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Let $H_1$ and $H_2$ be two instances of a finite Hidden Markov Model (HMM) $H$. That is, $H_1$ and $H_2$ have identical state spaces $Q$ as well as identical transition $A$ and emission probabilities $B$. However, they have different initial distributions $\pi_1$ and $\pi_2$, respectively. Let $(O_0, O_1, O_2\ldots)$ be a sequence of observations applied to both $H_1$ and $H_2$.

Has anyone considered the question how long does it take both $H_1$ and $H_2$ to forget their initial distributions? That means their belief states $b_1: Q \rightarrow [0, 1]$ and $b_2: Q \rightarrow [0, 1]$ become arbitrarily close to each other?

By "arbitrarily close" here I mean "Given any $\epsilon > 0$, what is $N(\epsilon) > 0$ such that, for all observations $O_n, n > N(\epsilon):\ dist(b_1, b_2) < \epsilon$ for some distance function $dist$?"

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  • $\begingroup$ If you really mean "arbitrarily close" then even with exponential decay of initial conditions it must take arbitrarily long. $\endgroup$
    – Glen_b
    Commented Feb 1, 2015 at 23:32
  • $\begingroup$ @Glen_b: I added clarification of that point. $\endgroup$
    – Ahmed Nassar
    Commented Feb 1, 2015 at 23:38

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