Hidden Markov Model Calculation

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Discussion Overview

The discussion revolves around calculating the probability P(X3=A) in a hidden Markov model defined with two states, A and B. Participants explore the application of the Chapman-Kolmogorov equations and the Markov property in this context.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • One participant requests assistance with calculating P(X3=A) given the initial probabilities and transition probabilities.
  • Another participant suggests that the question is aimed at understanding hidden Markov models and encourages the original poster to refer to their notes for examples.
  • A participant expresses familiarity with hidden Markov models but indicates confusion regarding the specific calculation, mentioning the use of the Chapman-Kolmogorov equations.
  • Another participant proposes a simplification of the problem by deriving additional transition probabilities and suggests representing the probabilities in matrix form to facilitate the calculation.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and approaches to the problem, with no consensus on the best method to calculate P(X3=A) or clarity on the steps involved.

Contextual Notes

Some participants reference the Chapman-Kolmogorov equations and the Markov property, but there is uncertainty regarding the application of these concepts to derive the desired probability. The discussion includes attempts to clarify the problem without resolving the calculation steps.

betsyrocamora
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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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