Bayesian Probabilistic Causal Model Principles and Code Practice

I. Introduction

In our daily lives, we often face situations where we need to understand cause and effect. For example, we might want to know whether a healthy diet leads to better heart health, or whether education level affects an individual's income level. However, determining these causal relationships can be a challenging problem. Fortunately, Bayesian probabilistic causal models provide us with a powerful tool to address this problem.

2. What is the Bayesian probabilistic causal model?

Bayesian probabilistic causal modeling is a method based on Bayesian statistics for determining causal relationships between variables. The core idea of this method is to use Bayes' theorem to calculate the posterior probability of each possible causal model, and then select the model with the highest posterior probability as the optimal model.

Bayes' theorem is a way of updating our uncertainty about an unknown quantity given some observed data. In Bayesian probabilistic causal models, we use Bayes' theorem to update our beliefs about each possible causal model.

3. Bayesian Structure Learning

Bayesian structure learning is an important component of Bayesian probabilistic causal models. Its goal is to find the structure of a probabilistic graphical model that best describes the observed data. This process includes defining the model space, calculating the posterior probability of each model, and selecting the optimal model.

1. Define the model space

In Bayesian structural learning, we need to define a model space, which is the set of all possible models. In Bayesian networks, each model is a specific network structure. This network structure is composed of nodes (representing variables) and directed edges (representing causal relationships).

2. Calculate the posterior probability of each model

Next, we need to calculate the posterior probability of each model. This requires knowing the prior probabilities of the model and the likelihood of the data.

The prior probability is our belief about each model before seeing the data. In general, we can assume that all models have equal prior probabilities, meaning we are fair to all models before we see the data.

The likelihood of data is the probability of observing the data given a model. We can use the data to calculate the likelihood of each model.

We can then use Bayes' theorem to calculate the posterior probability of each model. The posterior probability is our belief about each model after seeing the data.

3. Select the optimal model

Finally, we select the model with the highest posterior probability as the optimal model. This model is the one we believe best describes the observed data.

4. Simple implementation of understanding principles

This example assumes that the conditional probability table (CPT) of each model is known, and in actual Bayesian structure learning, the CPT usually needs to be learned from the data.

import numpy as np


# 假设我们有三个变量X, Y, Z，每个变量有两个可能的状态（0和1）
# 我们有一些观察到的数据
data = np.array([[0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1]])


# 我们的模型空间包括六个模型：X->Y->Z, X->Z->Y, Y->X->Z, Y->Z->X, Z->X->Y, Z->Y->X
# 我们可以用一个2x2x2的矩阵来表示每个模型的条件概率表（CPT）


# 为简化，我们假设所有CPTs都相同
cpt = np.array([[[0.5, 0.5],  # P(Z=0|X=0,Y=0), P(Z=1|X=0,Y=0)
                 [0.5, 0.5]], # P(Z=0|X=0,Y=1), P(Z=1|X=0,Y=1)
                [[0.5, 0.5],  # P(Z=0|X=1,Y=0), P(Z=1|X=1,Y=0)
                 [0.5, 0.5]]])# P(Z=0|X=1,Y=1), P(Z=1|X=1,Y=1)


# 我们可以计算每个模型的似然性
likelihoods = [np.product(cpt[data[:, i], data[:, j], data[:, k]]) for i in range(3) for j in range(3) if i != j for k in range(3) if k != i and k != j]


# 我们假设每个模型的先验概率都是相等的
priors = [1/6 for _ in range(6)]


# 我们可以使用贝叶斯定理来计算每个模型的后验概率
posteriors = [likelihood * prior for likelihood, prior in zip(likelihoods, priors)]


# 我们选择具有最高后验概率的模型作为最优模型
best_model = np.argmax(posteriors)


models = ["X->Y->Z", "X->Z->Y", "Y->X->Z", "Y->Z->X", "Z->X->Y", "Z->Y->X"]
print("The best model is " + models[best_model])

5. Practical Tutorial: Using Bayesian Structural Learning to Detect Causal Relationships

Next, we will walk through a hands-on tutorial demonstrating how to use Bayesian structural learning to detect causal relationships. In this tutorial, we will use Python's pgmpy library.

pgmpyis a Python library for implementing probabilistic graphical models, including Bayesian networks and Markov models. and pgmpyare two common tools used for Bayesian structure learning.HillClimbSearchBdeuScore

HillClimbSearch: Hill Climbing is an optimization algorithm used to find the optimal solution that satisfies certain constraints. In Bayesian structure learning, Hill Climbing search is used to search for the optimal model structure in the model space. Hill Climbing search starts with a random model structure, and then at each step, it attempts to modify the model structure (e.g., add, delete, or reverse edges) to improve the model's score. This process continues until no modifications can be found that would improve the score.
BdeuScore: BDeu (Bayesian Dirichlet equivalent uniform) score is a scoring function used to evaluate the structure of Bayesian network models. The BDeu score takes into account the fit and complexity of the model. Specifically, the BDeu score is the weighted sum of the model’s log-likelihood and the model’s complexity. In the BDeu score, the complexity of the model is estimated by counting the number of parameters of the model. An important feature of the BDeu score is that it has a parameter (called the equivalent sample size) that can be used to adjust the trade-off between fit and complexity.

In pgmpy, HillClimbSearchand BdeuScorecan be used together for Bayesian structure learning. Specifically, we can HillClimbSearchsearch the model space using , and BdeuScoreevaluate the score of each model using . Then, we select the model with the highest score as the optimal model.

1. Import necessary libraries

from pgmpy.models import BayesianModel
from pgmpy.estimators import BayesianEstimator, MaximumLikelihoodEstimator, BdeuScore, K2Score, BicScore
from pgmpy.estimators import HillClimbSearch, ExhaustiveSearch
from pgmpy.inference import BeliefPropagation
import numpy as np
import pandas as pd

2. Load and preprocess data

We will use a fictional data set that contains three variables: healthy diet, exercise, and heart health. Our goal is to determine the causal relationship between these three variables.

# 创建一个虚构的数据集
data = pd.DataFrame(np.random.randint(low=0, high=2, size=(5000, 3)), columns=['Diet','Exercise','Heart_Health'])

3. Define the model space

In Bayesian structural learning, we need to define a model space, which is the set of all possible models. In this example, our model space will include all possible directed acyclic graphs (DAGs).

4. Calculate the posterior probability of each model

Next, we need to calculate the posterior probability for each model. We can accomplish this HillClimbSearch using :BdeuScore

# 使用HillClimbSearch和BdeuScore
hc = HillClimbSearch(data, scoring_method=BdeuScore(data))
best_model = hc.estimate()
print(best_model.edges())

5. Select the optimal model

Finally, we select the model with the highest posterior probability as the optimal model. In this example, HillClimbSearch this task is already done for us, we just need to print out the edges of the optimal model.