MHB Hidden Markov Model Calculation

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The discussion revolves around calculating the probability P(X3=A) in a hidden Markov model with two states, A and B. The user is confused about applying the Chapman-Kolmogorov equations and total probability. Another participant suggests simplifying the approach by using a transition matrix to represent the probabilities. They recommend leveraging the Markov property to compute the probabilities iteratively. The conversation emphasizes understanding the foundational concepts of hidden Markov models for accurate calculations.
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Can someone help me with this question.

Let us define a Markov hidden model with 2-states A and B, such that

P(X0=A)=0.6,P(X0=B)=0.4,P(X1=A/X0=A)=0.3,P(X1=B/X0=B)=0.8

what is the value of P(X3=A) ??
 
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Hello betsyrocamora,

Welcome to MHB! (Wave)

Can you show what you have tried and where you are stuck so our helpers have a better idea how best to help you?
 
betsyrocamora said:
Can someone help me with this question.

Let us define a Markov hidden model with 2-states A and B, such that

P(X0=A)=0.6,P(X0=B)=0.4,P(X1=A/X0=A)=0.3,P(X1=B/X0=B)=0.8

what is the value of P(X3=A) ??

Welcome to MHB, betsyrocamora! :)

This looks like a question that is intended to learn what a hidden Markov model actually is.
Do your notes perhaps contain a worked example?
Or perhaps an example for a Markov chain?
 
I like Serena said:
Welcome to MHB, betsyrocamora! :)

This looks like a question that is intended to learn what a hidden Markov model actually is.
Do your notes perhaps contain a worked example?
Or perhaps an example for a Markov chain?

No, I know what is a hidden markov model, but with this one I am a little lost, I have tried to let j denote the state A. Used p3ij:=P(X3=j|X0=i) that satisfies the Chapman Kolmogorov equations and after that I think is the total probabilities ecuation but I am lost in there hahha
 
betsyrocamora said:
No, I know what is a hidden markov model, but with this one I am a little lost, I have tried to let j denote the state A. Used p3ij:=P(X3=j|X0=i) that satisfies the Chapman Kolmogorov equations and after that I think is the total probabilities ecuation but I am lost in there hahha

It seems to me you're making it unnecessarily complex.

From your given data we can deduce that $P(X_1=B\ |\ X_0=A)=0.7$ and $P(X_1=A\ |\ X_0=B)=0.2$.
You can write these numbers in a matrix M.
Then, assuming the Markov property being independent of n, we get:
$$\begin{bmatrix}{P(X_n=A) \\ P(X_n=B)}\end{bmatrix} = M^n \begin{bmatrix}{P(X_0=A) \\ P(X_0=B)}\end{bmatrix}$$
 
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