= Pseudo-marginal Metropolis–Hastings algorithm
= Pseudo-marginal algorithm
{synonym}
A <pseudo-marginal Metropolis–Hastings algorithm> replaces an intractable <likelihood function> by a nonnegative <unbiased likelihood estimator> and includes its auxiliary randomness in the state. If $\mathbb E[\widehat L(\theta,U)]=L(\theta)$ for $U\sim m_\theta$, the extended target is proportional to $p(\theta)\widehat L(\theta,u)m_\theta(u)$. An <independence> proposal for new auxiliary randomness cancels the $m_\theta$ factors in the <Metropolis–Hastings acceptance probability>. On rejection the old estimate must be retained. Integrating the extended target gives the desired marginal <posterior distribution> exactly.
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