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I am trying to understand what filter may be suitable for the following HMM:

The signal is a Wright-Fisher one-dimensional diffusion characterised by the SDE

$$dX_{t}=\frac{1}{2}\left(\alpha(1-x)-\beta x\right)dt+\sqrt{X_{t}(1-X_{t})}dB_{t}$$

with unknown parameters $\alpha$ and $\beta$. This process has stationary distribution $\mathrm{Beta}(\alpha,\beta)$. At discrete times Binomial observations are sampled conditionally on the value of the process $X_{t}=x$, $f_{Y_{t}}(y;x)=\text{Bin}(y,N;x)$.

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  • $\begingroup$ what a cool problem. so $X_t$ is bound on $[0 1]$ a binomial would have a probability of success. Could you elaborate on the observations? $\endgroup$ – Stanley Pawlukiewicz Jun 10 '17 at 1:35
  • $\begingroup$ Alright, then observations are drawn at discrete times (random) with distribution $\mathbb{P}(Y_{t}=y|X_{t}=x)=\binom{N}{y}x^{y}(1-x)^{N-y}$. I guess I should first make inference on the parameters of the diffusion ($\alpha$ and $\beta$), and then move forward with a particle filter, but that's just a guess. $\endgroup$ – ric_fog Jun 10 '17 at 11:00
  • $\begingroup$ what is $N$ related to? N possible observations? $\endgroup$ – Stanley Pawlukiewicz Jun 10 '17 at 15:53
  • $\begingroup$ Exactly, and it's arbitrary. $\endgroup$ – ric_fog Jun 11 '17 at 7:40
  • $\begingroup$ Arbitrary as ? 1) N unknown but some value N, 2) N random with a prior distribution. or 3) a N $in$ ( 0 ,1, 2, 3, ...) or a family of sampling distributions. It seems like you need to have some idea of N to solve this. $\endgroup$ – Stanley Pawlukiewicz Jun 11 '17 at 21:01

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