583 research outputs found
Bayes linear covariance matrix adjustment
In this thesis, a Bayes linear methodology for the adjustment of covariance
matrices is presented and discussed. A geometric framework for quantifying
uncertainties about covariance matrices is set up, and an inner-product for
spaces of random matrices is motivated and constructed. The inner-product on
this space captures aspects of our beliefs about the relationship between
covariance matrices of interest to us, providing a structure rich enough for us
to adjust beliefs about unknown matrices in the light of data such as sample
covariance matrices, exploiting second-order exchangeability and related
specifications to obtain representations allowing analysis.
Adjustment is associated with orthogonal projection, and illustrated with
examples of adjustments for some common problems. The problem of adjusting the
covariance matrices underlying exchangeable random vectors is tackled and
discussed. Learning about the covariance matrices associated with multivariate
time series dynamic linear models is shown to be amenable to a similar
approach. Diagnostics for matrix adjustments are also discussed.Comment: 146 pages. PhD thesis. Available as Postscript only. Also available
from http://fourier.dur.ac.uk:8000/~dma1djw/pub/thesis.html with figures.
More information about the Bayes linear programme can be found at
http://fourier.dur.ac.uk:8000/stats/bd
Scalable Inference for Markov Processes with Intractable Likelihoods
Bayesian inference for Markov processes has become increasingly relevant in
recent years. Problems of this type often have intractable likelihoods and
prior knowledge about model rate parameters is often poor. Markov Chain Monte
Carlo (MCMC) techniques can lead to exact inference in such models but in
practice can suffer performance issues including long burn-in periods and poor
mixing. On the other hand approximate Bayesian computation techniques can allow
rapid exploration of a large parameter space but yield only approximate
posterior distributions. Here we consider the combined use of approximate
Bayesian computation (ABC) and MCMC techniques for improved computational
efficiency while retaining exact inference on parallel hardware
Fast Bayesian parameter estimation for stochastic logistic growth models
The transition density of a stochastic, logistic population growth model with
multiplicative intrinsic noise is analytically intractable. Inferring model
parameter values by fitting such stochastic differential equation (SDE) models
to data therefore requires relatively slow numerical simulation. Where such
simulation is prohibitively slow, an alternative is to use model approximations
which do have an analytically tractable transition density, enabling fast
inference. We introduce two such approximations, with either multiplicative or
additive intrinsic noise, each derived from the linear noise approximation of
the logistic growth SDE. After Bayesian inference we find that our fast LNA
models, using Kalman filter recursion for computation of marginal likelihoods,
give similar posterior distributions to slow arbitrarily exact models. We also
demonstrate that simulations from our LNA models better describe the
characteristics of the stochastic logistic growth models than a related
approach. Finally, we demonstrate that our LNA model with additive intrinsic
noise and measurement error best describes an example set of longitudinal
observations of microbial population size taken from a typical, genome-wide
screening experiment.Comment: 24 pages, 5 figures and 2 table
- …