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6. Consider the following hidden Markov model which consists of three observations (x1, x2, x3) and three hidden states (y1, y2, y3). What is the joint probability distribution P(x1, x2, x3, y1, y2, y3)? (V1 V2 Уз X2 X2 Х3. (There are 6 nodes y1, y2 y3, x1, x2, and x3. The directed edges which have no weight are between Y1 to X1, Y2 to X2, Y3 to X3 and Y1 to Y2 and Y2 to Y3.) Ply1)P(x1|y1) P(x2|y2) P(x3|y3) P(x1|y1)Pſy2|y1)P(x2|y2)Ply3|y2)P(x3|y3) Ply1)P(x1|y1) Ply2|y1)P(x2|y2)Ply3|y2)P(x3|y3) P(x1|y1) P(x2|y2) P(x3|y3)
9. Given the following HMM model, where symbols X, Y and Z represent the possible observations in the states. In a circle representing a state, the symbols are equally possible observations in that state. How much is P(01 = X,O2 = X, O3 = Z S1S3S3)? 1/3 S2 1/3 XY ZY 2/3 2/3 1/3 ZX 1/3 1/3 (There are 3 nodes XY, ZY, ZX with the following configuration: XY is in state S1 and has a directed edge from XY to ZY with weight 1/3; there is another directed edge from XY to ZX with weight 2/3; ZY is in state 52 and has a directed edge from ZY to ZX with weight 2/3; there is a directed edge from ZY to XY with weight 1/3; ZX is in state 53 and there is a self loop at ZX with weight 1/3; there is a directed edge from ZX to ZY with weight 1/3; there is a directed edge from ZX to XY with weight 1/3.) 2/3 1/8 14 12
10. Given the following HMM model, where symbols X, Y and Z represent the possible observations in the states. In a circle representing a state, the symbols are equally possible observations in that state. 1/3 S S2 1/3 XY ZY 2/3 213 1/3 ZX 1/3 Sz 1/3 (There are 3 nodes XY, ZY, ZX with the following configuration: XY is in state S1 and has a directed edge from XY to ZY with weight 1/3; there is another directed edge from XY to ZX with weight 2/3; ZY is in state S2 and has a directed edge from ZY to ZX with weight 2/3; there is a directed edge from ZY to XY with weight 1/3; ZX is in state 53 and there is a self loop at ZX with weight 1/3; there is a directed edge from ZX to ZY with weight 1/3; there is a directed edge from ZX to XY with weight 1/3.) Suppose that the first state is S1, and we have a path of length 100, how many different possible state paths are there in total? 399 499 ООО 3100 4100