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# Vague Outline  ## Original Slice Sampling  \cite{Tran_2015} Slice sampling is a non-deterministic method used to sample from arbitrary curves. Specifically it is a Markov chain Monte-Carlo (McMC) method. The general principle behind such methods is to construct a Markov chain that has the desired distribution as an equilibrium solution. They are used to obtain numeric approximations from multi-dimensional integrals for which no closed-form solution is readily available.  Slice sampling in particular possesses several very desirable properties. It allows us to sample from integrals that are _proportional_ to our underlying distribution. That is, there is no assumption that our probability mass sums to one. The intractability of the denominator above is of no concern -- we simply proceed using the scaled posterior.  Additionally, slice sampling has no free parameters. This is useful from a practical perspective, as it removes the need to spend time tuning and updating hyper-parameters for more complex models. It also impacts on performance; the more general Metropolis-Hastings model is highly sensitive to step-size parameters, which means unless we get it right we're going to perform very poorly on evaluation. In contrast, slice sampling adjusts the step-size to match the local shape of the density function.  \cite{murray2009elliptical}  \cite{Tran_2015}  ## Cross Validation