121,640 research outputs found
On predictive probability matching priors
We revisit the question of priors that achieve approximate matching of
Bayesian and frequentist predictive probabilities. Such priors may be thought
of as providing frequentist calibration of Bayesian prediction or simply as
devices for producing frequentist prediction regions. Here we analyse the
term in the expansion of the coverage probability of a Bayesian
prediction region, as derived in [Ann. Statist. 28 (2000) 1414--1426]. Unlike
the situation for parametric matching, asymptotic predictive matching priors
may depend on the level . We investigate uniformly predictive matching
priors (UPMPs); that is, priors for which this term is zero for all
. It was shown in [Ann. Statist. 28 (2000) 1414--1426] that, in the
case of quantile matching and a scalar parameter, if such a prior exists then
it must be Jeffreys' prior. In the present article we investigate UPMPs in the
multiparameter case and present some general results about the form, and
uniqueness or otherwise, of UPMPs for both quantile and highest predictive
density matching.Comment: Published in at http://dx.doi.org/10.1214/074921708000000048 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
Tight Bounds for the Price of Anarchy of Simultaneous First Price Auctions
We study the Price of Anarchy of simultaneous first-price auctions for buyers
with submodular and subadditive valuations. The current best upper bounds for
the Bayesian Price of Anarchy of these auctions are e/(e-1) [Syrgkanis and
Tardos 2013] and 2 [Feldman et al. 2013], respectively. We provide matching
lower bounds for both cases even for the case of full information and for mixed
Nash equilibria via an explicit construction.
We present an alternative proof of the upper bound of e/(e-1) for first-price
auctions with fractionally subadditive valuations which reveals the worst-case
price distribution, that is used as a building block for the matching lower
bound construction.
We generalize our results to a general class of item bidding auctions that we
call bid-dependent auctions (including first-price auctions and all-pay
auctions) where the winner is always the highest bidder and each bidder's
payment depends only on his own bid.
Finally, we apply our techniques to discriminatory price multi-unit auctions.
We complement the results of [de Keijzer et al. 2013] for the case of
subadditive valuations, by providing a matching lower bound of 2. For the case
of submodular valuations, we provide a lower bound of 1.109. For the same class
of valuations, we were able to reproduce the upper bound of e/(e-1) using our
non-smooth approach.Comment: 37 pages, 5 figures, ACM Transactions on Economics and Computatio
Probability-Matching Predictors for Extreme Extremes
A location- and scale-invariant predictor is constructed which exhibits good
probability matching for extreme predictions outside the span of data drawn
from a variety of (stationary) general distributions. It is constructed via the
three-parameter {\mu, \sigma, \xi} Generalized Pareto Distribution (GPD). The
predictor is designed to provide matching probability exactly for the GPD in
both the extreme heavy-tailed limit and the extreme bounded-tail limit, whilst
giving a good approximation to probability matching at all intermediate values
of the tail parameter \xi. The predictor is valid even for small sample sizes
N, even as small as N = 3.
The main purpose of this paper is to present the somewhat lengthy derivations
which draw heavily on the theory of hypergeometric functions, particularly the
Lauricella functions. Whilst the construction is inspired by the Bayesian
approach to the prediction problem, it considers the case of vague prior
information about both parameters and model, and all derivations are undertaken
using sampling theory.Comment: 22 pages, 7 figure
Relational Entropic Dynamics of Particles
The general framework of entropic dynamics is used to formulate a relational
quantum dynamics. The main new idea is to use tools of information geometry to
develop an entropic measure of the mismatch between successive configurations
of a system. This leads to an entropic version of the classical best matching
technique developed by J. Barbour and collaborators. The procedure is
illustrated in the simple case of a system of N particles with global
translational symmetry. The generalization to other symmetries whether global
(rotational invariance) or local (gauge invariance) is straightforward. The
entropic best matching allows a quantum implementation Mach's principles of
spatial and temporal relationalism and provides the foundation for a method of
handling gauge theories in an informational framework.Comment: Presented at MaxEnt 2015, the 35th International Workshop on Bayesian
Inference and Maximum Entropy Methods in Science and Engineering (July
19--24, 2015, Potsdam NY, USA
Latent Bayesian melding for integrating individual and population models
In many statistical problems, a more coarse-grained model may be suitable for population-level behaviour, whereas a more detailed model is appropriate for accurate modelling of individual behaviour. This raises the question of how to integrate both types of models. Methods such as posterior regularization follow the idea of generalized moment matching, in that they allow matching expectations between two models, but sometimes both models are most conveniently expressed as latent variable models. We propose latent Bayesian melding, which is motivated by averaging the distributions over populations statistics of both the individual-level and the population-level models under a logarithmic opinion pool framework. ln a case study on electricity disaggregation, which is a type of single channel blind source separation problem, we show that latent Bayesian melding leads to significantly more accurate predictions than an approach based solely on generalized moment matching
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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