idea of how this can be used to construct the proposed PMH1 and PMH2 algorithms.

MH sampling with unbiased likelihoods {#sec:PMH}
-------------------------------------

The MH algorithm (see, e.g. @RobertCasella2004) is a member of the MCMC family for sampling from a target distribution $\pi(\theta)$ by simulating a carefully constructed Markov chain on $\Theta$. The chain is constructed in such a way that it admits the target as its unique stationary distribution.

The algorithm consists of two steps: (i) a new parameter $\theta''$ is sampled from a proposal distribution $q(\theta''|\theta')$ given the current state $\theta'$ and (ii) the current parameter is changed to $\theta''$ with probability $\alpha(\theta',\theta'')$, otherwise the chain remains at the current state. The acceptance probability is given by $$\begin{aligned}
    \alpha(\theta',\theta'') 
    = 
    1 \wedge
    \frac{ \pi(\theta'') }{\pi(\theta')} 
    \frac{ q(\theta' | \theta'')}{q(\theta'' | \theta')},
    \label{eq:MHacceptprob}\end{aligned}$$ where we use the notation $a \wedge b \triangleq \min\{a,b\}$.

In this paper, we have the parameter posterior distribution as the target distribution, i.e. $\pi(\theta) = p(\theta|\mathbf{y})$. This implies that the acceptance probability will depend explicitly on the intractable likelihood $p_\theta(\mathbf{y})$, preventing direct application of the MH algorithm to this problem. However, this difficulty can be circumvented by using a *pseudo-marginal* approach [@Beaumont2003; @AndrieuRoberts2009].

Assume that there 