X₁, X2, X₂ This network is shown in Fig. 1, which shows that three of the variables (X1, X2, X₁) have observed values, with the other three (Y, Y₁, Y₁) being unobserved. Your goal in this question is use variable elimination to compute p{} = END|X₁ = a, X₂ = b, X₂ = 4). All of these values are described below. MY WY Figure 1: Bayesian network for question 1 Assume that the Y, variables are all defined over three different values (Z. Z., and END), where the same conditional distribution (KY) is used to define the distribution of Y. given its parent Y.- (where there's a prior distribution over Y;). Further assume that the X, variables are all defined over three different values (a, b, and #), where the same conditional distribution (X,Y) is used to define the distribution of X, given its parent Y. These distributions are labeled (for case) on the arcs of the figure, and are provided below in Table 1 (in both sub-tables, each row is a different distribution): Y- 280 z, Z, END 15 805 16 35 .05 Z Z END 21 END 001 END 0 0 1 (c) The prior distribution for Y₁. Ca) The definition of p for 15 (b) The definition of p, for i 2.3 Table 1: The distributions needed for question 1. In order to compute ( NDX, X, X,), you'll need to eliminate both Y; and Y. To make the computations a bit easier, we'll start with Y, and then eliminate to Y. Assume that the factors have been defined as given in Table 2: Factor (X) Definition b) 1(K) P() fa(Y) P) f(YY) (135) Table 2: Initial factor definitions. (a) Eliminate Y. Name the resulting factor as f (i) Identify the factors that must be considered to eliminate Y. (ii) Identify the variable(s) that will be defined over. () Write the formula for f (in terms of the factors you've identified). (b) Eliminate Y. Name the resulting factor as g (i) Identify the factors that must be considered to eliminate Y₂. (i) Identify the variable(s) that will be defined over. () Write the formula for g (in terms of the factors you've identified). (c) Answer the original question: what is the posterior distribution p(Y, ENDX₁ = XbX) in terms of factors, after the variables Y, and are eliminated For example, in the first example in the class lecture, the final posterior distribution was: or Pampering Smokete Report true) (Tampering) × f„(Tampering) Etamping f(♥) × fu(V) P(Tampering Smoke-true / Report = true)=x f₁(Tampering) × f,(Tampering) In this assignment, you have been asked for a specific value of the query variable. So if I wanted the posterior probability of 'done' for tempering, the equation would be: P(Tampering done Smoke-true A Report = true)- ax fe(Tampering done) f(Tampering - done)

X₁, X2, X₂ This network is shown in Fig. 1, which shows that three of the variables (X1, X2, X₁) have observed values, with the other three (Y, Y₁, Y₁) being unobserved. Your goal in this question is use variable elimination to compute p{} = END|X₁ = a, X₂ = b, X₂ = 4). All of these values are described below. MY WY Figure 1: Bayesian network for question 1 Assume that the Y, variables are all defined over three different values (Z. Z., and END), where the same conditional distribution (KY) is used to define the distribution of Y. given its parent Y.- (where there's a prior distribution over Y;). Further assume that the X, variables are all defined over three different values (a, b, and #), where the same conditional distribution (X,Y) is used to define the distribution of X, given its parent Y. These distributions are labeled (for case) on the arcs of the figure, and are provided below in Table 1 (in both sub-tables, each row is a different distribution): Y- 280 z, Z, END 15 805 16 35 .05 Z Z END 21 END 001 END 0 0 1 (c) The prior distribution for Y₁. Ca) The definition of p for 15 (b) The definition of p, for i 2.3 Table 1: The distributions needed for question 1. In order to compute ( NDX, X, X,), you'll need to eliminate both Y; and Y. To make the computations a bit easier, we'll start with Y, and then eliminate to Y. Assume that the factors have been defined as given in Table 2: Factor (X) Definition b) 1(K) P() fa(Y) P) f(YY) (135) Table 2: Initial factor definitions. (a) Eliminate Y. Name the resulting factor as f (i) Identify the factors that must be considered to eliminate Y. (ii) Identify the variable(s) that will be defined over. () Write the formula for f (in terms of the factors you've identified). (b) Eliminate Y. Name the resulting factor as g (i) Identify the factors that must be considered to eliminate Y₂. (i) Identify the variable(s) that will be defined over. () Write the formula for g (in terms of the factors you've identified). (c) Answer the original question: what is the posterior distribution p(Y, ENDX₁ = XbX) in terms of factors, after the variables Y, and are eliminated For example, in the first example in the class lecture, the final posterior distribution was: or Pampering Smokete Report true) (Tampering) × f„(Tampering) Etamping f(♥) × fu(V) P(Tampering Smoke-true / Report = true)=x f₁(Tampering) × f,(Tampering) In this assignment, you have been asked for a specific value of the query variable. So if I wanted the posterior probability of 'done' for tempering, the equation would be: P(Tampering done Smoke-true A Report = true)- ax fe(Tampering done) f(Tampering - done)