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I have a discrete time Markov Chain $\{X_n: n \in \mathbb{N}_0\}$ with unknown transition matrix $P \in \mathbb{R}^{M \times M}$ on the state space $\mathcal{S}_X = \{1,2, \dots, M\}$, with $M \geq 2$. At each time step, the random variable $Y_n$ is defined by $Y_n = g(X_n)$, where $g: \mathcal{S}_X \to \{0,1\}$ is a deterministic function.

I can work out the emission probabilities $\mathbb{P}(Y_n = 0|X_n)$ and $ \mathbb{P}(Y_n=1|X_n)$, given $P$, i.e. these probabilities are known and computable functions of the entries of $P$.

Let's say I now observe $Z_n = \sum_{i =1}^k Y_n^i = \sum_{i =1}^k g(X_n^i)$, where $k \geq 1$ is unknown, where the sum is taken over $k$ iid Markov Chains $X_n^i$. Note that while I can write down the likelihood for $Z_1, Z_2, \dots$, the function is intractable for even moderately small values of $k$.

I am interested in estimating both $P$ and $k$ from my data $Z_n$. I know that I can use some sort of simulation based particle filter to estimate $P$ from the data in $Z$, however, I am wondering how I can estimate $k$. The value for $k$ is typically large (>1000). I am not sure that this problem requires advanced/transdimensional MCMC methods (e.g. reversible jump), since $P$ is independent of $k$, but I am not sure how to proceed as $k$ is large. It seems inefficient to fix $k$ and fit $P$ to each corresponding model, and then performing Bayesian model selection.

Does anyone know of any quick but accurate methods to perform model selection?

Thanks!

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