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<article class="md-content__inner md-typeset">
<h1 id="_1">贝叶斯方法与序列模型<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p><em>贝叶斯方法将先验信念与观测数据相结合，生成模型参数的后验分布。本文涵盖最大似然估计、最大后验估计、共轭先验、贝叶斯推断、隐马尔可夫模型和EM算法——这些技术是垃圾邮件过滤器、语言模型和不确定性感知机器学习的基础。</em></p>
<ul>
<li>
<p>到目前为止，我们介绍了各种分布以及如何计算概率。现在我们来处理机器学习的核心问题：给定观测数据，如何找到模型的最佳参数？</p>
</li>
<li>
<p><strong>最大似然估计 (MLE)</strong> 直接回答了这个问题：选择使观测数据概率最大的参数值。</p>
</li>
<li>
<p>形式上，给定数据 <span class="arithmatex">\(D = \{x_1, x_2, \ldots, x_n\}\)</span> 和带有参数 <span class="arithmatex">\(\theta\)</span> 的模型，<strong>似然函数</strong>为：</p>
</li>
</ul>
<div class="arithmatex">\[L(\theta | D) = P(D | \theta) = \prod_{i=1}^{n} P(x_i | \theta)\]</div>
<ul>
<li>乘积假设数据点独立同分布（i.i.d.）。MLE估计量为：</li>
</ul>
<div class="arithmatex">\[\hat{\theta}_{\text{MLE}} = \arg\max_\theta L(\theta | D)\]</div>
<ul>
<li>实践中我们最大化<strong>对数似然</strong>，因为对数将乘积转化为求和，并防止数值下溢：</li>
</ul>
<div class="arithmatex">\[\ell(\theta) = \log L(\theta | D) = \sum_{i=1}^{n} \log P(x_i | \theta)\]</div>
<ul>
<li>
<p>由于 <span class="arithmatex">\(\log\)</span> 是单调递增函数，使得 <span class="arithmatex">\(\ell(\theta)\)</span> 最大的 <span class="arithmatex">\(\theta\)</span> 也同样使得 <span class="arithmatex">\(L(\theta)\)</span> 最大。</p>
</li>
<li>
<p><strong>抛硬币示例</strong>：你抛一枚硬币10次，得到7次正面。硬币偏置 <span class="arithmatex">\(p\)</span>（正面概率）的MLE估计是多少？</p>
</li>
<li>
<p>每次抛掷服从 Bernoulli(<span class="arithmatex">\(p\)</span>)，因此10次抛掷中出现7次正面的似然为：</p>
</li>
</ul>
<div class="arithmatex">\[L(p) = \binom{10}{7} p^7 (1-p)^3\]</div>
<ul>
<li>
<p>取对数并求导：<span class="arithmatex">\(\frac{d\ell}{dp} = \frac{7}{p} - \frac{3}{1-p} = 0\)</span>，解得 <span class="arithmatex">\(\hat{p}_{\text{MLE}} = 7/10 = 0.7\)</span>。</p>
</li>
<li>
<p>MLE直观且简单。如果10次抛掷中得到7次正面，最可能的偏置是0.7。但注意一个问题：如果10次抛掷中得到10次正面，MLE会得出 <span class="arithmatex">\(\hat{p} = 1\)</span>，意味着硬币将永远正面朝上。仅凭10次观测就得出这样的结论似乎过于自信。</p>
</li>
<li>
<p><strong>最大后验估计 (MAP)</strong> 通过加入先验信念来修复这个问题。MAP不是仅最大化似然，而是最大化后验：</p>
</li>
</ul>
<div class="arithmatex">\[\hat{\theta}_{\text{MAP}} = \arg\max_\theta P(\theta | D) = \arg\max_\theta P(D | \theta) \cdot P(\theta)\]</div>
<ul>
<li>
<p>我们省略了分母 <span class="arithmatex">\(P(D)\)</span>，因为它不依赖于 <span class="arithmatex">\(\theta\)</span>，不影响argmax的结果。</p>
</li>
<li>
<p>先验 <span class="arithmatex">\(P(\theta)\)</span> 编码了我们在看到数据之前对 <span class="arithmatex">\(\theta\)</span> 的信念。如果我们使用 Beta(2, 2) 先验来表示硬币偏置（表达"硬币大致是公平的"这一温和信念），MAP估计就不再仅仅是正面的比例，而是被拉向0.5。</p>
</li>
</ul>
<p><img alt="MLE 找到似然的峰值；MAP 找到似然乘以先验的峰值" src="../../images/mle_vs_map.svg" /></p>
<ul>
<li>使用 Beta(<span class="arithmatex">\(\alpha\)</span>, <span class="arithmatex">\(\beta\)</span>) 先验，观测到 <span class="arithmatex">\(h\)</span> 次正面和 <span class="arithmatex">\(t\)</span> 次反面后，后验为 Beta(<span class="arithmatex">\(\alpha + h\)</span>, <span class="arithmatex">\(\beta + t\)</span>)，MAP估计为：</li>
</ul>
<div class="arithmatex">\[\hat{p}_{\text{MAP}} = \frac{\alpha + h - 1}{\alpha + \beta + h + t - 2}\]</div>
<ul>
<li>
<p>对于我们的示例，Beta(2,2)先验，7次正面，3次反面：<span class="arithmatex">\(\hat{p}_{\text{MAP}} = \frac{2 + 7 - 1}{2 + 2 + 10 - 2} = \frac{8}{12} = 0.667\)</span>。</p>
</li>
<li>
<p>注意MAP估计（0.667）相比MLE（0.7）如何被拉向0.5。先验起到了正则化的作用。在机器学习中，L2正则化（权重衰减）完全等价于在权重上使用高斯先验的MAP估计。</p>
</li>
<li>
<p><strong>完整的贝叶斯推断</strong>比MAP更进一步。它不是寻找单一的最佳 <span class="arithmatex">\(\theta\)</span>，而是维护整个后验分布 <span class="arithmatex">\(P(\theta | D)\)</span>。这不仅给你一个点估计，还给出了不确定性的度量。</p>
</li>
<li>
<p>对于具有Beta(2,2)先验和7次正面、3次反面的偏置硬币，完整的后验是 Beta(9, 5)。该分布的均值为 <span class="arithmatex">\(9/14 \approx 0.643\)</span>，其弥散程度告诉我们置信度的高低。数据越多，后验越窄。</p>
</li>
<li>
<p>三种方法形成了一个谱系：</p>
<ul>
<li><strong>MLE</strong>：无先验，仅依赖数据。速度快，但数据少时可能过拟合。</li>
<li><strong>MAP</strong>：带先验正则化的点估计。增加鲁棒性。</li>
<li><strong>完整贝叶斯</strong>：完整的后验分布。信息量最大，但通常计算成本高。</li>
</ul>
</li>
<li>
<p><strong>马尔可夫链</strong>对序列进行建模，其中下一状态仅依赖于当前状态，而不依赖于历史。这种"无记忆性"称为<strong>马尔可夫性</strong>：</p>
</li>
</ul>
<div class="arithmatex">\[P(X_{t+1} | X_t, X_{t-1}, \ldots, X_1) = P(X_{t+1} | X_t)\]</div>
<ul>
<li>
<p>以天气为例。明天的天气取决于今天的天气，但不取决于上周的天气（这是一个简化，但出奇地有用）。</p>
</li>
<li>
<p>马尔可夫链具有有限个<strong>状态</strong>和一个<strong>转移矩阵</strong> <span class="arithmatex">\(T\)</span>，其中元素 <span class="arithmatex">\(T_{ij}\)</span> 表示从状态 <span class="arithmatex">\(i\)</span> 转移到状态 <span class="arithmatex">\(j\)</span> 的概率。每一行之和为1。</p>
</li>
</ul>
<p><img alt="天气马尔可夫链，状态有雨天、晴天、多云，以及转移概率" src="../../images/markov_chain.svg" /></p>
<ul>
<li>对于上图的天气示例，转移矩阵为：</li>
</ul>
<div class="arithmatex">\[
T = \begin{pmatrix} 0.3 & 0.4 & 0.3 \\ 0.2 & 0.5 & 0.3 \\ 0.4 & 0.3 & 0.3 \end{pmatrix}
\]</div>
<ul>
<li>
<p>如果今天是雨天（状态向量 <span class="arithmatex">\(\mathbf{s}_0 = [1, 0, 0]\)</span>），明天天气的概率分布为 <span class="arithmatex">\(\mathbf{s}_1 = \mathbf{s}_0 T = [0.3, 0.4, 0.3]\)</span>。两天后：<span class="arithmatex">\(\mathbf{s}_2 = \mathbf{s}_0 T^2\)</span>。这使用了第一章中的矩阵乘法。</p>
</li>
<li>
<p>许多马尔可夫链会收敛到一个<strong>平稳分布</strong> <span class="arithmatex">\(\pi\)</span>，满足 <span class="arithmatex">\(\pi T = \pi\)</span>。无论从哪里出发，经过足够多的步数后，链会收敛到 <span class="arithmatex">\(\pi\)</span>。这一性质是MCMC（马尔可夫链蒙特卡罗）的基础，MCMC是贝叶斯机器学习中广泛使用的采样技术。</p>
</li>
<li>
<p><strong>隐马尔可夫模型 (HMM)</strong> 通过增加一层间接性来扩展马尔可夫链。真实状态是隐藏的（不可观测的），每个时间步隐藏状态会发出一个可观测的信号。</p>
</li>
</ul>
<p><img alt="HMM 结构：上方隐藏状态由转移连接，下方观测由发射连接" src="../../images/hmm_structure.svg" /></p>
<ul>
<li>
<p>HMM 有三个组成部分：</p>
<ul>
<li><strong>转移概率</strong> <span class="arithmatex">\(P(z_t | z_{t-1})\)</span>：隐藏状态如何演化（马尔可夫链）</li>
<li><strong>发射概率</strong> <span class="arithmatex">\(P(x_t | z_t)\)</span>：每个隐藏状态产生什么可观测输出</li>
<li><strong>初始分布</strong> <span class="arithmatex">\(P(z_1)\)</span>：起始隐藏状态的概率</li>
</ul>
</li>
<li>
<p><strong>雨伞示例</strong>：假设你不能直接看到天气，但可以观察到你的朋友是否带伞。隐藏状态为 {雨天, 晴天}，观测为 {带伞, 不带伞}。</p>
</li>
<li>
<p>转移概率：<span class="arithmatex">\(P(\text{雨天}|\text{雨天}) = 0.7\)</span>，<span class="arithmatex">\(P(\text{晴天}|\text{雨天}) = 0.3\)</span>，<span class="arithmatex">\(P(\text{雨天}|\text{晴天}) = 0.4\)</span>，<span class="arithmatex">\(P(\text{晴天}|\text{晴天}) = 0.6\)</span>。</p>
</li>
<li>
<p>发射概率：<span class="arithmatex">\(P(\text{带伞}|\text{雨天}) = 0.9\)</span>，<span class="arithmatex">\(P(\text{不带伞}|\text{雨天}) = 0.1\)</span>，<span class="arithmatex">\(P(\text{带伞}|\text{晴天}) = 0.2\)</span>，<span class="arithmatex">\(P(\text{不带伞}|\text{晴天}) = 0.8\)</span>。</p>
</li>
<li>
<p>HMM 的关键问题有：</p>
<ul>
<li><strong>解码</strong>：给定观测，最可能的隐藏状态序列是什么？由<strong>维特比算法</strong>求解。</li>
<li><strong>评估</strong>：观测序列的概率是多少？由<strong>前向算法</strong>求解。</li>
<li><strong>学习</strong>：给定观测，最佳模型参数是什么？由<strong>Baum-Welch算法</strong>求解（期望最大化算法的一个实例）。</li>
</ul>
</li>
<li>
<p><strong>维特比演算</strong>：假设你观测到 [带伞, 带伞, 不带伞]，想找到最可能的天气序列。</p>
</li>
<li>
<p>从初始概率开始。假设 <span class="arithmatex">\(P(R) = 0.5\)</span>，<span class="arithmatex">\(P(S) = 0.5\)</span>。</p>
</li>
<li>
<p><strong>第1天</strong>（观测到带伞）：</p>
<ul>
<li><span class="arithmatex">\(V_1(R) = P(R) \cdot P(U|R) = 0.5 \times 0.9 = 0.45\)</span></li>
<li><span class="arithmatex">\(V_1(S) = P(S) \cdot P(U|S) = 0.5 \times 0.2 = 0.10\)</span></li>
</ul>
</li>
<li>
<p><strong>第2天</strong>（观测到带伞）：</p>
<ul>
<li><span class="arithmatex">\(V_2(R) = \max(V_1(R) \cdot P(R|R), V_1(S) \cdot P(R|S)) \cdot P(U|R)\)</span></li>
<li><span class="arithmatex">\(= \max(0.45 \times 0.7, 0.10 \times 0.4) \times 0.9 = \max(0.315, 0.04) \times 0.9 = 0.2835\)</span></li>
<li><span class="arithmatex">\(V_2(S) = \max(V_1(R) \cdot P(S|R), V_1(S) \cdot P(S|S)) \cdot P(U|S)\)</span></li>
<li><span class="arithmatex">\(= \max(0.45 \times 0.3, 0.10 \times 0.6) \times 0.2 = \max(0.135, 0.06) \times 0.2 = 0.027\)</span></li>
</ul>
</li>
<li>
<p><strong>第3天</strong>（观测到不带伞）：</p>
<ul>
<li><span class="arithmatex">\(V_3(R) = \max(0.2835 \times 0.7, 0.027 \times 0.4) \times 0.1 = 0.1985 \times 0.1 = 0.01985\)</span></li>
<li><span class="arithmatex">\(V_3(S) = \max(0.2835 \times 0.3, 0.027 \times 0.6) \times 0.8 = 0.08505 \times 0.8 = 0.06804\)</span></li>
</ul>
</li>
<li>
<p>第3天的最大值在晴天。回溯：第3天 = 晴天（来自R），第2天 = 雨天（来自R），第1天 = 雨天。最可能的序列：<strong>雨天, 雨天, 晴天</strong>。</p>
</li>
<li>
<p><strong>前向-后向算法</strong>计算在给定整个观测序列条件下，每个时间步处于每个隐藏状态的概率。前向过程计算 <span class="arithmatex">\(P(z_t, x_{1:t})\)</span>，后向过程计算 <span class="arithmatex">\(P(x_{t+1:T} | z_t)\)</span>。两者相乘得到平滑后的状态概率。</p>
</li>
<li>
<p><strong>Baum-Welch算法</strong>在隐藏状态不可观测时从数据中学习HMM参数。它是一种期望最大化（EM）算法：E步使用前向-后向算法估计哪些隐藏状态生成了观测，M步更新转移概率和发射概率。</p>
</li>
<li>
<p>HMM在历史上主导了语音识别（隐藏的音素状态发出声学信号）和生物信息学（隐藏的基因状态发出DNA碱基对）。虽然深度学习在很大程度上已取代了这些领域中的HMM，但隐藏状态、发射和序列推断的思想仍然是序列模型的核心。</p>
</li>
<li>
<p><strong>条件随机场 (CRF)</strong> 通过去除发射独立假设来改进HMM。在HMM中，时间 <span class="arithmatex">\(t\)</span> 的观测仅依赖于时间 <span class="arithmatex">\(t\)</span> 的隐藏状态。CRF允许位置 <span class="arithmatex">\(t\)</span> 的标签依赖于整个输入序列。</p>
</li>
<li>
<p>线性链CRF对给定输入序列 <span class="arithmatex">\(\mathbf{x}\)</span> 条件下标签序列 <span class="arithmatex">\(\mathbf{y}\)</span> 的条件概率建模：</p>
</li>
</ul>
<div class="arithmatex">\[P(\mathbf{y} | \mathbf{x}) = \frac{1}{Z(\mathbf{x})} \exp\!\left(\sum_t \left[\sum_k \lambda_k f_k(y_t, y_{t-1}, \mathbf{x}, t)\right]\right)\]</div>
<ul>
<li>
<p>其中 <span class="arithmatex">\(f_k\)</span> 是特征函数（可以查看输入的任意部分），<span class="arithmatex">\(\lambda_k\)</span> 是学习到的权重，<span class="arithmatex">\(Z(\mathbf{x})\)</span> 是归一化常数。</p>
</li>
<li>
<p>CRF是判别式模型（直接建模 <span class="arithmatex">\(P(\mathbf{y}|\mathbf{x})\)</span>），而HMM是生成式模型（建模 <span class="arithmatex">\(P(\mathbf{x}, \mathbf{y})\)</span>）。这一区别与逻辑回归（判别式）和朴素贝叶斯（生成式）之间的区别相同。</p>
</li>
<li>
<p>在现代NLP中，CRF层通常被加在神经网络之上（BiLSTM-CRF、BERT-CRF），用于命名实体识别和词性标注等需要捕捉标签依赖关系的任务。</p>
</li>
</ul>
<h2 id="colab-notebook">编程练习（使用 CoLab 或 notebook）<a class="headerlink" href="#colab-notebook" title="Permanent link">&para;</a></h2>
<ol>
<li>
<p>实现抛硬币实验的MLE和MAP。观察MAP估计如何随不同的先验和不同的数据量而变化。
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># 数据：观测到的硬币抛掷结果</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">heads</span><span class="p">,</span> <span class="n">tails</span> <span class="o">=</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">3</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="c1"># MLE</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">p_mle</span> <span class="o">=</span> <span class="n">heads</span> <span class="o">/</span> <span class="p">(</span><span class="n">heads</span> <span class="o">+</span> <span class="n">tails</span><span class="p">)</span>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;MLE: </span><span class="si">{</span><span class="n">p_mle</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="c1"># 使用 Beta 先验的 MAP</span>
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="k">for</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span> <span class="ow">in</span> <span class="p">[(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">)]:</span>
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="n">p_map</span> <span class="o">=</span> <span class="p">(</span><span class="n">alpha</span> <span class="o">+</span> <span class="n">heads</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">alpha</span> <span class="o">+</span> <span class="n">beta</span> <span class="o">+</span> <span class="n">heads</span> <span class="o">+</span> <span class="n">tails</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span>
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;MAP (Beta(</span><span class="si">{</span><span class="n">alpha</span><span class="si">}</span><span class="s2">,</span><span class="si">{</span><span class="n">beta</span><span class="si">}</span><span class="s2">)): </span><span class="si">{</span><span class="n">p_map</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="c1"># 可视化 Beta(2,2) 先验下的后验</span>
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="n">theta</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.99</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="c1"># 后验为 Beta(alpha+heads, beta+tails)</span>
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="n">a_post</span><span class="p">,</span> <span class="n">b_post</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">heads</span><span class="p">,</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">tails</span>
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="n">posterior</span> <span class="o">=</span> <span class="n">theta</span><span class="o">**</span><span class="p">(</span><span class="n">a_post</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">theta</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">b_post</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="n">posterior</span> <span class="o">=</span> <span class="n">posterior</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">trapezoid</span><span class="p">(</span><span class="n">posterior</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">theta</span><span class="p">,</span> <span class="n">posterior</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;#e74c3c&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s2">&quot;后验 Beta(</span><span class="si">{</span><span class="n">a_post</span><span class="si">}</span><span class="s2">,</span><span class="si">{</span><span class="n">b_post</span><span class="si">}</span><span class="s2">)&quot;</span><span class="p">)</span>
<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">p_mle</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;#3498db&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s2">&quot;MLE = </span><span class="si">{</span><span class="n">p_mle</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a><span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">((</span><span class="n">a_post</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">a_post</span><span class="o">+</span><span class="n">b_post</span><span class="o">-</span><span class="mi">2</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;#e74c3c&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;--&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s2">&quot;MAP = </span><span class="si">{</span><span class="p">(</span><span class="n">a_post</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">a_post</span><span class="o">+</span><span class="n">b_post</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;θ (硬币偏置)&quot;</span><span class="p">)</span>
<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;密度&quot;</span><span class="p">)</span>
<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;7次正面、3次反面后 Beta(2,2) 先验下的后验分布&quot;</span><span class="p">)</span>
<a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a><span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a><span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div></p>
</li>
<li>
<p>为天气模型构建一个马尔可夫链并进行模拟。分别通过模拟和求解 <span class="arithmatex">\(\pi T = \pi\)</span> 计算平稳分布。
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="c1"># 转移矩阵：R（雨天）, S（晴天）, C（多云）</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="n">T</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a> <span class="p">[</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</span>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a> <span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a> <span class="p">[</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">]</span>
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="p">])</span>
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="n">states</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;雨天&quot;</span><span class="p">,</span> <span class="s2">&quot;晴天&quot;</span><span class="p">,</span> <span class="s2">&quot;多云&quot;</span><span class="p">]</span>
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a>
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="c1"># 模拟 100,000 步</span>
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="n">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="n">n_steps</span> <span class="o">=</span> <span class="mi">100_000</span>
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="n">state</span> <span class="o">=</span> <span class="mi">0</span> <span class="c1"># 从雨天开始</span>
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="n">counts</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a>
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_steps</span><span class="p">):</span>
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a> <span class="n">key</span><span class="p">,</span> <span class="n">subkey</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a> <span class="n">state</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">subkey</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="n">T</span><span class="p">[</span><span class="n">state</span><span class="p">])</span>
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a> <span class="n">counts</span> <span class="o">=</span> <span class="n">counts</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">state</span><span class="p">]</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a>
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="n">sim_stationary</span> <span class="o">=</span> <span class="n">counts</span> <span class="o">/</span> <span class="n">n_steps</span>
<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;模拟得到的平稳分布：&quot;</span><span class="p">)</span>
<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">states</span><span class="p">,</span> <span class="n">sim_stationary</span><span class="p">):</span>
<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot; </span><span class="si">{</span><span class="n">s</span><span class="si">}</span><span class="s2">: </span><span class="si">{</span><span class="n">p</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a>
<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="c1"># 解析法：找到特征值为1的左特征向量</span>
<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="n">eigenvalues</span><span class="p">,</span> <span class="n">eigenvectors</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="n">idx</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">eigenvalues</span> <span class="o">-</span> <span class="mf">1.0</span><span class="p">))</span>
<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="n">pi</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">eigenvectors</span><span class="p">[:,</span> <span class="n">idx</span><span class="p">])</span>
<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="n">pi</span> <span class="o">=</span> <span class="n">pi</span> <span class="o">/</span> <span class="n">pi</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">解析得到的平稳分布：&quot;</span><span class="p">)</span>
<a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">states</span><span class="p">,</span> <span class="n">pi</span><span class="p">):</span>
<a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot; </span><span class="si">{</span><span class="n">s</span><span class="si">}</span><span class="s2">: </span><span class="si">{</span><span class="n">p</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
<li>
<p>为雨伞HMM实现维特比算法，并解码一个观测序列。
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a>
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="c1"># HMM 参数</span>
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="n">states</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;雨天&quot;</span><span class="p">,</span> <span class="s2">&quot;晴天&quot;</span><span class="p">]</span>
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">obs_names</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;带伞&quot;</span><span class="p">,</span> <span class="s2">&quot;不带伞&quot;</span><span class="p">]</span>
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a>
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="n">trans</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">],</span> <span class="c1"># R-&gt;R, R-&gt;S</span>
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a> <span class="p">[</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">]])</span> <span class="c1"># S-&gt;R, S-&gt;S</span>
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a>
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="n">emit</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">],</span> <span class="c1"># R-&gt;带伞, R-&gt;不带伞</span>
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a> <span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">]])</span> <span class="c1"># S-&gt;带伞, S-&gt;不带伞</span>
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a>
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="n">init</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">])</span>
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a>
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="c1"># 观测：带伞=0，不带伞=1</span>
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="n">observations</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="c1"># 带伞, 带伞, 不带伞</span>
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a>
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="k">def</span><span class="w"> </span><span class="nf">viterbi</span><span class="p">(</span><span class="n">obs</span><span class="p">,</span> <span class="n">init</span><span class="p">,</span> <span class="n">trans</span><span class="p">,</span> <span class="n">emit</span><span class="p">):</span>
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a> <span class="n">n_states</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">init</span><span class="p">)</span>
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a> <span class="n">T</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">obs</span><span class="p">)</span>
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a> <span class="n">V</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">T</span><span class="p">,</span> <span class="n">n_states</span><span class="p">))</span>
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a> <span class="n">path</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">T</span><span class="p">,</span> <span class="n">n_states</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span>
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a>
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a> <span class="c1"># 初始化</span>
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a> <span class="n">V</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">init</span> <span class="o">*</span> <span class="n">emit</span><span class="p">[:,</span> <span class="n">obs</span><span class="p">[</span><span class="mi">0</span><span class="p">]])</span>
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a>
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a> <span class="c1"># 递推</span>
<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">T</span><span class="p">):</span>
<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_states</span><span class="p">):</span>
<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a> <span class="n">probs</span> <span class="o">=</span> <span class="n">V</span><span class="p">[</span><span class="n">t</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">trans</span><span class="p">[:,</span> <span class="n">j</span><span class="p">]</span>
<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a> <span class="n">V</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">probs</span><span class="p">)</span> <span class="o">*</span> <span class="n">emit</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">obs</span><span class="p">[</span><span class="n">t</span><span class="p">]])</span>
<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a> <span class="n">path</span> <span class="o">=</span> <span class="n">path</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">probs</span><span class="p">))</span>
<a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a>
<a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a> <span class="c1"># 回溯</span>
<a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a> <span class="n">best</span> <span class="o">=</span> <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">V</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]))]</span>
<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">T</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
<a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a> <span class="n">best</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">path</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="n">best</span><span class="p">[</span><span class="mi">0</span><span class="p">]]))</span>
<a id="__codelineno-2-38" name="__codelineno-2-38" href="#__codelineno-2-38"></a> <span class="k">return</span> <span class="n">best</span><span class="p">,</span> <span class="n">V</span>
<a id="__codelineno-2-39" name="__codelineno-2-39" href="#__codelineno-2-39"></a>
<a id="__codelineno-2-40" name="__codelineno-2-40" href="#__codelineno-2-40"></a><span class="n">decoded</span><span class="p">,</span> <span class="n">scores</span> <span class="o">=</span> <span class="n">viterbi</span><span class="p">(</span><span class="n">observations</span><span class="p">,</span> <span class="n">init</span><span class="p">,</span> <span class="n">trans</span><span class="p">,</span> <span class="n">emit</span><span class="p">)</span>
<a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;观测序列：&quot;</span><span class="p">,</span> <span class="p">[</span><span class="n">obs_names</span><span class="p">[</span><span class="n">o</span><span class="p">]</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">observations</span><span class="p">])</span>
<a id="__codelineno-2-42" name="__codelineno-2-42" href="#__codelineno-2-42"></a><span class="nb">print</span><span class="p">(</span><span class="s2">&quot;解码结果：&quot;</span><span class="p">,</span> <span class="p">[</span><span class="n">states</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">decoded</span><span class="p">])</span>
</code></pre></div></p>
</li>
<li>
<p>可视化随着观测更多抛硬币结果，后验如何演化。从 Beta(1,1) 先验（均匀分布）开始，每次抛掷后更新后验。
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a>
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="n">theta</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.99</span><span class="p">,</span> <span class="mi">300</span><span class="p">)</span>
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="n">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span>
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a>
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="c1"># 真实偏置 = 0.65</span>
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="n">flips</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mf">0.65</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">50</span><span class="p">,))</span>
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a>
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span> <span class="c1"># Beta(1,1) = 均匀分布</span>
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a>
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="k">for</span> <span class="n">n_obs</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">25</span><span class="p">,</span> <span class="mi">50</span><span class="p">]:</span>
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a> <span class="n">h</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">flips</span><span class="p">[:</span><span class="n">n_obs</span><span class="p">]</span><span class="o">.</span><span class="n">sum</span><span class="p">())</span>
<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a> <span class="n">t</span> <span class="o">=</span> <span class="n">n_obs</span> <span class="o">-</span> <span class="n">h</span>
<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a> <span class="n">a_post</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">h</span>
<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a> <span class="n">b_post</span> <span class="o">=</span> <span class="n">b</span> <span class="o">+</span> <span class="n">t</span>
<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a> <span class="n">y</span> <span class="o">=</span> <span class="n">theta</span><span class="o">**</span><span class="p">(</span><span class="n">a_post</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">theta</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">b_post</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a> <span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">trapezoid</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a> <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">theta</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s2">&quot;n=</span><span class="si">{</span><span class="n">n_obs</span><span class="si">}</span><span class="s2"> (h=</span><span class="si">{</span><span class="n">h</span><span class="si">}</span><span class="s2">)&quot;</span><span class="p">)</span>
<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a>
<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mf">0.65</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;black&quot;</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s2">&quot;:&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;真实 p=0.65&quot;</span><span class="p">)</span>
<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;θ&quot;</span><span class="p">)</span>
<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;密度&quot;</span><span class="p">)</span>
<a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;贝叶斯更新：数据越多后验越窄&quot;</span><span class="p">)</span>
<a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
<a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div></p>
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