30,579 research outputs found
Auxiliary Likelihood-Based Approximate Bayesian Computation in State Space Models
A computationally simple approach to inference in state space models is
proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation
of an intractable likelihood by matching summary statistics for the observed
data with statistics computed from data simulated from the true process, based
on parameter draws from the prior. Draws that produce a 'match' between
observed and simulated summaries are retained, and used to estimate the
inaccessible posterior. With no reduction to a low-dimensional set of
sufficient statistics being possible in the state space setting, we define the
summaries as the maximum of an auxiliary likelihood function, and thereby
exploit the asymptotic sufficiency of this estimator for the auxiliary
parameter vector. We derive conditions under which this approach - including a
computationally efficient version based on the auxiliary score - achieves
Bayesian consistency. To reduce the well-documented inaccuracy of ABC in
multi-parameter settings, we propose the separate treatment of each parameter
dimension using an integrated likelihood technique. Three stochastic volatility
models for which exact Bayesian inference is either computationally
challenging, or infeasible, are used for illustration. We demonstrate that our
approach compares favorably against an extensive set of approximate and exact
comparators. An empirical illustration completes the paper.Comment: This paper is forthcoming at the Journal of Computational and
Graphical Statistics. It also supersedes the earlier arXiv paper "Approximate
Bayesian Computation in State Space Models" (arXiv:1409.8363
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
Stochastic Volatility Filtering with Intractable Likelihoods
This paper is concerned with particle filtering for -stable
stochastic volatility models. The -stable distribution provides a
flexible framework for modeling asymmetry and heavy tails, which is useful when
modeling financial returns. An issue with this distributional assumption is the
lack of a closed form for the probability density function. To estimate the
volatility of financial returns in this setting, we develop a novel auxiliary
particle filter. The algorithm we develop can be easily applied to any hidden
Markov model for which the likelihood function is intractable or
computationally expensive. The approximate target distribution of our auxiliary
filter is based on the idea of approximate Bayesian computation (ABC). ABC
methods allow for inference on posterior quantities in situations when the
likelihood of the underlying model is not available in closed form, but
simulating samples from it is possible. The ABC auxiliary particle filter
(ABC-APF) that we propose provides not only a good alternative to state
estimation in stochastic volatility models, but it also improves on the
existing ABC literature. It allows for more flexibility in state estimation
while improving on the accuracy through better proposal distributions in cases
when the optimal importance density of the filter is unavailable in closed
form. We assess the performance of the ABC-APF on a simulated dataset from the
-stable stochastic volatility model and compare it to other currently
existing ABC filters
A Bayesian Approach toward Active Learning for Collaborative Filtering
Collaborative filtering is a useful technique for exploiting the preference
patterns of a group of users to predict the utility of items for the active
user. In general, the performance of collaborative filtering depends on the
number of rated examples given by the active user. The more the number of rated
examples given by the active user, the more accurate the predicted ratings will
be. Active learning provides an effective way to acquire the most informative
rated examples from active users. Previous work on active learning for
collaborative filtering only considers the expected loss function based on the
estimated model, which can be misleading when the estimated model is
inaccurate. This paper takes one step further by taking into account of the
posterior distribution of the estimated model, which results in more robust
active learning algorithm. Empirical studies with datasets of movie ratings
show that when the number of ratings from the active user is restricted to be
small, active learning methods only based on the estimated model don't perform
well while the active learning method using the model distribution achieves
substantially better performance.Comment: Appears in Proceedings of the Twentieth Conference on Uncertainty in
Artificial Intelligence (UAI2004
Inference via low-dimensional couplings
We investigate the low-dimensional structure of deterministic transformations
between random variables, i.e., transport maps between probability measures. In
the context of statistics and machine learning, these transformations can be
used to couple a tractable "reference" measure (e.g., a standard Gaussian) with
a target measure of interest. Direct simulation from the desired measure can
then be achieved by pushing forward reference samples through the map. Yet
characterizing such a map---e.g., representing and evaluating it---grows
challenging in high dimensions. The central contribution of this paper is to
establish a link between the Markov properties of the target measure and the
existence of low-dimensional couplings, induced by transport maps that are
sparse and/or decomposable. Our analysis not only facilitates the construction
of transformations in high-dimensional settings, but also suggests new
inference methodologies for continuous non-Gaussian graphical models. For
instance, in the context of nonlinear state-space models, we describe new
variational algorithms for filtering, smoothing, and sequential parameter
inference. These algorithms can be understood as the natural
generalization---to the non-Gaussian case---of the square-root
Rauch-Tung-Striebel Gaussian smoother.Comment: 78 pages, 25 figure
Approximate Bayesian Computation for a Class of Time Series Models
In the following article we consider approximate Bayesian computation (ABC)
for certain classes of time series models. In particular, we focus upon
scenarios where the likelihoods of the observations and parameter are
intractable, by which we mean that one cannot evaluate the likelihood even
up-to a positive unbiased estimate. This paper reviews and develops a class of
approximation procedures based upon the idea of ABC, but, specifically
maintains the probabilistic structure of the original statistical model. This
idea is useful, in that it can facilitate an analysis of the bias of the
approximation and the adaptation of established computational methods for
parameter inference. Several existing results in the literature are surveyed
and novel developments with regards to computation are given
Ensemble Kalman methods for high-dimensional hierarchical dynamic space-time models
We propose a new class of filtering and smoothing methods for inference in
high-dimensional, nonlinear, non-Gaussian, spatio-temporal state-space models.
The main idea is to combine the ensemble Kalman filter and smoother, developed
in the geophysics literature, with state-space algorithms from the statistics
literature. Our algorithms address a variety of estimation scenarios, including
on-line and off-line state and parameter estimation. We take a Bayesian
perspective, for which the goal is to generate samples from the joint posterior
distribution of states and parameters. The key benefit of our approach is the
use of ensemble Kalman methods for dimension reduction, which allows inference
for high-dimensional state vectors. We compare our methods to existing ones,
including ensemble Kalman filters, particle filters, and particle MCMC. Using a
real data example of cloud motion and data simulated under a number of
nonlinear and non-Gaussian scenarios, we show that our approaches outperform
these existing methods
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