5,619 research outputs found
Sequential Monte Carlo with Highly Informative Observations
We propose sequential Monte Carlo (SMC) methods for sampling the posterior
distribution of state-space models under highly informative observation
regimes, a situation in which standard SMC methods can perform poorly. A
special case is simulating bridges between given initial and final values. The
basic idea is to introduce a schedule of intermediate weighting and resampling
times between observation times, which guide particles towards the final state.
This can always be done for continuous-time models, and may be done for
discrete-time models under sparse observation regimes; our main focus is on
continuous-time diffusion processes. The methods are broadly applicable in that
they support multivariate models with partial observation, do not require
simulation of the backward transition (which is often unavailable), and, where
possible, avoid pointwise evaluation of the forward transition. When simulating
bridges, the last cannot be avoided entirely without concessions, and we
suggest an epsilon-ball approach (reminiscent of Approximate Bayesian
Computation) as a workaround. Compared to the bootstrap particle filter, the
new methods deliver substantially reduced mean squared error in normalising
constant estimates, even after accounting for execution time. The methods are
demonstrated for state estimation with two toy examples, and for parameter
estimation (within a particle marginal Metropolis--Hastings sampler) with three
applied examples in econometrics, epidemiology and marine biogeochemistry.Comment: 25 pages, 11 figure
Parallel resampling in the particle filter
Modern parallel computing devices, such as the graphics processing unit
(GPU), have gained significant traction in scientific and statistical
computing. They are particularly well-suited to data-parallel algorithms such
as the particle filter, or more generally Sequential Monte Carlo (SMC), which
are increasingly used in statistical inference. SMC methods carry a set of
weighted particles through repeated propagation, weighting and resampling
steps. The propagation and weighting steps are straightforward to parallelise,
as they require only independent operations on each particle. The resampling
step is more difficult, as standard schemes require a collective operation,
such as a sum, across particle weights. Focusing on this resampling step, we
analyse two alternative schemes that do not involve a collective operation
(Metropolis and rejection resamplers), and compare them to standard schemes
(multinomial, stratified and systematic resamplers). We find that, in certain
circumstances, the alternative resamplers can perform significantly faster on a
GPU, and to a lesser extent on a CPU, than the standard approaches. Moreover,
in single precision, the standard approaches are numerically biased for upwards
of hundreds of thousands of particles, while the alternatives are not. This is
particularly important given greater single- than double-precision throughput
on modern devices, and the consequent temptation to use single precision with a
greater number of particles. Finally, we provide auxiliary functions useful for
implementation, such as for the permutation of ancestry vectors to enable
in-place propagation.Comment: 21 pages, 6 figure
Delayed Sampling and Automatic Rao-Blackwellization of Probabilistic Programs
We introduce a dynamic mechanism for the solution of analytically-tractable
substructure in probabilistic programs, using conjugate priors and affine
transformations to reduce variance in Monte Carlo estimators. For inference
with Sequential Monte Carlo, this automatically yields improvements such as
locally-optimal proposals and Rao-Blackwellization. The mechanism maintains a
directed graph alongside the running program that evolves dynamically as
operations are triggered upon it. Nodes of the graph represent random
variables, edges the analytically-tractable relationships between them. Random
variables remain in the graph for as long as possible, to be sampled only when
they are used by the program in a way that cannot be resolved analytically. In
the meantime, they are conditioned on as many observations as possible. We
demonstrate the mechanism with a few pedagogical examples, as well as a
linear-nonlinear state-space model with simulated data, and an epidemiological
model with real data of a dengue outbreak in Micronesia. In all cases one or
more variables are automatically marginalized out to significantly reduce
variance in estimates of the marginal likelihood, in the final case
facilitating a random-weight or pseudo-marginal-type importance sampler for
parameter estimation. We have implemented the approach in Anglican and a new
probabilistic programming language called Birch.Comment: 13 pages, 4 figure
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