深度学习专栏（12）：结构化概率模型——解码复杂世界的因果密码

目录

​编辑

🌟 前言：当神经网络学会"推理"时会发生什么？

一、结构化概率模型：世界的因果罗盘

1.1 概率图模型的三原色

二、两大核心模型代码实战

2.1 贝叶斯网络：因果推理引擎

2.2 马尔可夫随机场：变量间的能量博弈

三、工业级应用案例解析

3.1 金融风险传染模拟

3.2 疫情传播预测

四、概率图模型工具包

4.1 动态概率传播可视化

🔥 模型选型指南

🚀 下期预告：蒙特卡罗方法——随机采样的艺术与科学

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🌟 前言：当神经网络学会"推理"时会发生什么？

"银行如何预测连锁违约风险？疾控中心怎样模拟病毒传播路径？答案藏在概率图的因果迷宫中！"
本文将用PyTorch+Pyro实现贝叶斯推理引擎，手把手构建金融风险传染模型与疫情传播模拟器，揭秘概率图如何让AI获得逻辑推理能力。文末附赠概率图可视化工具包，让你像福尔摩斯一样破解变量间的隐藏关系！

一、结构化概率模型：世界的因果罗盘

1.1 概率图模型的三原色

import networkx as nx
import matplotlib.pyplot as plt

# 构建心脏病预测贝叶斯网络
G = nx.DiGraph()
edges = [("年龄", "动脉硬化"), 
         ("吸烟", "动脉硬化"),
         ("动脉硬化", "心绞痛"),
         ("动脉硬化", "心肌梗塞"),
         ("运动量", "心绞痛")]

G.add_edges_from(edges)
pos = nx.spring_layout(G)

plt.figure(figsize=(10,6))
nx.draw(G, pos, with_labels=True, node_size=3000, node_color='#FFAAAA', 
        arrowsize=20, font_size=12)
plt.title("心脏疾病预测贝叶斯网络")
plt.show()

二、两大核心模型代码实战

2.1 贝叶斯网络：因果推理引擎

import torch
import pyro
import pyro.distributions as dist

def bayesian_network():
    # 先验概率
    age = pyro.sample("age", dist.Categorical(torch.tensor([0.3,0.7])))  # 0:年轻,1:老年
    smoke = pyro.sample("smoke", dist.Bernoulli(0.2))  # 20%吸烟
    
    # 条件概率表
    art_sclerosis_prob = torch.tensor([
        [[0.1, 0.9],   # 年轻不吸烟
         [0.3, 0.7]],  # 年轻吸烟
        [[0.4, 0.6],   # 老年不吸烟
         [0.8, 0.2]]   # 老年吸烟
    ])
    art_sclerosis = pyro.sample("art_sclerosis", 
        dist.Categorical(art_sclerosis_prob[age][smoke]))
    
    # 后验推断
    condition = {"age": torch.tensor(1), "art_sclerosis": torch.tensor(1)}
    posterior = pyro.infer.Importance(bayesian_network, num_samples=1000).run()
    smoke_prob = posterior.marginal("smoke").empirical["probs"][1]
    print(f"已知老年且动脉硬化，吸烟概率: {smoke_prob*100:.1f}%")

# 运行推理
bayesian_network()

2.2 马尔可夫随机场：变量间的能量博弈

import pgmpy.models
from pgmpy.inference import BeliefPropagation

# 构建社交网络信任模型
model = pgmpy.models.MarkovRandomField()
model.add_nodes_from(["用户A", "用户B", "用户C", "用户D"])
model.add_edges_from([("用户A","用户B"), ("用户B","用户C"), ("C","D"), ("A","D")])

# 定义势函数
factors = []
for node in model.nodes:
    factors.append(pgmpy.factors.discrete.DiscreteFactor(
        variables=[node],
        cardinality=[2],  # 0:不可信,1:可信
        values=[0.5, 0.5]))  # 均匀先验

for edge in model.edges:
    factors.append(pgmpy.factors.discrete.DiscreteFactor(
        variables=list(edge),
        cardinality=[2,2],
        values=[[2,1], [1,2]]))  # 相邻用户状态一致时能量低

model.add_factors(*factors)

# 信念传播推理
infer = BeliefPropagation(model)
infer.calibrate()
query = infer.query(variables=["用户D"], evidence={"用户A":0})
print(f"当用户A不可信时，用户D可信概率: {query.values[1]*100:.1f}%")

三、工业级应用案例解析

3.1 金融风险传染模拟

import pyro
import pyro.distributions as dist

def financial_network():
    # 定义银行节点
    banks = ["BankA", "BankB", "BankC"]
    exposures = {
        ("BankA", "BankB"): 0.3,
        ("BankB", "BankC"): 0.5,
        ("BankA", "BankC"): 0.2
    }
    
    # 初始违约概率
    with pyro.plate("banks", len(banks)):
        default = pyro.sample("default", dist.Bernoulli(0.1))  # 10%基础违约率
    
    # 风险传染过程
    for i, bank in enumerate(banks):
        neighbor_risk = sum(
            exposures[(n, bank)] * default[j] 
            for j, n in enumerate(banks) if (n, bank) in exposures
        )
        pyro.sample(f"default_{bank}", 
                    dist.Bernoulli(torch.sigmoid(2*neighbor_risk - 1)),
                    obs=default[i])

# 危机场景模拟
condition = {"default_BankA": torch.tensor(1.)}  # BankA已违约
posterior = pyro.infer.Importance(financial_network, num_samples=1000).run(condition)
print("BankB违约概率:", posterior.marginal("default_BankB").empirical["probs"][1])
print("BankC违约概率:", posterior.marginal("default_BankC").empirical["probs"][1])

3.2 疫情传播预测

class SEIRModel(pyro.nn.PyroModule):
    def __init__(self, population):
        super().__init__()
        self.population = population
        self.beta = pyro.param("beta", torch.tensor(0.3))  # 传播率
        self.gamma = pyro.param("gamma", torch.tensor(0.1)) # 康复率
    
    def forward(self, days=30):
        S = [self.population - 1]
        E = [0]
        I = [1]
        R = [0]
        
        for t in range(days):
            new_infected = pyro.sample(f"inf_{t}", 
                dist.Poisson(self.beta * I[-1] * S[-1]/self.population))
            new_exposed = pyro.sample(f"exp_{t}",
                dist.Poisson(0.5 * new_infected))
            new_recovered = pyro.sample(f"rec_{t}",
                dist.Poisson(self.gamma * I[-1]))
            
            S.append(S[-1] - new_infected)
            E.append(E[-1] + new_exposed - new_infected)
            I.append(I[-1] + new_infected - new_recovered)
            R.append(R[-1] + new_recovered)
            
        return {"S": S, "E": E, "I": I, "R": R}

# 参数推断
observations = {"I": [1,3,9,20,50,80,100,95,70,45]}  # 实际感染数据
guide = pyro.infer.autoguide.AutoDelta(SEIRModel(1000))
pyro.clear_param_store()
infer = pyro.infer.SVI(SEIRModel(1000), guide,
                       pyro.optim.Adam({"lr": 0.01}),
                       loss=pyro.infer.Trace_ELBO())
for epoch in range(1000):
    loss = infer.step(days=10)
    if epoch % 100 == 0:
        print(f"Epoch {epoch} Loss: {loss}")
        
print("推断传播率:", pyro.param("beta").item())
print("推断康复率:", pyro.param("gamma").item())

四、概率图模型工具包

4.1 动态概率传播可视化

import matplotlib.animation as animation

fig, ax = plt.subplots()
nx.draw(G, pos, with_labels=True, node_size=3000)

def update(frame):
    ax.clear()
    beliefs = infer.map_query(evidence=evidence_up_to(frame))
    node_colors = [beliefs[node][1] for node in G.nodes]
    nx.draw(G, pos, node_color=node_colors, cmap=plt.cm.Reds, 
            vmin=0, vmax=1, with_labels=True)
    return ax,

ani = animation.FuncAnimation(fig, update, frames=10, interval=500)
plt.show()

🔥 模型选型指南

模型类型	方向性	推理方式	适用场景	典型库
贝叶斯网络	有向	精确推断	因果推理	pgmpy, pyro
马尔可夫网络	无向	近似推断	关系建模	OpenGM, libDAI
因子图	混合	消息传递	复杂系统	ForneyLab
条件随机场	部分有向	梯度优化	序列标注	CRFsuite

🚀 下期预告：蒙特卡罗方法——随机采样的艺术与科学