全確率の定理とベイズの定理の関係をわかりやすく解説

import numpy as np
import matplotlib.pyplot as plt

# 工場のデータ
factories = ['Factory A', 'Factory B', 'Factory C']
production_share = [0.50, 0.30, 0.20]
defect_rate = [0.02, 0.03, 0.05]
colors = ['steelblue', 'coral', 'green']

# 全確率の定理による不良品率
contributions = [p * d for p, d in zip(production_share, defect_rate)]
total_defect = sum(contributions)

# ベイズの定理による事後確率
posterior = [c / total_defect for c in contributions]

fig, axes = plt.subplots(1, 3, figsize=(16, 5))

# (a) 全確率の定理の分解
ax = axes[0]
bottom = 0
for i, (factory, contrib) in enumerate(zip(factories, contributions)):
    bar = ax.bar('P(Defect)', contrib, bottom=bottom, color=colors[i],
                 alpha=0.8, label=f'{factory}: {contrib:.3f}',
                 edgecolor='gray')
    ax.text(0, bottom + contrib/2, f'{factory}\n{contrib:.3f}',
            ha='center', va='center', fontsize=10, fontweight='bold')
    bottom += contrib

ax.axhline(total_defect, color='black', linestyle='--', linewidth=1.5)
ax.text(0.35, total_defect + 0.001, f'Total = {total_defect:.3f}',
        fontsize=12, fontweight='bold')
ax.set_ylabel('Probability', fontsize=12)
ax.set_title('Law of Total Probability\n'
             r'$P(D) = \sum_i P(D|B_i)P(B_i)$', fontsize=13)
ax.legend(fontsize=9, loc='upper right')
ax.set_ylim(0, 0.04)
ax.grid(True, alpha=0.3, axis='y')

# (b) 事前確率 vs 事後確率
ax = axes[1]
x_pos = np.arange(3)
width = 0.35

bars1 = ax.bar(x_pos - width/2, production_share, width, color='lightblue',
               alpha=0.8, edgecolor='gray', label='Prior P(Factory)')
bars2 = ax.bar(x_pos + width/2, posterior, width, color='salmon',
               alpha=0.8, edgecolor='gray', label='Posterior P(Factory|Defect)')

ax.set_xticks(x_pos)
ax.set_xticklabels(factories, fontsize=10)
ax.set_ylabel('Probability', fontsize=12)
ax.set_title("Prior vs Posterior\n(Bayes' Theorem)", fontsize=13)
ax.legend(fontsize=10)
ax.grid(True, alpha=0.3, axis='y')

for bar in bars1:
    ax.text(bar.get_x() + bar.get_width()/2, bar.get_height() + 0.01,
            f'{bar.get_height():.2f}', ha='center', fontsize=10)
for bar in bars2:
    ax.text(bar.get_x() + bar.get_width()/2, bar.get_height() + 0.01,
            f'{bar.get_height():.3f}', ha='center', fontsize=10)

# (c) 不良品率と生産比率のトレードオフ
ax = axes[2]
# 不良品率を変化させたときの全体不良品率への寄与
defect_range = np.linspace(0.001, 0.10, 100)
for i, factory in enumerate(factories):
    contribution_curve = production_share[i] * defect_range
    ax.plot(defect_range * 100, contribution_curve * 100, linewidth=2,
            color=colors[i], label=f'{factory} (share={production_share[i]:.0%})')

ax.set_xlabel('Defect Rate (%)', fontsize=12)
ax.set_ylabel('Contribution to Total Defect Rate (%)', fontsize=12)
ax.set_title('Contribution = Defect Rate × Production Share', fontsize=13)
ax.legend(fontsize=10)
ax.grid(True, alpha=0.3)

# 実際の値をマーク
for i in range(3):
    ax.plot(defect_rate[i]*100, contributions[i]*100, 'o',
            color=colors[i], markersize=10, markeredgecolor='black',
            markeredgewidth=1.5)

plt.tight_layout()
plt.savefig('total_probability_theorem.png', dpi=150, bbox_inches='tight')
plt.show()

import numpy as np
import matplotlib.pyplot as plt

# チャネルパラメータ
p_send_0 = 0.6
p_send_1 = 0.4
p_correct_0 = 0.95  # P(R=0|S=0)
p_correct_1 = 0.90  # P(R=1|S=1)

# 遷移確率
p_r0_s0 = p_correct_0      # 0→0
p_r1_s0 = 1 - p_correct_0  # 0→1
p_r0_s1 = 1 - p_correct_1  # 1→0
p_r1_s1 = p_correct_1      # 1→1

fig, axes = plt.subplots(1, 3, figsize=(16, 5))

# (a) チャネル遷移図
ax = axes[0]
# 送信側
ax.text(0.1, 0.75, 'S=0', fontsize=16, ha='center', va='center',
        bbox=dict(boxstyle='round,pad=0.4', facecolor='lightblue'))
ax.text(0.1, 0.25, 'S=1', fontsize=16, ha='center', va='center',
        bbox=dict(boxstyle='round,pad=0.4', facecolor='lightsalmon'))

# 受信側
ax.text(0.9, 0.75, 'R=0', fontsize=16, ha='center', va='center',
        bbox=dict(boxstyle='round,pad=0.4', facecolor='lightblue'))
ax.text(0.9, 0.25, 'R=1', fontsize=16, ha='center', va='center',
        bbox=dict(boxstyle='round,pad=0.4', facecolor='lightsalmon'))

# 遷移の矢印
ax.annotate('', xy=(0.78, 0.75), xytext=(0.22, 0.75),
            arrowprops=dict(arrowstyle='->', color='blue', lw=2))
ax.text(0.5, 0.80, f'{p_r0_s0:.2f}', fontsize=13, ha='center',
        color='blue', fontweight='bold')

ax.annotate('', xy=(0.78, 0.25), xytext=(0.22, 0.75),
            arrowprops=dict(arrowstyle='->', color='red', lw=1.5, ls='--'))
ax.text(0.35, 0.45, f'{p_r1_s0:.2f}', fontsize=12, ha='center',
        color='red')

ax.annotate('', xy=(0.78, 0.75), xytext=(0.22, 0.25),
            arrowprops=dict(arrowstyle='->', color='red', lw=1.5, ls='--'))
ax.text(0.65, 0.55, f'{p_r0_s1:.2f}', fontsize=12, ha='center',
        color='red')

ax.annotate('', xy=(0.78, 0.25), xytext=(0.22, 0.25),
            arrowprops=dict(arrowstyle='->', color='blue', lw=2))
ax.text(0.5, 0.20, f'{p_r1_s1:.2f}', fontsize=13, ha='center',
        color='blue', fontweight='bold')

# 事前確率
ax.text(0.1, 0.95, f'P={p_send_0:.1f}', fontsize=11, ha='center')
ax.text(0.1, 0.05, f'P={p_send_1:.1f}', fontsize=11, ha='center')

ax.set_xlim(-0.05, 1.05)
ax.set_ylim(-0.05, 1.05)
ax.set_title('Binary Symmetric Channel', fontsize=13)
ax.axis('off')

# (b) 全確率の定理による受信確率
ax = axes[1]
# P(R=0)の分解
p_r0 = p_r0_s0 * p_send_0 + p_r0_s1 * p_send_1
p_r1 = p_r1_s0 * p_send_0 + p_r1_s1 * p_send_1

contributions_r0 = [p_r0_s0 * p_send_0, p_r0_s1 * p_send_1]
contributions_r1 = [p_r1_s0 * p_send_0, p_r1_s1 * p_send_1]

x_labels = ['P(R=0)', 'P(R=1)']
x_pos = [0, 1]

# スタック棒グラフ
bottom_0 = 0
colors_ch = ['steelblue', 'salmon']
labels_ch = ['from S=0', 'from S=1']
for i, (c0, c1) in enumerate(zip(contributions_r0, contributions_r1)):
    ax.bar(0, contributions_r0[i], bottom=sum(contributions_r0[:i]),
           color=colors_ch[i], alpha=0.8, edgecolor='gray',
           label=labels_ch[i] if True else '')
    ax.bar(1, contributions_r1[i], bottom=sum(contributions_r1[:i]),
           color=colors_ch[i], alpha=0.8, edgecolor='gray')

ax.text(0, p_r0 + 0.02, f'{p_r0:.3f}', ha='center', fontsize=12,
        fontweight='bold')
ax.text(1, p_r1 + 0.02, f'{p_r1:.3f}', ha='center', fontsize=12,
        fontweight='bold')

ax.set_xticks([0, 1])
ax.set_xticklabels(x_labels, fontsize=12)
ax.set_ylabel('Probability', fontsize=12)
ax.set_title('Law of Total Probability\nfor Received Bits', fontsize=13)
ax.legend(fontsize=10)
ax.grid(True, alpha=0.3, axis='y')
ax.set_ylim(0, 0.75)

# (c) ベイズの定理による復号
ax = axes[2]
# P(S=0|R=0) と P(S=1|R=0)
p_s0_r0 = (p_r0_s0 * p_send_0) / p_r0
p_s1_r0 = (p_r0_s1 * p_send_1) / p_r0
p_s0_r1 = (p_r1_s0 * p_send_0) / p_r1
p_s1_r1 = (p_r1_s1 * p_send_1) / p_r1

# 棒グラフ
x_pos2 = np.arange(2)
width = 0.35
ax.bar(x_pos2 - width/2, [p_s0_r0, p_s0_r1], width, color='steelblue',
       alpha=0.8, label='P(S=0|R)', edgecolor='gray')
ax.bar(x_pos2 + width/2, [p_s1_r0, p_s1_r1], width, color='salmon',
       alpha=0.8, label='P(S=1|R)', edgecolor='gray')

ax.set_xticks(x_pos2)
ax.set_xticklabels(['Received 0', 'Received 1'], fontsize=12)
ax.set_ylabel('Posterior Probability', fontsize=12)
ax.set_title("Bayes' Decoding\nP(Sent | Received)", fontsize=13)
ax.legend(fontsize=10)
ax.grid(True, alpha=0.3, axis='y')

# 値のアノテーション
for i, (v0, v1) in enumerate(zip([p_s0_r0, p_s0_r1], [p_s1_r0, p_s1_r1])):
    ax.text(i - width/2, v0 + 0.02, f'{v0:.3f}', ha='center', fontsize=10)
    ax.text(i + width/2, v1 + 0.02, f'{v1:.3f}', ha='center', fontsize=10)

plt.tight_layout()
plt.savefig('total_probability_channel.png', dpi=150, bbox_inches='tight')
plt.show()

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

# 混合正規分布のパラメータ
weights = [0.4, 0.35, 0.25]
means = [-2, 1, 4]
stds = [0.8, 1.2, 0.6]

x = np.linspace(-6, 8, 500)

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# (a) 混合分布の分解
ax = axes[0]
mixture = np.zeros_like(x)
colors_mix = ['steelblue', 'coral', 'green']

for i, (w, mu, sigma) in enumerate(zip(weights, means, stds)):
    component = w * stats.norm.pdf(x, mu, sigma)
    mixture += component
    ax.fill_between(x, component, alpha=0.3, color=colors_mix[i],
                    label=rf'$\pi_{i+1}\mathcal{{N}}(\mu_{i+1}={mu}, \sigma_{i+1}={sigma})$')
    ax.plot(x, component, color=colors_mix[i], linewidth=1.5)

ax.plot(x, mixture, 'k-', linewidth=2.5, label='Mixture $f(x)$')
ax.set_xlabel('x', fontsize=12)
ax.set_ylabel('Density', fontsize=12)
ax.set_title('Mixture of Gaussians\n'
             r'$f(x) = \sum_k \pi_k \mathcal{N}(x|\mu_k, \sigma_k^2)$',
             fontsize=13)
ax.legend(fontsize=9)
ax.grid(True, alpha=0.3)

# (b) 各コンポーネントの責任（事後確率）
ax = axes[1]
for i, (w, mu, sigma) in enumerate(zip(weights, means, stds)):
    component = w * stats.norm.pdf(x, mu, sigma)
    responsibility = component / np.maximum(mixture, 1e-10)
    ax.plot(x, responsibility, linewidth=2, color=colors_mix[i],
            label=rf'$P(Z={i+1}|x)$')

ax.set_xlabel('x', fontsize=12)
ax.set_ylabel('Posterior P(Component | x)', fontsize=12)
ax.set_title("Component Responsibility\n(Bayes' Theorem)", fontsize=13)
ax.legend(fontsize=10)
ax.grid(True, alpha=0.3)
ax.set_ylim(-0.05, 1.05)

plt.tight_layout()
plt.savefig('mixture_total_probability.png', dpi=150, bbox_inches='tight')
plt.show()