finished HW3

HW3/Q3/Q3.py

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+import matplotlib.pyplot as plt
+import numpy as np
+
+
+# H Y P E R P A R A M E T E R S
+mu_prior = 0      # prior mean
+sigma2_prior = 2   # prior variance (omega_0^2)
+sigma2_likelihood = 1  # likelihood variance (omega^2)
+n_samples = 1000000      # number of Monte Carlo samples
+
+# simulate θ ~ N(mu_0, omega_0^2) and y ~ N(θ, (omega)^2)
+theta = np.random.normal(mu_prior, np.sqrt(sigma2_prior), n_samples)
+y = np.random.normal(theta, np.sqrt(sigma2_likelihood))
+
+# posterior params for each y
+sigma2_posterior = 1 / (1 / sigma2_prior + 1 / sigma2_likelihood)
+mu_posterior = (mu_prior / sigma2_prior + y / sigma2_likelihood) * \
+    sigma2_posterior  # posterior mean
+
+# E[Var[θ|y]]
+expected_posterior_var = sigma2_posterior
+var_posterior_mean = np.var(mu_posterior)  # var[𝔼[θ|y]]
+prior_var = sigma2_prior                   # var[θ]
+
+# verify identity
+sum_terms = expected_posterior_var + var_posterior_mean
+
+print(f"Prior Variance (Var[θ]): {prior_var:.4f}")
+print(
+    f"Expected Posterior Variance (𝔼[Var[θ|y]]): {expected_posterior_var:.4f}")
+print(f"Variance of Posterior Mean (Var[𝔼[θ|y]]): {var_posterior_mean:.4f}")
+print(f"Sum of Terms: {sum_terms:.4f}")
+print(f"Identity Holds: {np.isclose(prior_var, sum_terms, atol=1e-3)}")
+
+# Plot posterior means and variances
+plt.figure(figsize=(10, 6))
+plt.hist(mu_posterior, bins=50, density=True,
+         alpha=0.6, label="Posterior Means")
+plt.axvline(mu_prior, color='r', linestyle='--', label="Prior Mean")
+plt.xlabel("Posterior Mean (𝔼[θ|y])")
+plt.ylabel("Density")
+plt.title("Distribution of Posterior Means vs. Prior")
+plt.legend()
+plt.grid(True)
+# plt.show()
+plt.savefig('part3.png')