Naïve Bayes Classifier. On April 15, 1912, the largest passenger liner ever made collided with an iceberg during her maiden voyage. When the Titanic sank it killed 1502 out of 2224 passengers and crew. This sensational tragedy shocked the international community and led to better safety regulations for ships. One of the reasons that the shipwreck resulted in such a loss of life was that there were not enough lifeboats for the passengers and crew. Although there was some element of luck involved in surviving the sinking, some groups of people were more likely to survive than others. Some demographic data are collected for 887 passengers and are provided in titanic. csv. The price class, gender (1 for Male and 0 for Female) and age are recorded for each survived (1) or killed (0) passenger. We use random variables S, C, G, and A for sur- vival status, price class, gender and age, respectively. In this problem, we are going to build a popular Naive Bayes Classifier to predict S given C, G, and A. (a) Estimate the PMFs for S, C, G, and A by finding the fraction of each realization of these random variables among all data. Plot these PMFs.
(b) We are interested in how C, G, and A affect S, in the context of the Naïve Bayes Classifier; the first step is to estimate the conditional PMF conditioning on the outcome of interest, i. e., survival or not. Estimate and plot the conditional PMFs for C, G, and A conditioned on S.
(c) In the Naïve Bayes Classifier, we use the conditional independence assump- tion. For example, if C, G, and A are conditionally independent on S, then PC, G, A S = 0) = P(C|S = 0)P(G|S = 0)P(A|S = 0) and P(C, G, A|S = 1) = P(C|S = 1)P(G|S = 1)P(A|S 1)P(A|S = 1). Using this assumption, compute P(S = 0, C = 1, G = 0, A < 40) and P(S = 1, C = 1, G = 0, A < 40) based on your estimations in (a) and (b).
(d) Based on your result in (c), compute P(S = 0C = 1, G = 0, A < 40) and P(S = 1 C = 1, G = 0, A < 40). Predict whether a female whose age is under 40 and who is in first class will survive or not.