965 research outputs found
Copula Processes
We define a copula process which describes the dependencies between
arbitrarily many random variables independently of their marginal
distributions. As an example, we develop a stochastic volatility model,
Gaussian Copula Process Volatility (GCPV), to predict the latent standard
deviations of a sequence of random variables. To make predictions we use
Bayesian inference, with the Laplace approximation, and with Markov chain Monte
Carlo as an alternative. We find both methods comparable. We also find our
model can outperform GARCH on simulated and financial data. And unlike GARCH,
GCPV can easily handle missing data, incorporate covariates other than time,
and model a rich class of covariance structures.Comment: 11 pages, 1 table, 1 figure. Submitted for publication. Since last
version: minor edits and reformattin
On choosing mixture components via non-local priors
Choosing the number of mixture components remains an elusive challenge. Model
selection criteria can be either overly liberal or conservative and return
poorly-separated components of limited practical use. We formalize non-local
priors (NLPs) for mixtures and show how they lead to well-separated components
with non-negligible weight, interpretable as distinct subpopulations. We also
propose an estimator for posterior model probabilities under local and
non-local priors, showing that Bayes factors are ratios of posterior to prior
empty-cluster probabilities. The estimator is widely applicable and helps set
thresholds to drop unoccupied components in overfitted mixtures. We suggest
default prior parameters based on multi-modality for Normal/T mixtures and
minimal informativeness for categorical outcomes. We characterise theoretically
the NLP-induced sparsity, derive tractable expressions and algorithms. We fully
develop Normal, Binomial and product Binomial mixtures but the theory,
computation and principles hold more generally. We observed a serious lack of
sensitivity of the Bayesian information criterion (BIC), insufficient parsimony
of the AIC and a local prior, and a mixed behavior of the singular BIC. We also
considered overfitted mixtures, their performance was competitive but depended
on tuning parameters. Under our default prior elicitation NLPs offered a good
compromise between sparsity and power to detect meaningfully-separated
components
Introduction to finite mixtures
Mixture models have been around for over 150 years, as an intuitively simple
and practical tool for enriching the collection of probability distributions
available for modelling data. In this chapter we describe the basic ideas of
the subject, present several alternative representations and perspectives on
these models, and discuss some of the elements of inference about the unknowns
in the models. Our focus is on the simplest set-up, of finite mixture models,
but we discuss also how various simplifying assumptions can be relaxed to
generate the rich landscape of modelling and inference ideas traversed in the
rest of this book.Comment: 14 pages, 7 figures, A chapter prepared for the forthcoming Handbook
of Mixture Analysis. V2 corrects a small but important typographical error,
and makes other minor edits; V3 makes further minor corrections and updates
following review; V4 corrects algorithmic details in sec 4.1 and 4.2, and
removes typo
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