60 research outputs found
Online Multi-task Learning with Hard Constraints
We discuss multi-task online learning when a decision maker has to deal
simultaneously with M tasks. The tasks are related, which is modeled by
imposing that the M-tuple of actions taken by the decision maker needs to
satisfy certain constraints. We give natural examples of such restrictions and
then discuss a general class of tractable constraints, for which we introduce
computationally efficient ways of selecting actions, essentially by reducing to
an on-line shortest path problem. We briefly discuss "tracking" and "bandit"
versions of the problem and extend the model in various ways, including
non-additive global losses and uncountably infinite sets of tasks
Bayesian prediction of jumps in large panels of time series data
We take a new look at the problem of disentangling the volatility and jumps
processes of daily stock returns. We first provide a computational framework
for the univariate stochastic volatility model with Poisson-driven jumps that
offers a competitive inference alternative to the existing tools. This
methodology is then extended to a large set of stocks for which we assume that
their unobserved jump intensities co-evolve in time through a dynamic factor
model. To evaluate the proposed modelling approach we conduct out-of-sample
forecasts and we compare the posterior predictive distributions obtained from
the different models. We provide evidence that joint modelling of jumps
improves the predictive ability of the stochastic volatility models.Comment: 49 pages, 27 figures, 4 table
Markov chain Monte Carlo for exact inference for diffusions
We develop exact Markov chain Monte Carlo methods for discretely-sampled,
directly and indirectly observed diffusions. The qualification "exact" refers
to the fact that the invariant and limiting distribution of the Markov chains
is the posterior distribution of the parameters free of any discretisation
error. The class of processes to which our methods directly apply are those
which can be simulated using the most general to date exact simulation
algorithm. The article introduces various methods to boost the performance of
the basic scheme, including reparametrisations and auxiliary Poisson sampling.
We contrast both theoretically and empirically how this new approach compares
to irreducible high frequency imputation, which is the state-of-the-art
alternative for the class of processes we consider, and we uncover intriguing
connections. All methods discussed in the article are tested on typical
examples.Comment: 23 pages, 6 Figures, 3 Table
Analysis of the Gibbs sampler for hierarchical inverse problems
Many inverse problems arising in applications come from continuum models
where the unknown parameter is a field. In practice the unknown field is
discretized resulting in a problem in , with an understanding
that refining the discretization, that is increasing , will often be
desirable. In the context of Bayesian inversion this situation suggests the
importance of two issues: (i) defining hyper-parameters in such a way that they
are interpretable in the continuum limit and so that their
values may be compared between different discretization levels; (ii)
understanding the efficiency of algorithms for probing the posterior
distribution, as a function of large Here we address these two issues in
the context of linear inverse problems subject to additive Gaussian noise
within a hierarchical modelling framework based on a Gaussian prior for the
unknown field and an inverse-gamma prior for a hyper-parameter, namely the
amplitude of the prior variance. The structure of the model is such that the
Gibbs sampler can be easily implemented for probing the posterior distribution.
Subscribing to the dogma that one should think infinite-dimensionally before
implementing in finite dimensions, we present function space intuition and
provide rigorous theory showing that as increases, the component of the
Gibbs sampler for sampling the amplitude of the prior variance becomes
increasingly slower. We discuss a reparametrization of the prior variance that
is robust with respect to the increase in dimension; we give numerical
experiments which exhibit that our reparametrization prevents the slowing down.
Our intuition on the behaviour of the prior hyper-parameter, with and without
reparametrization, is sufficiently general to include a broad class of
nonlinear inverse problems as well as other families of hyper-priors.Comment: to appear, SIAM/ASA Journal on Uncertainty Quantificatio
SMC^2: an efficient algorithm for sequential analysis of state-space models
We consider the generic problem of performing sequential Bayesian inference
in a state-space model with observation process y, state process x and fixed
parameter theta. An idealized approach would be to apply the iterated batch
importance sampling (IBIS) algorithm of Chopin (2002). This is a sequential
Monte Carlo algorithm in the theta-dimension, that samples values of theta,
reweights iteratively these values using the likelihood increments
p(y_t|y_1:t-1, theta), and rejuvenates the theta-particles through a resampling
step and a MCMC update step. In state-space models these likelihood increments
are intractable in most cases, but they may be unbiasedly estimated by a
particle filter in the x-dimension, for any fixed theta. This motivates the
SMC^2 algorithm proposed in this article: a sequential Monte Carlo algorithm,
defined in the theta-dimension, which propagates and resamples many particle
filters in the x-dimension. The filters in the x-dimension are an example of
the random weight particle filter as in Fearnhead et al. (2010). On the other
hand, the particle Markov chain Monte Carlo (PMCMC) framework developed in
Andrieu et al. (2010) allows us to design appropriate MCMC rejuvenation steps.
Thus, the theta-particles target the correct posterior distribution at each
iteration t, despite the intractability of the likelihood increments. We
explore the applicability of our algorithm in both sequential and
non-sequential applications and consider various degrees of freedom, as for
example increasing dynamically the number of x-particles. We contrast our
approach to various competing methods, both conceptually and empirically
through a detailed simulation study, included here and in a supplement, and
based on particularly challenging examples.Comment: 27 pages, 4 figures; supplementary material available on the second
author's web pag
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