Update app.py

app.py

@@ -7,7 +7,6 @@ from numpyro.infer import MCMC, NUTS
 from sklearn.datasets import make_regression
 from jax import random
 import streamlit as st
-import sympy as sp
 
 
 # Define the model
@@ -32,6 +31,21 @@ def run_linear_regression(X, y, alpha_prior, beta_prior, sigma_prior):
     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.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)
@@ -49,48 +63,47 @@ def run_linear_regression(X, y, alpha_prior, beta_prior, sigma_prior):
 # User Input
 st.write("# Bayesian Linear Regression")
 
-st.latex(r'''\begin{aligned}
-\textbf{Prior:} \hspace{0.5 cm} p(\alpha) &= N(\mu_0, \Sigma_0),\hspace{0.2 cm} p(\beta) = N(\mu_1, \Sigma_1),\hspace{0.2 cm} p(\sigma) = N(\mu_2, \Sigma_2)\\
-\textbf{Likelihood:} \hspace{0.5 cm} p(y | X, \beta,\alpha, \sigma) &= N(X\beta + \alpha, \sigma)\\
-\textbf{Posterior:} \hspace{0.5 cm} p(\alpha | X, y) &\propto p(y | X, \beta, \sigma) p(\alpha),\hspace{0.2 cm} p(\beta | X, y) \propto p(y | X, \alpha, \sigma) p(\beta),\hspace{0.2 cm} p(\sigma | X, y) \propto p(y | X, \beta, \alpha) p(\sigma)
-\end{aligned}''')
+
+""" 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 alpha", -10.0, 10.0, 0.0, 0.1)
-    alpha_scale = st.slider("Select a standard deviation value for alpha", 0.01, 10.0, 1.0, 0.1)
+    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 alpha", -10.0, 10.0, 0.0, 0.1)
-    alpha_scale = st.slider("Select a scale value for alpha", 0.01, 10.0, 1.0, 0.1)
+    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 location value for alpha", -10.0, 10.0, 0.0, 0.1)
-    alpha_scale = st.slider("Select a scale value for alpha", 0.01, 10.0, 1.0, 0.1)
+    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 beta", -10.0, 10.0, 0.0, 0.1)
-    beta_scale = st.slider("Select a standard deviation value for beta", 0.01, 10.0, 1.0, 0.1)
+    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 beta", -10.0, 10.0, 0.0, 0.1)
-    beta_scale = st.slider("Select a scale value for beta", 0.01, 10.0, 1.0, 0.1)
+    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 location value for beta", -10.0, 10.0, 0.0, 0.1)
-    beta_scale = st.slider("Select a scale value for beta", 0.01, 10.0, 1.0, 0.1)
+    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 sigma", 0.01, 10.0, 1.0, 0.1)
+    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 sigma", 0.01, 10.0, 1.0, 0.1)
+    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)