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()
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=1500,node_color='grey',font_size=10,font_weight='bold',arrowsize=20) 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("\n Approximate Probabilities of {}".format(query_variable)) print(query_result)
Thus, Gibb's Sampling( Approximate Inference method) is succuessfully implemented using python.