To construct a python program to implement approximate inference using Gibbs Sampling.
Step 1: Bayesian Network Definition and CPDs:
- Define the Bayesian network structure using the BayesianNetwork class from pgmpy.models.
- Define Conditional Probability Distributions (CPDs) for each variable using the TabularCPD class.
- Add the CPDs to the network.
- Print the structure of the Bayesian network using the print(network) statement.
- Import the necessary libraries (networkx and matplotlib).
- Create a directed graph using networkx.DiGraph().
- Define the nodes and edges of the graph.
- Add nodes and edges to the graph.
- Optionally, define positions for the nodes.
- Use nx.draw() to visualize the graph using matplotlib.
- Initialize Gibbs Sampling for MCMC using the GibbsSampling class and provide the Bayesian network.
- Set the number of samples to be generated using num_samples.
- Use the sample() method of the GibbsSampling instance to perform MCMC sampling.
- Store the generated samples in the samples variable.
- Specify the variable for which you want to calculate the approximate probabilities (query_variable).
- Use .value_counts(normalize=True) on the samples of the query_variable to calculate approximate probabilities.
- Print the calculated approximate probabilities for the specified query_variable.
# Importing required libraries
from pgmpy.models import BayesianNetwork
from pgmpy.factors.discrete import TabularCPD
from pgmpy.sampling import GibbsSampling
import networkx as nx
import matplotlib.pyplot as plt
# Define bayesian network structure
network=BayesianNetwork([
('Burglary','Alarm'),
('Earthquake','Alarm'),
('Alarm','JohnCalls'),
('Alarm','MaryCalls')
])
# Define the conditional probability distributions
cpd_burglary = TabularCPD(variable='Burglary',variable_card=2,values=[[0.999],[0.001]])
cpd_earthquake = TabularCPD(variable='Earthquake',variable_card=2,values=[[0.998],[0.002]])
cpd_alarm = TabularCPD(variable ='Alarm',variable_card=2, values=[[0.999, 0.71, 0.06, 0.05],[0.001, 0.29, 0.94, 0.95]],evidence=['Burglary','Earthquake'],evidence_card=[2,2])
cpd_john_calls = TabularCPD(variable='JohnCalls',variable_card=2,values=[[0.95,0.1],[0.05,0.9]],evidence=['Alarm'],evidence_card=[2])
cpd_mary_calls = TabularCPD(variable='MaryCalls',variable_card=2,values=[[0.99,0.3],[0.01,0.7]],evidence=['Alarm'],evidence_card=[2])
# Add CPDs to the network
network.add_cpds(cpd_burglary,cpd_earthquake,cpd_alarm,cpd_john_calls,cpd_mary_calls)
# Print the Bayesian network structure
print("Bayesian Network Structure :")
print(network)
# Create a directed graph
G = nx.DiGraph()
# Define nodes and edges
nodes=['Burglary','Earthquake','Alarm','JohnCalls','MaryCalls']
edges=[('Burglary','Alarm'),('Earthquake','Alarm'),('Alarm','JohnCalls'),('Alarm','MaryCalls')]
# Add nodes and edges to the graph
G.add_nodes_from(nodes)
G.add_edges_from(edges)
# Set positions for nodes(optional)
pos ={
'Burglary':(0,0),
'Earthquake':(2,0),
'Alarm':(1,-2),
'JohnCalls':(0,-4),
'MaryCalls':(2,-4)
}
# Draw the graph
nx.draw(G,pos,with_labels=True,node_size=5000,node_color='yellow',font_size=10,font_weight='bold',arrowsize=12)
plt.title("Bayesian Network: Alarm Problem")
plt.show()
# Initialize Gibbs sampling for MCMC
gibbs_sampler =GibbsSampling(network)
# Set the number of samples
num_samples=10000
# Perform MCMC sampling
samples=gibbs_sampler.sample(size=num_samples)
# Calculate approximate probabilities based on the samples
query_variable='Burglary'
query_result= samples[query_variable].value_counts(normalize=True)
# Print the approximate probabilities
print("\n Approximate Probabilities of {}".format(query_variable))
print(query_result)
Thus, Gibb's Sampling( Approximate Inference method) is succuessfully implemented using python.