diff --git a/.translate/state/likelihood_bayes.md.yml b/.translate/state/likelihood_bayes.md.yml index eb6d5b01..c7a451ed 100644 --- a/.translate/state/likelihood_bayes.md.yml +++ b/.translate/state/likelihood_bayes.md.yml @@ -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 diff --git a/lectures/likelihood_bayes.md b/lectures/likelihood_bayes.md index bcaffe5b..a97e0de8 100644 --- a/lectures/likelihood_bayes.md +++ b/lectures/likelihood_bayes.md @@ -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
@@ -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 @@ -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 @@ -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。 @@ -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 @@ -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) ``` @@ -449,16 +446,16 @@ $$ ```{code-cell} ipython3 @jit -def simulate_mixture_path(x_true, T): +def simulate_mixture_path(x_true, T, rng): """ 模拟混合时序协议下的 T 个观测值。 """ w = np.empty(T) for t in range(T): - if np.random.rand() < x_true: - w[t] = np.random.beta(F_a, F_b) + if rng.random() < x_true: + w[t] = rng.beta(F_a, F_b) else: - w[t] = np.random.beta(G_a, G_b) + w[t] = rng.beta(G_a, G_b) return w ``` @@ -479,8 +476,8 @@ prior_params = [(1, 3), (1, 1), (3, 1)] prior_means = [a/(a+b) for a, b in prior_params] # 从混合模型生成一条观测路径 -set_seed() -w_mix = simulate_mixture_path(x_true, T_mix) +rng = np.random.default_rng(142857) +w_mix = simulate_mixture_path(x_true, T_mix, rng) ``` ### 在错误模型下 $\pi_t$ 的行为 @@ -736,7 +733,7 @@ $$ ```{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)) @@ -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) @@ -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() ``` @@ -826,11 +823,11 @@ 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 @@ -838,7 +835,7 @@ 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() @@ -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() ``` @@ -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 ``` @@ -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()