Inference

The basic concept of variable elimination is same as doing marginalization over Joint Distribution. But variable elimination avoids computing the Joint Distribution by doing marginalization over much smaller factors. So basically if we want to eliminate X from our distribution, then we compute the product of all the factors involving X and marginalize over them, thus allowing us to work on much smaller factors.

Inference Algorithms

The main categories for inference algorithms:

1. Exact Inference: These algorithms find the exact probability values for our queries.

2. Approximate Inference: These algorithms try to find approximate values by saving on computation.

Two common Inference algorithms with variable Elimination

1. Clique Tree Belief Propagation

2. Variable Elimination

Examples Inference

Example (1)

Lets load the Sprinkler data set and make some inferences.

What is the probability of wet grass given that it Rains, and the sprinkler is off and its cloudy: P(wet grass | rain=1, sprinkler=0, cloudy=1)?

import bnlearn
model = bnlearn.import_DAG('sprinkler')
q1 = bnlearn.inference.fit(model, variables=['Wet_Grass'], evidence={'Rain':1, 'Sprinkler':0, 'Cloudy':1})

The probability having wet grass is 0.9 and not-wet-gras is 0.1.

Wet_Grass	phi(Wet_Grass)
Wet_Grass(0)	0.1000
Wet_Grass(1)	0.9000

Example (2)

What is the probability of wet grass given and Rain given that the Sprinkler is on?

q2 = bnlearn.inference.fit(model, variables=['Wet_Grass','Rain'], evidence={'Sprinkler':1})

The highest probability is that in these condition, there is wet grass and no rain (P=0.63)

Wet_Grass	Rain	phi(Wet_Grass,Rain)
Wet_Grass(0)	Rain(0)	0.0700
Wet_Grass(0)	Rain(1)	0.0030
Wet_Grass(1)	Rain(0)	0.6300
Wet_Grass(1)	Rain(1)	0.2970

Example (3)

Given our model, what is the probability on lung cancer given that the person is a smoker and xray is negative? P(lung | smoker=1, xray=0)

# Lets create the dataset
model = bnlearn.import_DAG('asia')

Lets make the inference:

 q1 = bnlearn.inference.fit(model, variables=['lung'], evidence={'xray':0, 'smoke':1})

+---------+-------------+
| lung    |   phi(lung) |
+=========+=============+
| lung(0) |      0.1423 |
+---------+-------------+
| lung(1) |      0.8577 |
+---------+-------------+

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