Part B

Bayesian Statistics over Bayesian Networks and Inferencing them using Pgmpy

Python library: $!$pip install pgmpy

Theorem of Bayes

Bayes' theorem is a fundamental concept in Bayesian statistics, as it is used

by Bayesian methods to update probabilities, or degrees of belief, following the

acquisition of new data. The conditional probability of A occurring given that

$B$ occurs is represented as follows:

$P(A|B)=\frac{P(B|A)*P(A)}{P(B)}$

The law of total probability can be used to calculate the probability of the

evidence $P(B).$ If $\{A_1,A_2,$dots,$A_n\}$ is a partition of the sample space, which is

the collection of all experimental results, then:

$P(B)={\displaystyle \underset{i}{\overset{?}{}}}P(B|A_i)*P(A_i)$

Bayesian Network

A Bayesian network is a probabilistic graphical model, a kind of statistical model

that utilizes a directed acyclic graph $($DAG$)$ to describe a set of variables and

their conditional relationships. Bayesian networks are perfect for analyzing past

occurrences and estimating the probability that any of numerous known causes

had a role.

We can use Bayes' rule and the total probability theorem to infer probabili$-$

ties in a Bayesian network. Let's consider an example: students are required to

submit a project for Introduction in AI$,$ and the Professor attempts to predict

if students copied $($plagiarized$)$ the content. For this, the Professor would like

to create a statistical method that can take preemptive measures based on the

information given. Let's assume we have the following information:

L: A prediction from a NLP ML $($Natural Language Processing Machine

Learning$)$ model that can read the content of the Project and give a score

$($signalled$)($probability$)$ that this content is copied$/$plagiarized$.$

I: Another student $($colleague$)$ marks the material as inappropriate$/$plagiarism$.$

S: The Project was suspended before for any bad remarks.

N: Score $($Probability$)$ that the Project should not be considered $($not

scored$).$

R: Score $($Probability$)$ that the Student should be restricted for this lecture

$($ratrappage$).$

Using Pgmpy Python Library

We will use the pgmpy Python library to compute the probabilities. You can

follow the tutorial and analyze the existing functions at:

https:$//$pgmpy$.$org$/$models$/$bayesiannetwork$.$html

Import the libraries :

from pgmpy$.$models import BayesianModel

from pgmpy$.$factors.discrete import TabularCPD

from pgmpy$.$ inference import VariableElimination

import numpy as np

Question $1$

Create the Bayesian network using: bayesNet $=$ BayesianModel $()$

For all nodes: bayesNet.add$\_$node $("$Name$\_$Node"$)$

For all edges: bayesNet.add$\_$edge$("$Name$\_$Node$1",$ "Name$\_$Node$2")$

Add CPDs to each node using the function TabularCPD $().$

Check if the model is correctly created using bayesNet.check$\_$model$().$

Create a solver that uses variable elimination internally for inference:

solver $=$ VariableElimination $($bayesNet$)$