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#Vague Outline  ## Slice Sampling  Slice sampling is a non-deterministic method used to sample from arbitrary curves. Specifically slice sampling belongs to the family of Markov chain Monte-Carlo (McMC) methods. 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. We can then use the samples to construct estimations of statistics on the underlying distribution (mean, mode, variance etc.) by simply calculating the equivalent sample-statistic.   Slice sampling in particular possesses several very desirable properties. It allows us to sample from integrals that are _proportional_ to our underlying distribution. There is no assumption that our probability mass sums to one. Consequently our intractable 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}  ## kkk  \cite{rennie2005regularized}  ## Bayesian Optimisation  ##  Bayesian Formulation Define the logistic sigmoid function \(\sigma(\alpha) = \frac{1}{1 + \exp(-\alpha)}\). A logistic regression models a binary-response variable with the following density function: