3,204 research outputs found
Maximum a Posteriori Estimation by Search in Probabilistic Programs
We introduce an approximate search algorithm for fast maximum a posteriori
probability estimation in probabilistic programs, which we call Bayesian ascent
Monte Carlo (BaMC). Probabilistic programs represent probabilistic models with
varying number of mutually dependent finite, countable, and continuous random
variables. BaMC is an anytime MAP search algorithm applicable to any
combination of random variables and dependencies. We compare BaMC to other MAP
estimation algorithms and show that BaMC is faster and more robust on a range
of probabilistic models.Comment: To appear in proceedings of SOCS1
Particle-kernel estimation of the filter density in state-space models
Sequential Monte Carlo (SMC) methods, also known as particle filters, are
simulation-based recursive algorithms for the approximation of the a posteriori
probability measures generated by state-space dynamical models. At any given
time , a SMC method produces a set of samples over the state space of the
system of interest (often termed "particles") that is used to build a discrete
and random approximation of the posterior probability distribution of the state
variables, conditional on a sequence of available observations. One potential
application of the methodology is the estimation of the densities associated to
the sequence of a posteriori distributions. While practitioners have rather
freely applied such density approximations in the past, the issue has received
less attention from a theoretical perspective. In this paper, we address the
problem of constructing kernel-based estimates of the posterior probability
density function and its derivatives, and obtain asymptotic convergence results
for the estimation errors. In particular, we find convergence rates for the
approximation errors that hold uniformly on the state space and guarantee that
the error vanishes almost surely as the number of particles in the filter
grows. Based on this uniform convergence result, we first show how to build
continuous measures that converge almost surely (with known rate) toward the
posterior measure and then address a few applications. The latter include
maximum a posteriori estimation of the system state using the approximate
derivatives of the posterior density and the approximation of functionals of
it, for example, Shannon's entropy.
This manuscript is identical to the published paper, including a gap in the
proof of Theorem 4.2. The Theorem itself is correct. We provide an {\em
erratum} at the end of this document with a complete proof and a brief
discussion.Comment: IMPORTANT: This manuscript is identical to the published paper,
including a gap in the proof of Theorem 4.2. The Theorem itself is correct.
We provide an erratum at the end of this document. Published at
http://dx.doi.org/10.3150/13-BEJ545 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Parameter estimation by implicit sampling
Implicit sampling is a weighted sampling method that is used in data
assimilation, where one sequentially updates estimates of the state of a
stochastic model based on a stream of noisy or incomplete data. Here we
describe how to use implicit sampling in parameter estimation problems, where
the goal is to find parameters of a numerical model, e.g.~a partial
differential equation (PDE), such that the output of the numerical model is
compatible with (noisy) data. We use the Bayesian approach to parameter
estimation, in which a posterior probability density describes the probability
of the parameter conditioned on data and compute an empirical estimate of this
posterior with implicit sampling. Our approach generates independent samples,
so that some of the practical difficulties one encounters with Markov Chain
Monte Carlo methods, e.g.~burn-in time or correlations among dependent samples,
are avoided. We describe a new implementation of implicit sampling for
parameter estimation problems that makes use of multiple grids (coarse to fine)
and BFGS optimization coupled to adjoint equations for the required gradient
calculations. The implementation is "dimension independent", in the sense that
a well-defined finite dimensional subspace is sampled as the mesh used for
discretization of the PDE is refined. We illustrate the algorithm with an
example where we estimate a diffusion coefficient in an elliptic equation from
sparse and noisy pressure measurements. In the example, dimension\slash
mesh-independence is achieved via Karhunen-Lo\`{e}ve expansions
Multi-scale uncertainty quantification in geostatistical seismic inversion
Geostatistical seismic inversion is commonly used to infer the spatial
distribution of the subsurface petro-elastic properties by perturbing the model
parameter space through iterative stochastic sequential
simulations/co-simulations. The spatial uncertainty of the inferred
petro-elastic properties is represented with the updated a posteriori variance
from an ensemble of the simulated realizations. Within this setting, the
large-scale geological (metaparameters) used to generate the petro-elastic
realizations, such as the spatial correlation model and the global a priori
distribution of the properties of interest, are assumed to be known and
stationary for the entire inversion domain. This assumption leads to
underestimation of the uncertainty associated with the inverted models. We
propose a practical framework to quantify uncertainty of the large-scale
geological parameters in seismic inversion. The framework couples
geostatistical seismic inversion with a stochastic adaptive sampling and
Bayesian inference of the metaparameters to provide a more accurate and
realistic prediction of uncertainty not restricted by heavy assumptions on
large-scale geological parameters. The proposed framework is illustrated with
both synthetic and real case studies. The results show the ability retrieve
more reliable acoustic impedance models with a more adequate uncertainty spread
when compared with conventional geostatistical seismic inversion techniques.
The proposed approach separately account for geological uncertainty at
large-scale (metaparameters) and local scale (trace-by-trace inversion)
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