likelihood based on the actual measurements raised to the power $K$. Now, it is clear that the MLE of $\theta$ is the argmax for both $L(\theta;y^{(K)})$ and $L(\theta;y)$. By considering a prior distribution $\pi(\theta)$, we have the posterior distribution resulting from a data-cloned likelihood $$\pi(\theta|y^{(K)})\propto L(\theta;y^{(K)})\pi(\theta)=(L(\theta;y))^K\pi(\theta).
\label{eq:cloned-data-posterior}$$ It is easy to prove that for a large enough $K$ the mean of the posterior $\pi(\theta|y^{(K)})$ approaches the MLE of $\theta$ regardless the specific choice of $\pi(\theta)$ [@lele-dennis-lutscher] and a central limit theorem can be derived [@jacquier-johannes-polson; @lele2010estimability]. However, simulations experiments in [@lele-dennis-lutscher] show that using informative priors enable a more rapid convergence to the MLE.\
In algorithm \[alg:generalized-lele-algorithm\] we consider a generalization of the data-cloning Metropolis-Hastings algorithm for sampling from $\pi(\theta|y^{(K)})$ (we call it “generalization” simply because in [@lele-dennis-lutscher] the proposal distribution for $\theta$ is $\pi(\theta)$, while we consider a general proposal $u(\cdot)$). Consider a proposed value $\theta^{\#}$ generated from a distribution having density $u(\theta^{\#}|\theta^*)$ and a proposal $X^{\#(k)}$ generated from $v(X^{\#(k)}|\theta^{\#})$ which we set for convenience to be $v(X^{\#(k)}|\theta^{\#})\equiv p(X^{\#(k)}|\theta^{\#})$.

1\. Initialization: Fix a starting value $\theta^*$ or generate it from its prior $\pi(\theta)$ and set $\theta_1=\theta^*$. Set $j=1$. 2. Generate $K$ independent values of $X$, denoted $X^{*(1)},...,X^{*(K)}$ from $p(X|\theta^{*})$. 