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🌐 [translation-sync] [likelihood_bayes.md] Update np.random → Generator API #248

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8bb79e0
Update translation: lectures/likelihood_bayes.md
mmcky Aug 12, 2026
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Update translation: .translate/state/likelihood_bayes.md.yml
mmcky Aug 12, 2026
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8 changes: 4 additions & 4 deletions .translate/state/likelihood_bayes.md.yml
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Original file line number Diff line number Diff line change
@@ -1,6 +1,6 @@
source-sha: a1cd90c5ef6d6f05277318f47fe21eda60fa5e6b
synced-at: "2026-07-18"
source-sha: 06c784f5e5939bd1379adeb2289488d84dfa7c8d
synced-at: "2026-08-12"
model: claude-sonnet-5
mode: RESYNC
mode: UPDATE
section-count: 8
tool-version: 0.17.0
tool-version: 0.25.0
111 changes: 55 additions & 56 deletions lectures/likelihood_bayes.md
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Expand Up @@ -13,20 +13,20 @@ translation:
title: 似然比过程和贝叶斯学习
headings:
Overview: 概述
The Setting: 背景设置
Likelihood Ratio Processes and Bayes’ Law: 似然比过程和贝叶斯定律
Likelihood Ratio Processes and Bayes’ Law::A recursive formula: 递归公式
The setting: 背景设置
Likelihood ratio processes and Bayes’ law: 似然比过程和贝叶斯定律
Likelihood ratio processes and Bayes’ law::A recursive formula: 递归公式
Another timing protocol: 另一种时序协议
Another timing protocol::Behavior of $\pi_t$ under wrong model: 在错误模型下 $\pi_t$ 的行为
Behavior of Posterior Probability $\{\pi_t\}$ Under Subjective Probability Distribution: 后验概率 $\{\pi_t\}$ 在主观概率分布下的行为
Behavior of Posterior Probability $\{\pi_t\}$ Under Subjective Probability Distribution::Mechanical details again: 再谈机械细节
Behavior of Posterior Probability $\{\pi_t\}$ Under Subjective Probability Distribution::Some simulations: 一些模拟
Initial Prior is Verified by Paths Drawn from Subjective Conditional Densities: 通过从主观条件密度中抽取的路径验证初始先验
Drilling Down a Little Bit: 深入分析
Related Lectures: 相关讲座
Behavior of posterior probability $\{\pi_t\}$ under subjective probability distribution: 后验概率 $\{\pi_t\}$ 在主观概率分布下的行为
Behavior of posterior probability $\{\pi_t\}$ under subjective probability distribution::Mechanical details again: 再谈机械细节
Behavior of posterior probability $\{\pi_t\}$ under subjective probability distribution::Some simulations: 一些模拟
Initial prior is verified by paths drawn from subjective conditional densities: 通过从主观条件密度中抽取的路径验证初始先验
Drilling down a little bit: 深入分析
Related lectures: 相关讲座
---

(likelihood_ratio_process)=
(likelihood_bayes)=
```{raw} jupyter
<div id="qe-notebook-header" align="right" style="text-align:right;">
<a href="https://quantecon.org/" title="quantecon.org">
Expand Down Expand Up @@ -67,7 +67,7 @@ mpl.font_manager.fontManager.addfont(FONTPATH)
plt.rcParams['font.family'] = ['Source Han Serif SC']

import numpy as np
from numba import vectorize, jit, prange
from numba import vectorize, jit
from math import gamma
import pandas as pd
from scipy.integrate import quad
Expand All @@ -76,10 +76,7 @@ from scipy.integrate import quad
import seaborn as sns
colors = sns.color_palette()

@jit
def set_seed():
np.random.seed(142857)
set_seed()
rng = np.random.default_rng(142857)
```

## 背景设置
Expand Down Expand Up @@ -162,7 +159,7 @@ g = jit(lambda x: p(x, G_a, G_b))

```{code-cell} ipython3
@jit
def simulate(a, b, T=50, N=500):
def simulate(a, b, rng, T=50, N=500):
'''
生成N组T个似然比观测值，
以N x T矩阵形式返回。
Comment on lines 161 to 165
Expand All @@ -174,7 +171,7 @@ def simulate(a, b, T=50, N=500):
for i in range(N):

for j in range(T):
w = np.random.beta(a, b)
w = rng.beta(a, b)
l_arr[i, j] = f(w) / g(w)

return l_arr
Expand All @@ -183,12 +180,12 @@ def simulate(a, b, T=50, N=500):
我们还将使用以下Python代码来准备一些信息丰富的模拟

```{code-cell} ipython3
l_arr_g = simulate(G_a, G_b, N=50000)
l_arr_g = simulate(G_a, G_b, rng, N=50000)
l_seq_g = np.cumprod(l_arr_g, axis=1)
```

```{code-cell} ipython3
l_arr_f = simulate(F_a, F_b, N=50000)
l_arr_f = simulate(F_a, F_b, rng, N=50000)
l_seq_f = np.cumprod(l_arr_f, axis=1)
```

Expand Down Expand Up @@ -449,16 +446,16 @@ $$

```{code-cell} ipython3
@jit
def simulate_mixture_path(x_true, T):
def simulate_mixture_path(x_true, T, rng):
"""
模拟混合时序协议下的 T 个观测值。
"""
w = np.empty(T)
for t in range(T):
if np.random.rand() < x_true:
w[t] = np.random.beta(F_a, F_b)
if rng.random() < x_true:
w[t] = rng.beta(F_a, F_b)
else:
w[t] = np.random.beta(G_a, G_b)
w[t] = rng.beta(G_a, G_b)
return w
```

Expand All @@ -479,8 +476,8 @@ prior_params = [(1, 3), (1, 1), (3, 1)]
prior_means = [a/(a+b) for a, b in prior_params]

# 从混合模型生成一条观测路径
set_seed()
w_mix = simulate_mixture_path(x_true, T_mix)
rng = np.random.default_rng(142857)
w_mix = simulate_mixture_path(x_true, T_mix, rng)
```

### 在错误模型下 $\pi_t$ 的行为
Expand Down Expand Up @@ -736,7 +733,7 @@ $$

```{code-cell} ipython3
@jit
def martingale_simulate(π0, N=5000, T=200):
def martingale_simulate(π0, rng, N=5000, T=200):

π_path = np.empty((N,T+1))
w_path = np.empty((N,T))
Expand All @@ -746,27 +743,27 @@ def martingale_simulate(π0, N=5000, T=200):
π = π0
for t in range(T):
# draw w
if np.random.rand() <= π:
w = np.random.beta(F_a, F_b)
if rng.random() <= π:
w = rng.beta(F_a, F_b)
else:
w = np.random.beta(G_a, G_b)
w = rng.beta(G_a, G_b)
π = π*f(w)/g(w)/(π*f(w)/g(w) + 1 - π)
π_path[n,t+1] = π
w_path[n,t] = w

return π_path, w_path

def fraction_0_1(π0, N, T, decimals):
def fraction_0_1(π0, rng, N, T, decimals):

π_path, w_path = martingale_simulate(π0, N=N, T=T)
π_path, w_path = martingale_simulate(π0, rng, N=N, T=T)
values, counts = np.unique(np.round(π_path[:,-1], decimals=decimals), return_counts=True)
return values, counts

def create_table(π0s, N=10000, T=500, decimals=2):
def create_table(π0s, rng, N=10000, T=500, decimals=2):

outcomes = []
for π0 in π0s:
values, counts = fraction_0_1(π0, N=N, T=T, decimals=decimals)
values, counts = fraction_0_1(π0, rng, N=N, T=T, decimals=decimals)
freq = counts/N
outcomes.append(dict(zip(values, freq)))
table = pd.DataFrame(outcomes).sort_index(axis=1).fillna(0)
Expand All @@ -777,15 +774,15 @@ def create_table(π0s, N=10000, T=500, decimals=2):
T = 200
π0 = .5

π_path, w_path = martingale_simulate(π0=π0, T=T, N=10000)
π_path, w_path = martingale_simulate(π0=π0, rng=rng, T=T, N=10000)
```

```{code-cell} ipython3
fig, ax = plt.subplots()
for i in range(100):
ax.plot(range(T+1), π_path[i, :])
ax.plot(range(T+1), π_path[i, :], lw=2)

ax.set_xlabel('$t$')
ax.set_xlabel('time')
ax.set_ylabel(r'$\pi_t$')
plt.show()
```
Expand Down Expand Up @@ -826,19 +823,19 @@ plt.show()
那么让我们把 $\pi_0$ 改为 $.3$，看看对于不同的 $t$ 值，$\pi_t$ 集合的分布会发生什么变化。

```{code-cell} ipython3
# 模拟
# simulate
T = 200
π0 = .3

π_path3, w_path3 = martingale_simulate(π0=π0, T=T, N=10000)
π_path3, w_path3 = martingale_simulate(π0=π0, rng=rng, T=T, N=10000)
```

```{code-cell} ipython3
fig, ax = plt.subplots()
for t in [1, 10, T-1]:
ax.hist(π_path3[:,t], bins=20, alpha=0.4, label=f'T={t}')

ax.set_ylabel('计数')
ax.set_ylabel('count')
ax.set_xlabel(r'$\pi_T$')
ax.legend(loc='upper right')
plt.show()
Expand All @@ -848,19 +845,21 @@ plt.show()

注意其中一条路径涉及系统性更高的 $w_t$ 值，这些结果将 $\pi_t$ 向上推动。

在模拟早期的随机抽样会将主观分布推向更频繁地从 $F$ 中抽样的方向，这会将 $\pi_t$ 推向 $0$。
在模拟早期的随机抽样会将主观分布推向更频繁地从 $F$ 中抽样的方向，这会将 $\pi_t$ 推向 $1$。

```{code-cell} ipython3
fig, ax = plt.subplots()
for i, j in enumerate([10, 100]):
ax.plot(range(T+1), π_path[j,:], color=colors[i], label=fr'$\pi$_path, {j}-th simulation')
ax.plot(range(1,T+1), w_path[j,:], color=colors[i], label=fr'$w$_path, {j}-th simulation', alpha=0.3)
ax.plot(range(T+1), π_path[j,:], color=colors[i],
label=fr'${{\pi_t}}$, {j}-th simulation', lw=2)
ax.plot(range(1,T+1), w_path[j,:], color=colors[i],
label=fr'${{w_t}}$, {j}-th simulation', alpha=0.3, lw=2)

ax.legend(loc='upper right')
ax.set_xlabel('$t$')
ax.set_xlabel('time')
ax.set_ylabel(r'$\pi_t$')
ax2 = ax.twinx()
ax2.set_ylabel("$w_t$")
ax2.set_ylabel(r"$w_t$")
plt.show()
```

Expand All @@ -878,7 +877,7 @@ plt.show()

```{code-cell} ipython3
# create table
table = create_table(list(np.linspace(0,1,11)), N=10000, T=500)
table = create_table(list(np.linspace(0,1,11)), rng, N=10000, T=500)
table
```

Expand All @@ -903,27 +902,27 @@ $$

```{code-cell} ipython3
@jit
def compute_cond_var(pi, mc_size=int(1e6)):
# create monte carlo draws
def compute_cond_var(π, rng, mc_size=int(1e6)):
# Create Monte Carlo draws
mc_draws = np.zeros(mc_size)

for i in prange(mc_size):
if np.random.rand() <= pi:
mc_draws[i] = np.random.beta(F_a, F_b)
for i in range(mc_size):
if rng.random() <= π:
mc_draws[i] = rng.beta(F_a, F_b)
else:
mc_draws[i] = np.random.beta(G_a, G_b)
mc_draws[i] = rng.beta(G_a, G_b)

dev = pi*f(mc_draws)/(pi*f(mc_draws) + (1-pi)*g(mc_draws)) - pi
dev = π*f(mc_draws)/(π*f(mc_draws) + (1-π)*g(mc_draws)) - π
return np.mean(dev**2)

pi_array = np.linspace(0, 1, 40)
π_array = np.linspace(0, 1, 40)
cond_var_array = []

for pi in pi_array:
cond_var_array.append(compute_cond_var(pi))
for π in π_array:
cond_var_array.append(compute_cond_var(π, rng))

fig, ax = plt.subplots()
ax.plot(pi_array, cond_var_array)
ax.plot(π_array, cond_var_array, lw=2)
ax.set_xlabel(r'$\pi_{t-1}$')
ax.set_ylabel(r'$\sigma^{2}(\pi_{t}\vert \pi_{t-1})$')
plt.show()
Expand Down
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