10 research outputs found
Computing Entropies With Nested Sampling
The Shannon entropy, and related quantities such as mutual information, can
be used to quantify uncertainty and relevance. However, in practice, it can be
difficult to compute these quantities for arbitrary probability distributions,
particularly if the probability mass functions or densities cannot be
evaluated. This paper introduces a computational approach, based on Nested
Sampling, to evaluate entropies of probability distributions that can only be
sampled. I demonstrate the method on three examples: a simple gaussian example
where the key quantities are available analytically; (ii) an experimental
design example about scheduling observations in order to measure the period of
an oscillating signal; and (iii) predicting the future from the past in a
heavy-tailed scenario.Comment: Accepted for publication in Entropy. 21 pages, 3 figures. Software
available at https://github.com/eggplantbren/InfoNes
Rare Event Simulation and Splitting for Discontinuous Random Variables
Multilevel Splitting methods, also called Sequential Monte-Carlo or
\emph{Subset Simulation}, are widely used methods for estimating extreme
probabilities of the form where is a deterministic
real-valued function and can be a random finite- or
infinite-dimensional vector. Very often, is supposed to be
a continuous random variable and a lot of theoretical results on the
statistical behaviour of the estimator are now derived with this hypothesis.
However, as soon as some threshold effect appears in and/or is
discrete or mixed discrete/continuous this assumption does not hold any more
and the estimator is not consistent.
In this paper, we study the impact of discontinuities in the \emph{cdf} of
and present three unbiased \emph{corrected} estimators to handle them.
These estimators do not require to know in advance if is actually
discontinuous or not and become all equal if is continuous. Especially, one
of them has the same statistical properties in any case. Efficiency is shown on
a 2-D diffusive process as well as on the \emph{Boolean SATisfiability problem}
(SAT).Comment: 16 pages (12 + Appendix 4 pages), 6 figure
Unbiased and Consistent Nested Sampling via Sequential Monte Carlo
We introduce a new class of sequential Monte Carlo methods called Nested
Sampling via Sequential Monte Carlo (NS-SMC), which reframes the Nested
Sampling method of Skilling (2006) in terms of sequential Monte Carlo
techniques. This new framework allows convergence results to be obtained in the
setting when Markov chain Monte Carlo (MCMC) is used to produce new samples. An
additional benefit is that marginal likelihood estimates are unbiased. In
contrast to NS, the analysis of NS-SMC does not require the (unrealistic)
assumption that the simulated samples be independent. As the original NS
algorithm is a special case of NS-SMC, this provides insights as to why NS
seems to produce accurate estimates despite a typical violation of its
assumptions. For applications of NS-SMC, we give advice on tuning MCMC kernels
in an automated manner via a preliminary pilot run, and present a new method
for appropriately choosing the number of MCMC repeats at each iteration.
Finally, a numerical study is conducted where the performance of NS-SMC and
temperature-annealed SMC is compared on several challenging and realistic
problems. MATLAB code for our experiments is made available at
https://github.com/LeahPrice/SMC-NS .Comment: 45 pages, some minor typographical errors fixed since last versio
A randomized Multi-index sequential Monte Carlo method
We consider the problem of estimating expectations with respect to a target
distribution with an unknown normalizing constant, and where even the
unnormalized target needs to be approximated at finite resolution. Under such
an assumption, this work builds upon a recently introduced multi-index
Sequential Monte Carlo (SMC) ratio estimator, which provably enjoys the
complexity improvements of multi-index Monte Carlo (MIMC) and the efficiency of
SMC for inference. The present work leverages a randomization strategy to
remove bias entirely, which simplifies estimation substantially, particularly
in the MIMC context, where the choice of index set is otherwise important.
Under reasonable assumptions, the proposed method provably achieves the same
canonical complexity of MSE^(-1) as the original method, but without
discretization bias. It is illustrated on examples of Bayesian inverse
problems.Comment: 26 pages 6 figure
Nested Sampling for Uncertainty Quantification and Rare Event Estimation
Nested Sampling is a method for computing the Bayesian evidence, also called
the marginal likelihood, which is the integral of the likelihood with respect
to the prior. More generally, it is a numerical probabilistic quadrature rule.
The main idea of Nested Sampling is to replace a high-dimensional likelihood
integral over parameter space with an integral over the unit line by employing
a push-forward with respect to a suitable transformation. Practically, a set of
active samples ascends the level sets of the integrand function, with the
measure contraction of the super-level sets being statistically estimated. We
justify the validity of this approach for integrands with non-negligible
plateaus, and demonstrate Nested Sampling's practical effectiveness in
estimating the (log-)probability of rare events.Comment: 24 page
Nested Sampling Methods
Nested sampling (NS) computes parameter posterior distributions and makes
Bayesian model comparison computationally feasible. Its strengths are the
unsupervised navigation of complex, potentially multi-modal posteriors until a
well-defined termination point. A systematic literature review of nested
sampling algorithms and variants is presented. We focus on complete algorithms,
including solutions to likelihood-restricted prior sampling, parallelisation,
termination and diagnostics. The relation between number of live points,
dimensionality and computational cost is studied for two complete algorithms. A
new formulation of NS is presented, which casts the parameter space exploration
as a search on a tree. Previously published ways of obtaining robust error
estimates and dynamic variations of the number of live points are presented as
special cases of this formulation. A new on-line diagnostic test is presented
based on previous insertion rank order work. The survey of nested sampling
methods concludes with outlooks for future research.Comment: Updated version incorporating constructive input from four(!)
positive reports (two referees, assistant editor and editor). The open-source
UltraNest package and astrostatistics tutorials can be found at
https://johannesbuchner.github.io/UltraNest
Recommended from our members
Bayesian Methods and Machine Learning in Astrophysics
This thesis is concerned with methods for Bayesian inference and their applications in astrophysics. We principally discuss two related themes: advances in nested sampling (Chapters 3 to 5), and Bayesian sparse reconstruction of signals from noisy data (Chapters 6 and 7).
Nested sampling is a popular method for Bayesian computation which is widely used in astrophysics. Following the introduction and background material in Chapters 1 and 2, Chapter 3 analyses the sampling errors in nested sampling parameter estimation and presents a method for estimating them numerically for a single nested sampling calculation. Chapter 4 introduces diagnostic tests for detecting when software has not performed the nested sampling algorithm accurately, for example due to missing a mode in a multimodal posterior. The uncertainty estimates and diagnostics in Chapters 3 and 4 are implemented in the software package, and both chapters describe an astronomical application of the techniques introduced.
Chapter 5 describes dynamic nested sampling: a generalisation of the nested sampling algorithm which can produce large improvements in computational efficiency compared to standard nested sampling. We have implemented dynamic nested sampling in the and software packages.
Chapter 6 presents a principled Bayesian framework for signal reconstruction, in which the signal is modelled by basis functions whose number (and form, if required) is determined by the data themselves. This approach is based on a Bayesian interpretation of conventional sparse reconstruction and regularisation techniques, in which sparsity is imposed through priors via Bayesian model selection. We demonstrate our method for noisy 1- and 2-dimensional signals, including examples of processing astronomical images. The numerical implementation uses dynamic nested sampling, and uncertainties are calculated using the methods introduced in Chapters 3 and 4. Chapter 7 applies our Bayesian sparse reconstruction framework to artificial neural networks, where it allows the optimum network architecture to be determined by treating the number of nodes and hidden layers as parameters.
We conclude by suggesting possible areas of future research in Chapter 8