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4 Naive Bayes [10 points] Below is a table relating two input variables X1 and X; with one output value Y. Assume that X1 is conditionally independent of X2 given Y. Assume further that range(X1) = {0, l} and range(X2) = {0,1,2} X1 X2 Y |X1 X2 Y |X1 X2 Y 0 1 0 1 2 1 1 0 2 0 0 0 1 0 1 1 0 2 0 1 0 0 2 1 1 0 2 1 2 0 D 0 1 1 1 1 A Categorical random variable is a generalization of a Bernoulli random variable. Whereas a Bernoulli has two possible outcomes, a Categorical has it possible outcomes and is parameterized by p1...\" subject to the constraint 25' pi = 1. When I: = 2, it is equivalent to the Bernoulli. For example, W ~ Categorieal(0.3,0.7) is the same as W ~ Bernoulh'(0.7). a. (3 points) Assume X1 conditioned on Y has the following distributions. Note that the superscripts (e_g_ the "a" in pf\") do not indicate an exponent, but are just used to distinguish the value of Y to A Categorical random variable is a generalization of a Bernoulli random variable. Whereas a Bernoulli has two possible outcomes, a Categorical has k possible outcomes and is parameterized by P1...Pk subject to the constraint _, p; = 1. When k = 2, it is equivalent to the Bernoulli. For example, W ~ Categorical(0.3, 0.7) is the same as W ~ Bernoulli(0.7). a. (3 points) Assume X1 conditioned on Y has the following distributions. Note that the superscripts (e.g. the "a" in p,") do not indicate an exponent, but are just used to distinguish the value of Y to which the parameter corresponds. (a) (XilY =0) ~ Categorical(p,", P2 (b) (Xily = 1) ~ Categorical(p, , P2) (c) (XilY =2) ~ Categorical(P,", P2) Compute the values for P2 , P2 , P2 (b ) DC)using Laplace add-one smoothing. Answer. b. (3 points) Assume X2 conditioned on Y has the following distributions. (Yoly 1((d) (d) (d))b. (3 points) Assume X2 conditioned on Y has the following distributions. (d) (X2|Y =0) ~ Categorical(P, , P2 , P3 p (d) ) (e) (X2|Y = 1) ~ Categorical(p, , P2 , P3 ) (f ) (X2|Y = 2) ~ Categorical(P, , P2 , P3 (f) ) Compute the values for p,", p3 , p3 using Laplace add-one smoothing. Answer.- 6 - c. (4 points) Compute argmaxy P(Y = y(X1 = 0, X2 = 2) using Laplace add-one smoothing. As part of your calculation, we provide here the values of the following expressions to reduce the algebra you require. 15 P(Y = 1, X1 = 0, X2 = 2) = 224 1 P(Y = 2, X1 = 0, X2 = 2) = 120