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.
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
alarm_model = BayesianNetwork(
[
("Burglary", "Alarm"),
("Earthquake", "Alarm"),
("Alarm", "JohnCalls"),
("Alarm", "MaryCalls"),
]
)
# Defining the parameters using CPT
from pgmpy.factors.discrete import TabularCPD
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_johncalls = TabularCPD(
variable="JohnCalls",
variable_card=2,
values=[[0.95, 0.1], [0.05, 0.9]],
evidence=["Alarm"],
evidence_card=[2],
)
cpd_marycalls = TabularCPD(
variable="MaryCalls",
variable_card=2,
values=[[0.1, 0.7], [0.9, 0.3]],
evidence=["Alarm"],
evidence_card=[2],
)
# Associating the parameters with the model structure
alarm_model.add_cpds(
cpd_burglary, cpd_earthquake, cpd_alarm, cpd_johncalls, cpd_marycalls
)
print("Bayesian Network Structure")
print(alarm_model)
G=nx.DiGraph()
nodes=['Burglary','Earthquake','JohnCalls','MaryCalls']
edges=[('Burglary','Alarm'),('Earthquake','Alarm'),('Alarm','JohnCalls'),('Alarm','MaryCalls')]
G.add_nodes_from(nodes)
G.add_edges_from(edges)
pos={
'Burglary':(0,0),
'Earthquake':(2,0),
'Alarm':(1,-2),
'JohnCalls':(0,-4),
'MaryCalls':(2,-4)
}
nx.draw(G,pos,with_labels=True,node_size=1500,node_color="skyblue",font_size=10,font_weight="bold",arrowsize=20)
plt.title("Bayesian Network: Burglar Alarm Problem")
plt.show()
gibbssampler=GibbsSampling(alarm_model)
num_samples=10000
samples=gibbssampler.sample(size=num_samples)
query_variable="Burglary"
query_result=samples[query_variable].value_counts(normalize=True)
print("\n Approximate probabilities of {}:".format(query_variable))
print(query_result)
i.)
ii.)
iii.)
Thus, Gibb's Sampling( Approximate Inference method) is succuessfully implemented using python.
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