In this lab, you will explore Bayesian networks by solving a probabilistic problem mathematically and verifying your results using Python code. The task involves a medical diagnosis scenario.

Scenario:

Consider a Bayesian network for diagnosing a disease based on various symptoms and test results. The network includes the following variables:

Disease $($D$)$: True if the patient has the disease.

Test $($T$)$: True if the test result is positive.

Symptom$1($S$1)$: True if the patient exhibits Symptom$1.$

Symptom$2($S$2)$: True if the patient exhibits Symptom$2.$

Network Structure:

Disease $($D$)$ is connected to Test $($T$).$

Disease $($D$)$ is connected to Symptom$1($S$1).$

Disease $($D$)$ is connected to Symptom$2($S$2).$

Conditional Probability Tables $($CPTs$)$:

$P(D)=0\mathrm{.}01$

P$($T$|$D$)$:

$P(T|D)=0\mathrm{.}95$

$P(T|notD)=0\mathrm{.}1$

P$($S$1|$D$)$:

$P(S1|D)=0\mathrm{.}8$

$P(S1|notD)=0\mathrm{.}3$

P$($S$2|$D$)$:

@$P(S2|D)=0\mathrm{.}6$

$P(S2|notD)=0\mathrm{.}2$

Task:

Mathematical Calculation:

Calculate the probability of the patient having the disease given that both symptoms and the test are positive, i$.$e$.,($ P$($D$|$S$1,$ S$2,$ T$)).$

Use the following steps for the calculation:

Calculate the joint probabilities $(P(S1,S2,T|D)).$

Use marginalization to find the probabilities of the hidden variable $($Disease$).$

Normalize the results to find the posterior probability $(P(D|S1,S2,T)).$

Python Code Implementation:

Adjust the provided Python code in the lecture to reflect the new network structure and conditional probabilities.

Calculate $($ P$($D$|$S$1,$ S$2,$ T$))$ using the code and verify it against your mathematical solution.