Likelihood weighting

Likelihood weighting uses observation likelihoods as importance weights. If parameters are drawn independently from their prior distribution $\rho$, weight a draw by $L(\theta )$ and estimate a posterior expectation of $h$ by ${\sum }_{j}L({\theta }_j)h({\theta }_j)/{\sum }_{j}L({\theta }_j)$. With positive finite evidence and integrable weighted quantities, the strong law of large numbers proves consistency of this self-normalized importance sampling ratio. Finite-sample bias and concentration of the weights remain possible. A different proposal requires the additional prior density-to-proposal density ratio.