jnp.array(X), jnp.array(y), alpha_prior=alpha_prior, beta_prior=beta_prior, sigma_prior=sigma_prior) | |
| mcmc.print_summary() | |
| # Get posterior samples | |
| samples = mcmc.get_samples() | |
| # Plot the results | |
| fig, ax = plt.subplots(1, 3, figsize=(12, 4)) | |
| ax[0].hist(samples['alpha'], bins=20, density=True) | |
| ax[0].set_title('alpha') | |
| ax[1].hist(samples['beta'], bins=20, density=True) | |
| ax[1].set_title('beta') | |
| ax[2].hist(samples['sigma'], bins=20, density=True) | |
| ax[2].set_title('sigma') | |
| st.write("The plot of posterior samples is shown below:") | |
| st.pyplot(fig) | |
| st.write("The mean of alpha is", np.mean(samples['alpha'])) | |
| st.write("The mean of beta is", np.mean(samples['beta'])) | |
| st.write("The mean of sigma is", np.mean(samples['sigma'])) | |
| st.write("The plot of predicted line is shown below using mean values of alpha and beta (Xβ+α):") | |
| fig, ax = plt.subplots(figsize=(8, 6)) | |
| ax.scatter(X, y, color='blue', alpha=0.5, label='data') | |
| light_color = (1.0, 0.5, 0.5, 0.7) | |
| for i in range(499): | |
| alpha_i = samples['alpha'][i] | |
| beta_i = samples['beta'][i] | |
| ax.plot(X, alpha_i + beta_i * X, color=light_color) | |
| alpha_i = samples['alpha'][498] | |
| beta_i = samples['beta'][498] | |
| ax.plot(X, alpha_i + beta_i * X, color=light_color,label='MCMC samples') | |
| ax.plot(X, np.mean(samples['alpha']) + np.mean(samples['beta']) * X, color='red', label='mean') | |
| ax.legend(loc='upper left') | |
| st.pyplot(fig) | |
| # User Input | |
| st.write("# Bayesian Linear Regression") | |
| """ Prior: p(α) = N(μ0,Σ0), p(β) = N(μ1,Σ1), p(σ) = N(Σ2)""" | |
| """ Likelihood: p(y|X,β,α,σ) = N(Xβ+α,σ)""" | |
| """ Posterior: p(α|X,y) ∝ p(y|X,β,σ)p(α), p(β|X,y) ∝ p(y|X,α,σ)p(β), p(σ|X,y) ∝ p(y|X,β,α)p(σ)""" | |
| alpha_prior_option = st.selectbox("Choose an option for alpha prior:", ["Normal", "Laplace", "Cauchy"]) | |
| if alpha_prior_option == "Normal": | |
| alpha_loc = st.slider("Select a mean value for prior of alpha(α) (μ0)", -10.0, 10.0, 0.0, 0.1) | |
| alpha_scale = st.slider("Select a standard deviation value for prior of alpha(α) (Σ0)", 0.01, 10.0, 1.0, 0.1) | |
| alpha_prior = dist.Normal(alpha_loc, alpha_scale) | |
| elif alpha_prior_option == "Laplace": | |
| alpha_loc = st.slider("Select a mean value for prior of alpha(α) (μ0)", -10.0, 10.0, 0.0, 0.1) | |
| alpha_scale = st.slider("Select a standard deviation value for prior of alpha(α) (Σ0)", 0.01, 10.0, 1.0, 0.1) | |
| alpha_prior = dist.Laplace(alpha_loc, alpha_scale) | |
| elif alpha_prior_option == "Cauchy": | |
| alpha_loc = st.slider("Select a mean value for prior of alpha(α) (μ0)", -10.0, 10.0, 0.0, 0.1) | |
| alpha_scale = st.slider("Select a standard deviation value for prior of alpha(α) (Σ0)", 0.01, 10.0, 1.0, 0.1) | |
| alpha_prior = dist.Cauchy(alpha_loc, alpha_scale) | |
| beta_prior_option = st.selectbox("Choose an option for beta prior:", ["Normal", "Laplace", "Cauchy"]) | |
| if beta_prior_option == "Normal": | |
| beta_loc = st.slider("Select a mean value for prior of beta(β) (μ1)", -10.0, 10.0, 0.0, 0.1) | |
| beta_scale = st.slider("Select a standard deviation value for prior of beta(β) (Σ1)", 0.01, 10.0, 1.0, 0.1) | |
| beta_prior = dist.Normal(beta_loc, beta_scale) | |
| elif beta_prior_option == "Laplace": | |
| beta_loc = st.slider("Select a mean value for prior of beta(β) (μ1)", -10.0, 10.0, 0.0, 0.1) | |
| beta_scale = st.slider("Select a standard deviation value for prior of beta(β) (Σ1)", 0.01, 10.0, 1.0, 0.1) | |
| beta_prior = dist.Laplace(beta_loc, beta_scale) | |
| elif beta_prior_option == "Cauchy": | |
| beta_loc = st.slider("Select a mean value for prior of beta(β) (μ1)", -10.0, 10.0, 0.0, 0.1) | |
| beta_scale = st.slider("Select a standard deviation value for prior of beta(β) (Σ1)", 0.01, 10.0, 1.0, 0.1) | |
| beta_prior = dist.Cauchy(beta_loc, beta_scale) | |
| sigma_prior_option = st.selectbox("Choose an option for sigma prior:", ["HalfNormal", "HalfCauchy"]) | |
| if sigma_prior_option == "HalfNormal": | |
| sigma_scale = st.slider("Select a scale value for prior of sigma(σ) (Σ2)", 0.01, 10.0, 1.0, 0.1) | |
| sigma_prior = dist.HalfNormal(sigma_scale) | |
| elif sigma_prior_option == "HalfCauchy": | |
| sigma_scale = st.slider("Select a scale value for prior of sigma(σ) (Σ2)", 0.01, 10.0, 1.0, 0.1) | |
| sigma_prior = dist.HalfCauchy(sigma_scale) | |
| rng_key = random.PRNGKey(0) | |
| X, y = make_regression(n_samples=50, n_features=1, noise=10.0, random_state=0) | |
| X = X.reshape(50) | |
| if alpha_prior and beta_prior and sigma_prior: | |
| run_linear_regression(X, y, alpha_prior, beta_prior, sigma_prior) |