Problem $1$: Na$$ve Bayes and Bayesian Networks $(30$ points$)$

A team of health researchers are investigating lifestyle factors influencing the risk of heart disease. They have gathered the following dataset, which captures an individual's education level, alcohol consumption habits, and physical activity. You can assume this dataset is an accurate representation of the broader population.

$\backslash$table$[[\backslash$table$[[$Education$],[($Educated$/$Uneducated$)]],\backslash$table$[[$Alcohol Consumption$],[($Regular$/$Non$-$Regular$/$Never$)]],\backslash$table$[[$Physical Activity$],[($High$/$Low$)]],\backslash$table$[[$Heart Disease$],[($Yes$/$No$)]]],[$Educated$,$Regular,High,No$],[$Uneducated$,$Non$-$Regular,Low,Yes$],[$Educated$,$Never,High,No$],[$Educated$,$Regular,Low,Yes$],[$Uneducated$,$Non$-$Regular,High,No$],[$Uneducated$,$Never,Low,Yes$],[$Educated$,$Non$-$Regular,Low,Yes$],[$Uneducated$,$Regular,High,No$],[$Educated$,$Never,Low,Yes$],[$Uneducated$,$Regular,High,Yes$]]$

$(10$ points$)$ Implement the Na$$ve Bayes method to compute the probability that an individual has heart disease given that they are educated, consume alcohol regularly, and have low physical activity.

$(10$ points$)$ Assume that $(1)$ alcohol consumption and physical activity are dependent on education and $(2)$ heart disease is dependent on alcohol consumption and physical activity but not directly on education. Construct a Bayesian network for this dataset, specifying the Conditional Probability Table $($CPT$)$ for each node. Based on your network, compute the probability that an individual has heart disease given the same conditions as above. $($You should not use observed value to calculate the probability.$)$

$1$

$(5$ points$)$ Compare the results obtained from Na$$ve Bayes and Bayesian networks. Discuss the assumptions made by each method and why it may or may not be reasonable for this dataset.

$(5$ points$)$ Modify the Bayesian network by adding a new variable "Diet" with levels "Healthy" and "Unhealthy". Discuss how this change might influence the risk of heart disease.