scripts for coins and stars

coins-max-entropy.py

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+## Finds the most likely p, by maximising the entropy given k heads out of n tosses subject to that constraint of k being heads out of n tosses
+
+from scipy.optimize import minimize
+from scipy.special import comb
+import numpy as np
+
+# Define Shannon entropy for Binomial distribution
+def shannon_entropy(p, n):
+    p = p[0]  # Extract p from the array
+    terms = [comb(n, k) * (p ** k) * ((1 - p) ** (n - k)) for k in range(n + 1)]
+    entropy = -np.sum(term * np.log2(term) for term in terms if term > 0)  # Exclude zero terms
+    return -entropy  # Minimize the negative entropy to maximize entropy
+
+# Define constraint for the mean (x/n)
+def constraint_mean(p, n, x):
+    return n * p[0] - x
+
+# Number of trials (n) and observed successes (x)
+n = 10
+x = 3
+
+# Initial guess for p
+initial_p = [0.5]
+
+# Constraint for mean
+con = {'type': 'eq', 'fun': constraint_mean, 'args': (n, x)}
+
+# Perform optimization to maximize entropy
+result = minimize(shannon_entropy, initial_p, args=(n,), constraints=[con], bounds=[(0, 1)])
+
+# Extract optimized p value
+optimized_p = result.x[0]
+
+print(f"Optimized p to maximize entropy: {optimized_p}")

stars-max-entropy.py

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+import matplotlib.pyplot as plt
+from scipy.optimize import minimize
+import numpy as np
+
+# Define Shannon entropy for Multinomial distribution with a fixed number of samples
+def shannon_entropy_multinomial_fixed_samples(p, n):
+    entropy = -np.sum(p_i * np.log2(p_i) for p_i in p if p_i > 0)  # Exclude zero terms
+    return -n * entropy  # Account for the number of samples
+
+# Define constraint for the average rating with a fixed number of samples
+def constraint_average_fixed_samples(p, avg, n):
+    weighted_sum = np.sum(i * p_i for i, p_i in enumerate(p, start=1))
+    return n * weighted_sum - n * avg  # Account for the number of samples
+
+# Function to perform optimization for given number of reviews and average rating
+def optimize_probs(num_reviews, avg_review):
+    # Initial guess for p
+    initial_p = [0.2, 0.2, 0.2, 0.2, 0.2]
+
+    # Constraint for average review
+    con_avg = {'type': 'eq', 'fun': constraint_average_fixed_samples, 'args': (avg_review, num_reviews)}
+
+    # Additional constraint for probabilities to sum to 1
+    con_sum = {'type': 'eq', 'fun': lambda p: np.sum(p) - 1}
+
+    # Perform optimization to maximize entropy
+    result = minimize(shannon_entropy_multinomial_fixed_samples, initial_p, args=(num_reviews,), constraints=[con_avg, con_sum], bounds=[(0, 1)]*5)
+
+    # Extract optimized p values
+    return result.x
+
+# Known average review
+avg_review = 4.0
+
+# Different number of reviews
+num_reviews_list = [4, 8, 16, float('inf')]
+
+# Store optimized probabilities
+optimized_probs = {}
+
+# Perform optimization for each number of reviews
+for num_reviews in num_reviews_list:
+    optimized_p = optimize_probs(num_reviews, avg_review) if num_reviews != float('inf') else optimize_probs(10000, avg_review)  # Use a large number to approximate infinity
+    optimized_probs[num_reviews] = optimized_p
+
+# Adjusted plot to show a dot plot for each case
+fig, ax = plt.subplots()
+
+# Different markers for better visibility
+markers = ['o', 's', '^', 'x']
+
+for i, (num_reviews, optimized_p) in enumerate(optimized_probs.items()):
+    ax.plot(range(1, 6), optimized_p, marker=markers[i], linestyle='-', label=f'{num_reviews} Reviews')
+
+# Labels and title
+ax.set_xlabel('Star Rating')
+ax.set_ylabel('Probability')
+ax.set_title('Optimized Probabilities for Different Numbers of Reviews')
+ax.legend()
+
+plt.show()