one-doc-one-pos.py

import numpy as np
import matplotlib.pyplot as plt

# Parameters
T = 35  # Number of trials
beta = 0.5  # Probability parameter for z
alpha_1 = 0.2  # Probability parameter for x if z=0
alpha_2 = 0.6  # Probability parameter for x if z=1

# Simulate z
z = np.random.binomial(1, beta)  # z ~ Bernoulli(beta)

# Simulate x based on z
if z == 0:
    x = np.random.binomial(1, alpha_1, T)  # x ~ Bernoulli(alpha_1) for T trials
else:
    x = np.random.binomial(1, alpha_2, T)  # x ~ Bernoulli(alpha_2) for T trials

# Display simulated data
print(f"Sampled value of z: {z}")
print(f"Sampled values of x: {x}")

# Create plots for T trials
t_values = np.arange(1, T + 1)

posterior_values = []
entropy_values = []
variance_values = []

for t in t_values:  # Loop includes 0 to T

    # Calculate posterior measures
    S = np.sum(x[:t])  # Total successes

    P_z0_given_x = (alpha_1 ** S * (1 - alpha_1) ** (t - S) * (1 - beta)) / (
        alpha_1 ** S * (1 - alpha_1) ** (t - S) * (1 - beta) +
        alpha_2 ** S * (1 - alpha_2) ** (t - S) * beta
    )
    P_z1_given_x = 1 - P_z0_given_x

    posterior_values.append(1 - P_z0_given_x)
    variance_values.append(P_z0_given_x * P_z1_given_x)
    entropy_values.append(-P_z0_given_x * np.log(P_z0_given_x + 1e-10) - P_z1_given_x * np.log(P_z1_given_x + 1e-10))



# Posterior probability plot
plt.figure(figsize=(12, 4))
plt.subplot(1, 3, 1)
plt.scatter(t_values, posterior_values, color='blue', label=r'$P(z=1|\mathbf{x})$', alpha=0.6)
plt.title('Posterior Probability $P(z=1|\mathbf{x})$')
plt.xlabel('Trial')
plt.ylabel('Probability')
plt.legend()
plt.grid()

# Posterior variance plot
plt.subplot(1, 3, 2)
plt.scatter(t_values, variance_values, color='green', label=r'Var$(z|\mathbf{x})$', alpha=0.6)
plt.title('Posterior Variance')
plt.xlabel('Trial')
plt.ylabel('Variance')
plt.legend()
plt.grid()

# Posterior entropy plot
plt.subplot(1, 3, 3)
plt.scatter(t_values, entropy_values, color='red', label=r'$H(z|\mathbf{x})$', alpha=0.6)
plt.title('Posterior Entropy $H(z|\mathbf{x})$')
plt.xlabel('Trial')
plt.ylabel('Entropy')
plt.legend()
plt.grid()

plt.tight_layout()
plt.show()