MetropolisHastings is a Markov Chain Monte Carlo (MCMC) algorithm for approximate inference. It allows for sampling from a probability distribution where direct sampling is difficult  usually owing to the presence of an intractable integral.
MH consists of a proposal distribution $q\left(\theta^{'}\mid\theta\right)$ to draw a parameter value. To decide whether $\theta^{'}$ is accepted or rejected, we then calculate a ratio:
$$ \frac{p\left(\theta^{'}\mid{D}\right)}{p\left(\theta\mid{D}\right)} $$
We then draw a random number $r \in \left[0, 1\right]$ and accept if it is under the ratio, reject otherwise. If we accept, we set $\theta_{i} = \theta^{'}$ and repeat.
By the end we have a sample of $\theta$ values that we can use to form quantities over an approximate posterior, such as the expectation and uncertainty bounds. In practice, we typically have a period of tuning to achieve an acceptable acceptance ratio for the algorithm, as well as a warmup period to reduce bias towards initialization values.
Image: Samuel Hudec
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