exists an unbiased, non-negative estimator of the likelihood $\widehat{p}_{\theta}(\mathbf{y}|u)$. We introduce explicitly the random variable $u \in \mathsf{U}$ used to construct this estimator, and we let $m_{\theta}(u)$ denote the probability density of $u$ on $\mathsf{U}$.

The pseudo-marginal method is then a standard MH algorithm operating in a non-standard extended space $\Theta \times \mathsf{U}$, with the *extended target* $$\begin{aligned}
    \pi( \theta, u | \mathbf{y} ) 
    = 
    \frac{ \widehat{p}_{\theta}( \mathbf{y} | u ) m_{\theta}(u) p(\theta) } { p(\mathbf{y}) } 
    =
    \frac{ \widehat{p}_{\theta}( \mathbf{y} | u ) m_{\theta}(u) p(\theta | \mathbf{y}) } { p_{\theta}(\mathbf{y}) },\end{aligned}$$ and proposal distribution $m_{\theta''}(u'') q(\theta'' | \theta')$.

Since the likelihood estimator is unbiased, $\mathbb{E}_{u|\theta}[\widehat{p}_{\theta}(\mathbf{y}|u)] = p_{\theta}( \mathbf{y} )$, it follows that the extended target admits $p(\theta | \mathbf{y})$ as a marginal. Hence, by simulating from the extended target $\pi( \theta, u | \mathbf{y} ) $ we obtain samples from the original target distribution $p(\theta | \mathbf{y})$ as a byproduct.

If the likelihood is estimated by using SMC (see Section \[sec:SMC\]) we obtain the PMH algorithm. The random variable $u$ then corresponds to all the weighted particles generated by the SMC algorithm. However, these random variables carry useful information, not only about the likelihood, but also about the geometry 