#StackBounty: #mcmc #metropolis-hastings #marginal #particle-filter #sequential-monte-carlo Using the pseudo marginal approach for esti…

Bounty: 50

I have $$K>0$$ iid unknown Markov chains $${X_n^k : n in mathbb{N}}, k=1, dots, K$$ on a discrete state space $$S_X = {1,2,3}$$, each chain runs and gives rise to observations of the form $${Y_n^k : n in mathbb{N}}, k=1, dots, K$$ on a discrete state space $$S_Y = {1,2}$$.

Given a static parameter $$theta in mathbb{R}$$, completely independent of $$K$$ (also static), the transition probability matrix $$P_{theta}:= P_{i,j} = mathbb{P}(X_n=j|X_{n-1}=i)$$ and emission matrix $$B_{theta}:= B_{i,j} = mathbb{P}(Y_n = j|X_n=i)$$ of each chain is known.

At each time point I observe $$z_n = sum_{k=1}^K y_n^k$$. Over a time series of length $$N$$, I am able to estimate the marginal log-likelihood $$log hat{p}(z_1, dots, z_N|theta,K)$$ given any $$theta$$ and $$K$$ using the bootstrap particle filter with $$N_{p,k}$$ particles for any given $$k$$.

I would like to sample from the posterior $$p(theta,K|z_1,dots,z_N)$$.
Now, I know that I can use a pseudo-marginal MCMC approach to sample from the posterior $$p(theta|z_1,dots,z_N,K)$$ using a particle estimate of the marginal log-likelihood. At the moment, I am also using this chain to sample from $$K$$ (using a random walk on the integers), and it seems to be recovering $$K$$. However, I am unsure as to whether or not this part needs a reversible jump element. The state space of my target distribution is $$mathbb{R} times mathbb{N}$$ and does therefore not vary in dimension, however, since the particle filter requires $$k times N_{p,k}$$ samples of Markov chains at each iteration, I am still unsure whether the algorithm itself reaches the correct stationary distribution in this way.

Any justification of a reversible jump element or clarification of my current method would be very highly appreciated. Thanks!

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