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I asked this yesterday on math.stackexchange. It didn't get any attention so I'm reposting here. Hope this is in line with policy.

I have data which I model as $50$ rounds of a Markov chain with linear transition probabilities $$ P_{ij} = \begin{cases} \alpha_1 i + \alpha_0 & \mbox{ if } j=i-1 \\ \beta_1 i + \beta_0 & \mbox{ if } j=i+1 \\ 1-(\alpha_1+\beta_1) i - (\alpha_0+\beta_0) & \mbox{ if } j=i \\ 0 & \mbox{ otherwise} \end{cases} $$

Note:

  1. The linear probabilities mean this can't be well defined on $\mathbb{Z}$, to resolve that I bound $0\le i\le 50$.

  2. The resultant constraints on $\alpha_{0,1},\beta_{0,1}$ are dependent and not very nice. I don't mind making do with neater box constraints because I have some initial guesses for them.

What I have are samples drawn from the probability distribution $P^{50} e_k$ resulting from this process for several known starting points $k$.

I currently use (somewhat arbitrarily) a simulated annealing type process to minimize the covariance of the empirical data and the computed probability distribution given $\alpha_{0,1},\beta_{0,1}$, but I would like something more robust. My question is:

  • Has this type of fitting problem been studied? If so, where can I find results on it?
  • Are there any similar results on fitting Markov chains that I can use to get ideas on how I could look for a solution?
  • Are there other optimization methods (not necessarily assuming the data results from a Markov chain) I should look into?
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  • $\begingroup$ This is a Birth-Death chain. You need to fix formulas for i = 1 and i = last row, since i=i has no j-1 and i = last row has no j = i+ 1. Then impose constraints on entries of $P_{ij}$ to be in [0,1] .Then minimize some criterion, for example, (possibly weighted) least squares, of results vs. $P^{50}e_k$. So essentially a linearly constrained nonlinear least squares problem. It may be non-convex and have non-global local minima. $\endgroup$
    – Mark L. Stone
    Commented Aug 29, 2016 at 14:58

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