from pgmpy.models import BayesianModel

cancer_model = BayesianModel([('Pollution', 'Cancer'), 
                              ('Smoker', 'Cancer'),
                              ('Cancer', 'Xray'),
                              ('Cancer', 'Dyspnoea')])

这个网络中有五个节点: Pollution, Cancer, Smoker, Xray, Dyspnoea.

('Pollution', 'Cancer'): 一条有向边, 从 Pollution 指向 Cancer, 表示环境污染有可能导致癌症.

('Smoker', 'Cancer'): 吸烟有可能导致癌症.

('Cancer', 'Xray'): 得癌症的人可能会去照X射线.

('Cancer', 'Dyspnoea'): 得癌症的人可能会呼吸困难.

from pgmpy.factors.discrete import TabularCPD

cpd_poll = TabularCPD(variable='Pollution', variable_card=2,
                      values=[[0.9], [0.1]])
cpd_smoke = TabularCPD(variable='Smoker', variable_card=2,
                       values=[[0.3], [0.7]])
cpd_cancer = TabularCPD(variable='Cancer', variable_card=2,
                        values=[[0.03, 0.05, 0.001, 0.02],
                                [0.97, 0.95, 0.999, 0.98]],
                        evidence=['Smoker', 'Pollution'],
                        evidence_card=[2, 2])
cpd_xray = TabularCPD(variable='Xray', variable_card=2,
                      values=[[0.9, 0.2], [0.1, 0.8]],
                      evidence=['Cancer'], evidence_card=[2])
cpd_dysp = TabularCPD(variable='Dyspnoea', variable_card=2,
                      values=[[0.65, 0.3], [0.35, 0.7]],
                      evidence=['Cancer'], evidence_card=[2])

这些代码, 主要是建立一些概率表, 然后往表里面填入了一些参数.

Pollution: 有两种概率, 分别是 0.9 和 0.1.

Smoker: 有两种概率, 分别是 0.3 和 0.7. (意思是在一个人群里, 有 30% 的人吸烟, 有 70% 的人不吸烟)

Cancer: envidence 表示有 Smoker 和 Pollution 两个节点指向 Cancer 节点;

[0.03, 0.05, 0.001, 0.02] 的意思, 通过下表可以较容易看出来:

可以看出, 环境不好, 又抽烟的人, 有 0.98 的概率得癌症.

Xray: 通过 envidence 可以看出, 只有 Cancer 指向它; [0.9, 0.2] 表示, 未得癌症的人去照X光, 有 0.9 的概率正常, 而得了癌症的人去照X光, 有 0.2 的概率正常, [0.1, 0.8] 表示, 未得癌症的人去照X光, 有 0.1 的概率不正常, 而得癌症的人有 0.8 的概率不正常.

Dyspnoea: 未得癌症的人有 0.65 的可能性呼吸正常, 得癌症的人只有 0.3 的可能性呼吸正常; 未得癌症的人有 0.35 的可能性呼吸困难, 得癌症的人有 0.7 的可能性呼吸困难.

# Associating the parameters with the model structure.
cancer_model.add_cpds(cpd_poll, cpd_smoke, cpd_cancer, cpd_xray, cpd_dysp)

# Checking if the cpds are valid for the model.
cancer_model.check_model()

第一行代码表示将概率分布表加入网络中; 第二行代码验证模型数据的正确性(至于怎么验证的, 我没有看源码, 但是猜测是验证相关数据的和是否为1, 验证数据中是否有负数, 等等).

cancer_model.is_active_trail('Pollution', 'Smoker')  # False
cancer_model.is_active_trail('Pollution', 'Smoker', observed=['Cancer'])  # True
cancer_model.local_independencies('Xray')  # (Xray _|_ Dyspnoea, Smoker, Pollution | Cancer)
cancer_model.get_independencies()

第一行代码表示, Pollution 节点是否指向 Smoker 节点, 返回的是 False.

第二行代码表示, Cancer 节点的计算, 是否与 Pollution 和 Smoker 两个节点有关, 返回的是 True.

第三行代码的输出表示, Xray 与 Dyspnoea, Smoker, Pollution 不相关, 与 Cancer 相关.

第四行代码输出所有的依赖关系.

变量消除法是精确推断的一种方法.

from pgmpy.inference import VariableElimination
asia_infer = VariableElimination(asia_model)
q = asia_infer.query(variables=['bronc'], evidence={'smoke': 0})
print(q['bronc'])

结果如下:

+---------+--------------+
| bronc   |   phi(bronc) |
|---------+--------------|
| bronc_0 |       0.6000 |
| bronc_1 |       0.4000 |
+---------+--------------+

意思是, 在不吸烟的情况下, 得支气管炎的概率是 0.4, 未得支气管炎的概率是 0.6.

在在吸烟情况下, 得支气管炎的概率和未得支气管炎的概率可以这样查询:

q = asia_infer.query(variables=['bronc'], evidence={'smoke': 1})
print(q['bronc'])

结果如下:

+---------+--------------+
| bronc   |   phi(bronc) |
|---------+--------------|
| bronc_0 |       0.3000 |
| bronc_1 |       0.7000 |
+---------+--------------+

学生例子中包含5个随机变量, 如下所示:

生成数据:

import numpy as np
import pandas as pd
raw_data = np.random.randint(low=0, high=2, size=(1000, 5))
data = pd.DataFrame(raw_data, columns=['D', 'I', 'G', 'L', 'S'])

定义模型:

from pgmpy.models import BayesianModel
from pgmpy.estimators import MaximumLikelihoodEstimator, BayesianEstimator
model = BayesianModel([('D', 'G'), ('I', 'G'), ('I', 'S'), ('G', 'L')])
model.fit(data, estimator=MaximumLikelihoodEstimator)
for cpd in model.get_cpds():
    print("CPD of {variable}:".format(variable=cpd.variable))
    print(cpd)

结果如下:

CPD of D:
+------+-------+
| D(0) | 0.501 |
+------+-------+
| D(1) | 0.499 |
+------+-------+
CPD of G:
+------+------+----------------+----------------+----------------+
| D    | D(0) | D(0)           | D(1)           | D(1)           |
+------+------+----------------+----------------+----------------+
| I    | I(0) | I(1)           | I(0)           | I(1)           |
+------+------+----------------+----------------+----------------+
| G(0) | 0.48 | 0.509960159363 | 0.444915254237 | 0.551330798479 |
+------+------+----------------+----------------+----------------+
| G(1) | 0.52 | 0.490039840637 | 0.555084745763 | 0.448669201521 |
+------+------+----------------+----------------+----------------+
CPD of I:
+------+-------+
| I(0) | 0.486 |
+------+-------+
| I(1) | 0.514 |
+------+-------+
CPD of L:
+------+----------------+----------------+
| G    | G(0)           | G(1)           |
+------+----------------+----------------+
| L(0) | 0.489959839357 | 0.501992031873 |
+------+----------------+----------------+
| L(1) | 0.510040160643 | 0.498007968127 |
+------+----------------+----------------+
CPD of S:
+------+----------------+----------------+
| I    | I(0)           | I(1)           |
+------+----------------+----------------+
| S(0) | 0.512345679012 | 0.468871595331 |
+------+----------------+----------------+
| S(1) | 0.487654320988 | 0.531128404669 |
+------+----------------+----------------+

参数学习:

pseudo_counts = {'D': [300, 700], 'I': [500, 500], 'G': [800, 200], 'L': [500, 500], 'S': [400, 600]}
model.fit(data, estimator=BayesianEstimator, prior_type='dirichlet', pseudo_counts=pseudo_counts)
for cpd in model.get_cpds():
    print("CPD of {variable}:".format(variable=cpd.variable))
    print(cpd)

结果如下:

CPD of D:
+------+--------+
| D(0) | 0.4005 |
+------+--------+
| D(1) | 0.5995 |
+------+--------+
CPD of G:
+------+-------+----------------+----------------+----------------+
| D    | D(0)  | D(0)           | D(1)           | D(1)           |
+------+-------+----------------+----------------+----------------+
| I    | I(0)  | I(1)           | I(0)           | I(1)           |
+------+-------+----------------+----------------+----------------+
| G(0) | 0.736 | 0.741806554756 | 0.732200647249 | 0.748218527316 |
+------+-------+----------------+----------------+----------------+
| G(1) | 0.264 | 0.258193445244 | 0.267799352751 | 0.251781472684 |
+------+-------+----------------+----------------+----------------+
CPD of I:
+------+-------+
| I(0) | 0.493 |
+------+-------+
| I(1) | 0.507 |
+------+-------+
CPD of L:
+------+----------------+----------------+
| G    | G(0)           | G(1)           |
+------+----------------+----------------+
| L(0) | 0.496662216288 | 0.500665778961 |
+------+----------------+----------------+
| L(1) | 0.503337783712 | 0.499334221039 |
+------+----------------+----------------+
CPD of S:
+------+----------------+----------------+
| I    | I(0)           | I(1)           |
+------+----------------+----------------+
| S(0) | 0.436742934051 | 0.423381770145 |
+------+----------------+----------------+
| S(1) | 0.563257065949 | 0.576618229855 |
+------+----------------+----------------+