Impara Inference in Markov Random Fields | Markov Random Fields: Undirected Models

To compute marginal probabilities in a Markov Random Field (MRF), you use the method of summing over all possible configurations of the hidden variables. This process is often called brute-force enumeration because it involves evaluating the joint probability for every combination of variable assignments, then summing the probabilities where the variable of interest takes on a specific value. For example, if you want the marginal probability of variable $A$ , you sum the joint probabilities for all values of the other variables ( $B$ and $C$ ), while keeping $A$ fixed at a particular value. This approach is simple to implement for small graphs, but quickly becomes computationally expensive as the number of variables increases.


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import numpy as np

# Possible values for each variable (binary, for simplicity)
values = [0, 1]

# Define factor tables for the MRF with variables A, B, C
# phi1: factor on (A, B)
phi1 = np.array([[0.9, 0.1],   # phi1[A=0, B=0], phi1[A=0, B=1]
                 [0.2, 0.8]])  # phi1[A=1, B=0], phi1[A=1, B=1]

# phi2: factor on (B, C)
phi2 = np.array([[0.3, 0.7],   # phi2[B=0, C=0], phi2[B=0, C=1]
                 [0.6, 0.4]])  # phi2[B=1, C=0], phi2[B=1, C=1]

# phi3: factor on (C, A)
phi3 = np.array([[0.5, 0.5],   # phi3[C=0, A=0], phi3[C=0, A=1]
                 [0.4, 0.6]])  # phi3[C=1, A=0], phi3[C=1, A=1]

def joint_prob(a, b, c):
    # Multiply the values from each factor table for the given assignment
    return phi1[a, b] * phi2[b, c] * phi3[c, a]

# Compute the unnormalized marginal for A
unnormalized_marginal_A = np.zeros(2)
for a in values:
    total = 0.0
    for b in values:
        for c in values:
            total += joint_prob(a, b, c)
    unnormalized_marginal_A[a] = total

# Normalize to get probabilities
marginal_A = unnormalized_marginal_A / np.sum(unnormalized_marginal_A)
print("Marginal probability P(A):", marginal_A)

This code demonstrates how to compute the marginal probability of variable A in a simple MRF with three binary variables: A, B, and C. The code defines factor tables for each edge in the undirected graph. The function joint_prob(a, b, c) multiplies the relevant factor values for a particular assignment. To find the marginal for A, the code loops over all possible values of B and C for each possible value of A, sums the resulting joint probabilities, and then normalizes the results so they sum to one. This brute-force approach ensures that every possible configuration is considered when calculating the marginal.

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Sezione 3. Capitolo 3

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              1234567891011121314151617181920212223242526272829303132333435
            
import numpy as np

# Possible values for each variable (binary, for simplicity)
values = [0, 1]

# Define factor tables for the MRF with variables A, B, C
# phi1: factor on (A, B)
phi1 = np.array([[0.9, 0.1],   # phi1[A=0, B=0], phi1[A=0, B=1]
                 [0.2, 0.8]])  # phi1[A=1, B=0], phi1[A=1, B=1]

# phi2: factor on (B, C)
phi2 = np.array([[0.3, 0.7],   # phi2[B=0, C=0], phi2[B=0, C=1]
                 [0.6, 0.4]])  # phi2[B=1, C=0], phi2[B=1, C=1]

# phi3: factor on (C, A)
phi3 = np.array([[0.5, 0.5],   # phi3[C=0, A=0], phi3[C=0, A=1]
                 [0.4, 0.6]])  # phi3[C=1, A=0], phi3[C=1, A=1]

def joint_prob(a, b, c):
    # Multiply the values from each factor table for the given assignment
    return phi1[a, b] * phi2[b, c] * phi3[c, a]

# Compute the unnormalized marginal for A
unnormalized_marginal_A = np.zeros(2)
for a in values:
    total = 0.0
    for b in values:
        for c in values:
            total += joint_prob(a, b, c)
    unnormalized_marginal_A[a] = total

# Normalize to get probabilities
marginal_A = unnormalized_marginal_A / np.sum(unnormalized_marginal_A)
print("Marginal probability P(A):", marginal_A)

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Grazie per i tuoi commenti!

Sezione 3. Capitolo 3