Many problems depend on the evolution of states over time. Consider the state space diagram below (inspired
Question:
Many problems depend on the evolution of states over time. Consider the state space diagram below (inspired by the Hidden Markov Model) and answer the following questions. This week we are just getting comfortable with conditional probabilities. In this example, the only unknown parameter is θ.
(a) Write P (Xt = 1|Yt = 5, Xt+1 = 1) in terms of θ. You do not need to simplify your answer. Recall the directed graphical model in the figure above implies the conditional independence of Yt and Xt+1 on X, or equivalently P(Yt,Xt+1|Xt) = P(Yt|Xt)P(Xt+1|Xt).
(b) Express P(Yt+1) as a summation over x′,x ∈ {0,1} in terms of conditional probabilities of Yt+1, Xt+1, and Xt.
(c) Write P(Yt+1 = 5|Xt = 0) and P(Yt+1 = 5|Xt = 1)
(d) Write P (Yt = 5, Yt+1 = 5) in terms of θ. You do not need to simplify your answer. Recall the directed graphical model in the figure above implies the conditional independence of Yt and Yt+1 on X, or equivalently P(Yt,Yt+1|Xt) = P(Yt|Xt)P(Yt+1|Xt).
(e) Write P (Xt = 1|Yt = 5, Yt+1 = 5) in terms of θ. You do not need to simplify your answer. Recall the directed graphical model in the figure above implies the conditional independence of Yt and Yt+1 on X, or equivalently P(Yt,Yt+1|Xt) = P(Yt|Xt)P(Yt+1|Xt).
(f) Compare your answers in terms of θ from [Part 1] P(Xt = 1|Yt = 5), [Part 2] P(Xt = 1|Yt = 5,Xt+1 = 1), and [Part 3] P(Xt = 1|Yt = 5,Yt+1 = 5). Evaluate each answer at θ = 0.5, and explain how different information changes the probability. How would our results change if Xt+1|Xt ∼ Bernoulli(0.9)?
(g) What value of θ maximizes P(Yt = 5) and P(Yt = 5,Yt+1 = 5)? Note the maximum of a line with a non-zero slope over [0, 1] will be at either end-point.
Expert Answer:
It seems like youve provided a sequence of random variables and expressions involving them Lets brea... View the full answer
