39,950 research outputs found
Sequential Bayesian inference for static parameters in dynamic state space models
A method for sequential Bayesian inference of the static parameters of a
dynamic state space model is proposed. The method is based on the observation
that many dynamic state space models have a relatively small number of static
parameters (or hyper-parameters), so that in principle the posterior can be
computed and stored on a discrete grid of practical size which can be tracked
dynamically. Further to this, this approach is able to use any existing
methodology which computes the filtering and prediction distributions of the
state process. Kalman filter and its extensions to non-linear/non-Gaussian
situations have been used in this paper. This is illustrated using several
applications: linear Gaussian model, Binomial model, stochastic volatility
model and the extremely non-linear univariate non-stationary growth model.
Performance has been compared to both existing on-line method and off-line
methods
Stable Recovery Of Sparse Vectors From Random Sinusoidal Feature Maps
Random sinusoidal features are a popular approach for speeding up
kernel-based inference in large datasets. Prior to the inference stage, the
approach suggests performing dimensionality reduction by first multiplying each
data vector by a random Gaussian matrix, and then computing an element-wise
sinusoid. Theoretical analysis shows that collecting a sufficient number of
such features can be reliably used for subsequent inference in kernel
classification and regression.
In this work, we demonstrate that with a mild increase in the dimension of
the embedding, it is also possible to reconstruct the data vector from such
random sinusoidal features, provided that the underlying data is sparse enough.
In particular, we propose a numerically stable algorithm for reconstructing the
data vector given the nonlinear features, and analyze its sample complexity.
Our algorithm can be extended to other types of structured inverse problems,
such as demixing a pair of sparse (but incoherent) vectors. We support the
efficacy of our approach via numerical experiments
Ecological non-linear state space model selection via adaptive particle Markov chain Monte Carlo (AdPMCMC)
We develop a novel advanced Particle Markov chain Monte Carlo algorithm that
is capable of sampling from the posterior distribution of non-linear state
space models for both the unobserved latent states and the unknown model
parameters. We apply this novel methodology to five population growth models,
including models with strong and weak Allee effects, and test if it can
efficiently sample from the complex likelihood surface that is often associated
with these models. Utilising real and also synthetically generated data sets we
examine the extent to which observation noise and process error may frustrate
efforts to choose between these models. Our novel algorithm involves an
Adaptive Metropolis proposal combined with an SIR Particle MCMC algorithm
(AdPMCMC). We show that the AdPMCMC algorithm samples complex, high-dimensional
spaces efficiently, and is therefore superior to standard Gibbs or Metropolis
Hastings algorithms that are known to converge very slowly when applied to the
non-linear state space ecological models considered in this paper.
Additionally, we show how the AdPMCMC algorithm can be used to recursively
estimate the Bayesian Cram\'er-Rao Lower Bound of Tichavsk\'y (1998). We derive
expressions for these Cram\'er-Rao Bounds and estimate them for the models
considered. Our results demonstrate a number of important features of common
population growth models, most notably their multi-modal posterior surfaces and
dependence between the static and dynamic parameters. We conclude by sampling
from the posterior distribution of each of the models, and use Bayes factors to
highlight how observation noise significantly diminishes our ability to select
among some of the models, particularly those that are designed to reproduce an
Allee effect
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