Question

СО C1 C2 C3 HI H2 H3 Suppose the following conditional probability tables: Ct-1 Plq Ct-1) true .6 false .4 Ct true false P(ht|C) .8 .4 P(Co) .5 Given the observation sequence Hi = true, H, = false, and Hz = true, compute the following: (a) P(C3|h1:3) (b) P(C4|h1:3)

Answer 1

To compute the probabilities, we will use the Forward Algorithm, which is an iterative method for computing the probability of a sequence of observations.

Let's denote the forward probabilities as α_t(c_t) = P(c_t, h_1:t), where h_1:t is the sequence of observations from h_1 to h_t. We will compute the forward probabilities for each time step and then normalize them to obtain the desired probabilities.

1. Initialization: α_0(c_0) = P(c_0)
α_0(true) = 0.5
α_0(false) = 0.5

2. Iteration for t = 1 to 3:
a) t = 1 (h_1 = true)
α_1(true) = P(h_1=true|c_1=true) * [P(c_1=true|c_0=true) * α_0(true) + P(c_1=true|c_0=false) * α_0(false)]
= 0.8 * [0.6 * 0.5 + 0.4 * 0.5]
= 0.8 * 0.5
= 0.4
α_1(false) = P(h_1=true|c_1=false) * [P(c_1=false|c_0=true) * α_0(true) + P(c_1=false|c_0=false) * α_0(false)]
= 0.2 * [0.4 * 0.5 + 0.6 * 0.5]
= 0.2 * 0.5
= 0.1
Normalize α_1:
α_1(true) = 0.4 / (0.4 + 0.1) = 0.8
α_1(false) = 0.1 / (0.4 + 0.1) = 0.2

b) t = 2 (h_2 = false)
α_2(true) = P(h_2=false|c_2=true) * [P(c_2=true|c_1=true) * α_1(true) + P(c_2=true|c_1=false) * α_1(false)]
= 0.2 * [0.6 * 0.8 + 0.4 * 0