I was learning Hidden Markov model, and encountered this theory about convergence of Markov model.

For example, consider a weather model, where on a first-day probability of weather being sunny was 0.9 while that of being rainy - 0.1. The transition distribution probabilities were - probability of sunny on current day given weather was sunny on previous day - i.e P(sun/sun) = 0.9. Similarly other probabilities were - P(sun/rain) = 0.3, P(rain/rain) = 0.7, and P(rain/sun) = 0.1.

Using bayes inference, we can find probability of distribution of weather on day Xt. It was mentioned that, after some point (i.e at state t = infinite), the probability distribution of weather on a day converges to p(sun) = 0.75 and p(rain) = 0.25.

Can someone explain this convergence mathematically? How does it converge to this distribution? It doesn't matter what's the distribution on day 1, the model always converges to the same values.


This isn't a hidden Markov model; this is an ordinary Markov model. Take a look at Wikipedia's article on Markov chains and specifically the notion of a steady-state distribution (or stationary distribution), or read about the subject in your favorite textbook -- there are many that cover Markov chains.

  • $\begingroup$ @mihir-thatte: To complete what is said above, I think you are confusing two common examples for Markov chains and hidden Markov models. Here is a good reading presenting both: di.ubi.pt/~jpaulo/competence/tutorials/hmm-tutorial-1.pdf. In the HMM case, the variable of interest is hidden and observed through another variable (like the use of an umbrella or the number of ice-creams sold could be observed to infer some information about the weather using an HMM) $\endgroup$ – Eskapp Nov 4 '16 at 17:02

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