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I am defining a model on a vector, $T$, of size $n$, wherein each element $t_i \in T$ is independent and either $0$ or $1$. This model depends on 3 other parameters, $q$ (also a vector of size $n$), $\theta$, and $\phi$ (both $\theta$ and $\phi$ are vectors of size $m$). All elements of all vectors are independent. The vectors $T$ and $q$ have all been observed so I am going to assume that $P(T)=c_1$ and $P(q) = c_2$ where $c_1$ and $c_2$ are constants. Additionally, I assume a uniform prior on $\theta$, and some arbitrary prior on $\phi$ that is easy to sample from.

I define my model as follows:

\begin{align*} P\left(T\vert q,\theta,\phi\right)=\prod\limits_{k=1}^m\prod\limits_{\{i\mid h(i)=k\}}\left(q_i\theta_k\right)^{t_i}\left(1-q_i\theta_k\right)^{1-t_i}\tag{1} \end{align*} Where $h$ is some function that maps an index in $T$ to an index in $\phi$.

Since the $t_i$ are $0$ or $1$, this is essentially a product of bernoulli distributions each with a different parameter $p=q_i\theta_k$.

Ultimately, my goal is to estimate \begin{align*} P\left(\phi\vert T, q\right) \propto P\left(T\vert q,\phi\right)P\left(\phi\right)\tag{2} \end{align*}

The actual values of the $\theta$ are not relevant to me so I would like to take the marginal distribution. \begin{align*} P\left(T\vert q, \phi\right)=\prod\limits_{k=1}^m\int_1^0\prod\limits_{\{i\mid h(i)=k\}}\left(q_i\theta_k\right)^{t_i}\left(1-q_i\theta_k\right)^{1-t_i}d\theta_k\tag{3} \end{align*} This integral is quite difficult to do analytically (unless I can justify switching the order of the product and the integral) so my idea would be to try and sample directly from the distribution.

I am already doing MCMC to estimate the distribution in equation $(2)$ So I'd like to sample \begin{align*} \theta_k\sim U(0,1) \end{align*} Then plug this into equation $(1)$ and use that in my MCMC simulation for estimating equation $(2)$.

Does this sound like it would work? Is there a better way to estimate equation ${3}$?

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  • $\begingroup$ There is no $\phi$ in (1)... $\endgroup$
    – Xi'an
    Nov 12, 2021 at 20:43
  • $\begingroup$ @Xi'an, oops. thank you for pointing that out. The $\phi$ is encoded in the limit of the left most product and also in the function $h$ that maps indices of $T$ to indices of $\phi$. I'm not sure how best to notate this relation. $\endgroup$ Nov 13, 2021 at 22:25

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Using an unbiased estimate $\hat\pi(\phi)$ of the target density $\pi(\phi)$ in a Metropolis-Hastings algorithm is called the pseudo-marginal Metropolis–Hastings algorithm, first introduced by Beaumont (Genetics, 2003) and later extended by Andrieu and Roberts (Annals of Statistics, 2009). It is validated by treating $\hat\pi(\phi)$ as an auxiliary variable.

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