Markov chain sampling methods that adapt to characteristics of the distribution being sampled can be constructed using the principle that one can sample from a distribution by sampling uniformly from the region under the plot of its density function. A Markov chain that converges to this uniform distribution can be constructed by alternating uniform sampling in the vertical direction with uniform sampling from the horizontal "slice" defined by the current vertical position, or more generally, with some update that leaves the uniform distribution over this slice invariant. Such "slice sampling" methods are easily implemented for univariate distributions, and can be used to sample from a multivariate distribution by updating each variable in turn. This approach is often easier to implement than Gibbs sampling and more efficient than simple Metropolis updates, due to the ability of slice sampling to adaptively choose the magnitude of changes made. It is therefore attractive for routine and automated use. Slice sampling methods that update all variables simultaneously are also possible. These methods can adaptively choose the magnitudes of changes made to each variable, based on the local properties of the density function. More ambitiously, such methods could potentially adapt to the dependencies between variables by constructing local quadratic approximations. Another approach is to improve sampling efficiency by suppressing random walks. This can be done for univariate slice sampling by "overrelaxation," and for multivariate slice sampling by "reflection" from the edges of the slice.

SummaryWe propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data‐generating mechanism. For instance, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our framework uses loss functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.

We propose a more efficient version of the slice sampler for Dirichlet process mixture models described by Walker (2007). This sampler allows the fitting of infinite mixture models with a wide-range of prior specification. To illustrate this flexiblity we develop a new nonparametric prior for mixture models by normalizing an infinite sequence of independent positive random variables and show how the slice sampler can be applied to make inference in this model. Two submodels are studied in detail. The first one assumes that the positive random variables are Gamma distributed and the second assumes that they are inverseGaussian distributed. Both priors have two hyperparameters and we consider their effect on the prior distribution of the number of occupied clusters in a sample. Extensive computational comparisons with alternative "conditional" simulation techniques for mixture models using the standard Dirichlet process prior and our new prior are made. The properties of the new prior are illustrated on a density estimation problem.

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