46,378 research outputs found
Bayesian nonparametric multivariate convex regression
In many applications, such as economics, operations research and
reinforcement learning, one often needs to estimate a multivariate regression
function f subject to a convexity constraint. For example, in sequential
decision processes the value of a state under optimal subsequent decisions may
be known to be convex or concave. We propose a new Bayesian nonparametric
multivariate approach based on characterizing the unknown regression function
as the max of a random collection of unknown hyperplanes. This specification
induces a prior with large support in a Kullback-Leibler sense on the space of
convex functions, while also leading to strong posterior consistency. Although
we assume that f is defined over R^p, we show that this model has a convergence
rate of log(n)^{-1} n^{-1/(d+2)} under the empirical L2 norm when f actually
maps a d dimensional linear subspace to R. We design an efficient reversible
jump MCMC algorithm for posterior computation and demonstrate the methods
through application to value function approximation
Optimal Belief Approximation
In Bayesian statistics probability distributions express beliefs. However,
for many problems the beliefs cannot be computed analytically and
approximations of beliefs are needed. We seek a loss function that quantifies
how "embarrassing" it is to communicate a given approximation. We reproduce and
discuss an old proof showing that there is only one ranking under the
requirements that (1) the best ranked approximation is the non-approximated
belief and (2) that the ranking judges approximations only by their predictions
for actual outcomes. The loss function that is obtained in the derivation is
equal to the Kullback-Leibler divergence when normalized. This loss function is
frequently used in the literature. However, there seems to be confusion about
the correct order in which its functional arguments, the approximated and
non-approximated beliefs, should be used. The correct order ensures that the
recipient of a communication is only deprived of the minimal amount of
information. We hope that the elementary derivation settles the apparent
confusion. For example when approximating beliefs with Gaussian distributions
the optimal approximation is given by moment matching. This is in contrast to
many suggested computational schemes.Comment: made improvements on the proof and the languag
Computational aspects of Bayesian spectral density estimation
Gaussian time-series models are often specified through their spectral
density. Such models present several computational challenges, in particular
because of the non-sparse nature of the covariance matrix. We derive a fast
approximation of the likelihood for such models. We propose to sample from the
approximate posterior (that is, the prior times the approximate likelihood),
and then to recover the exact posterior through importance sampling. We show
that the variance of the importance sampling weights vanishes as the sample
size goes to infinity. We explain why the approximate posterior may typically
multi-modal, and we derive a Sequential Monte Carlo sampler based on an
annealing sequence in order to sample from that target distribution.
Performance of the overall approach is evaluated on simulated and real
datasets. In addition, for one real world dataset, we provide some numerical
evidence that a Bayesian approach to semi-parametric estimation of spectral
density may provide more reasonable results than its Frequentist counter-parts
Approximate Bayesian Computation in State Space Models
A new approach to inference in state space models is proposed, based on
approximate Bayesian computation (ABC). ABC avoids evaluation of the likelihood
function by matching observed summary statistics with statistics computed from
data simulated from the true process; exact inference being feasible only if
the statistics are sufficient. With finite sample sufficiency unattainable in
the state space setting, we seek asymptotic sufficiency via the maximum
likelihood estimator (MLE) of the parameters of an auxiliary model. We prove
that this auxiliary model-based approach achieves Bayesian consistency, and
that - in a precise limiting sense - the proximity to (asymptotic) sufficiency
yielded by the MLE is replicated by the score. In multiple parameter settings a
separate treatment of scalar parameters, based on integrated likelihood
techniques, is advocated as a way of avoiding the curse of dimensionality. Some
attention is given to a structure in which the state variable is driven by a
continuous time process, with exact inference typically infeasible in this case
as a result of intractable transitions. The ABC method is demonstrated using
the unscented Kalman filter as a fast and simple way of producing an
approximation in this setting, with a stochastic volatility model for financial
returns used for illustration
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