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 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 for 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 realization and understanding principle

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


# Suppose we have three variables X, Y, Z, each variable has two possible states (0 and 1)
# We have some observed data
data = np.array([[0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1]])


# Our model space includes six models: X->Y->Z, X->Z->Y, Y->X->Z, Y->Z->X, Z->X->Y, Z->Y->X
# We can use a 2x2x2 matrix to represent the conditional probability table (CPT) of each model


# For simplicity, we assume that all CPTs are the same
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)


# We can calculate the likelihood of each model
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]


# We assume that the prior probabilities of each model are equal
priors = [1/6 for _ in range(6)]


# We can use Bayes' theorem to calculate the posterior probability of each model
posteriors = [likelihood * prior for likelihood, prior in zip(likelihoods, priors)]


# We select the model with the highest posterior probability as the optimal model
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 Structure 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.

pgmpy is a Python library for implementing probabilistic graphical models, including Bayesian networks and Markov models. HillClimbSearch and BdeuScore in pgmpy are two common tools for Bayesian structure learning.

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, HillClimbSearch and BdeuScore can be used together for Bayesian structure learning. Specifically, we can use HillClimbSearch to search the model space and BdeuScore to evaluate the score of each model. 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. Loading and preprocessing 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.

# Create a fictitious data set
data = pd.DataFrame(np.random.randint(low=0, high=2, size=(5000, 3)), columns=['Diet','Exercise','Heart_Health'] )

3. Define 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 use HillClimbSearch and BdeuScore to accomplish this task:

# Use HillClimbSearch and 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 has already completed this task for us, we only need to print out the edges of the optimal model.