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Particle Metropolis-Hastings using gradient and Hessian information
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-9424-1272
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
Dept. of Information Technology, Uppsala University, Uppsala, Sweden.
2015 (English)In: Statistics and computing, ISSN 0960-3174, E-ISSN 1573-1375, Vol. 25, no 1, 81-92 p.Article in journal (Other academic) Published
Abstract [en]

Particle Metropolis-Hastings (PMH) allows for Bayesian parameter inference in nonlinear state space models by combining MCMC and particle filtering. The latter is used to estimate the intractable likelihood. In its original formulation, PMH makes use of a marginal MCMC proposal for the parameters, typically a Gaussian random walk. However, this can lead to a poor exploration of the parameter space and an inefficient use of the generated particles.

We propose two alternative versions of PMH that incorporate gradient and Hessian information about the posterior into the proposal. This information is more or less obtained as a byproduct of the likelihood estimation. Indeed, we show how to estimate the required information using a fixed-lag particle smoother, with a computational cost growing linearly in the number of particles. We conclude that the proposed methods can: (i) decrease the length of the burn-in phase, (ii) increase the mixing of the Markov chain at the stationary phase, and (iii) make the proposal distribution scale invariant which simplifies tuning.

Place, publisher, year, edition, pages
Springer, 2015. Vol. 25, no 1, 81-92 p.
National Category
Control Engineering Signal Processing Probability Theory and Statistics
URN: urn:nbn:se:liu:diva-106749DOI: 10.1007/s11222-014-9510-0ISI: 000349028500013OAI: diva2:718416
Probabilistic modelling of dynamical systems
Swedish Research Council, 621-2013-5524

On the day of the defence date the status of this article was Manuscript.

Available from: 2014-05-21 Created: 2014-05-21 Last updated: 2016-05-04Bibliographically approved
In thesis
1. Sequential Monte Carlo for inference in nonlinear state space models
Open this publication in new window or tab >>Sequential Monte Carlo for inference in nonlinear state space models
2014 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Nonlinear state space models (SSMs) are a useful class of models to describe many different kinds of systems. Some examples of its applications are to model; the volatility in financial markets, the number of infected persons during an influenza epidemic and the annual number of major earthquakes around the world. In this thesis, we are concerned with state inference, parameter inference and input design for nonlinear SSMs based on sequential Monte Carlo (SMC) methods.

The state inference problem consists of estimating some latent variable that is not directly observable in the output from the system. The parameter inference problem is concerned with fitting a pre-specified model structure to the observed output from the system. In input design, we are interested in constructing an input to the system, which maximises the information that is available about the parameters in the system output. All of these problems are analytically intractable for nonlinear SSMs. Instead, we make use of SMC to approximate the solution to the state inference problem and to solve the input design problem. Furthermore, we make use of Markov chain Monte Carlo (MCMC) and Bayesian optimisation (BO) to solve the parameter inference problem.

In this thesis, we propose new methods for parameter inference in SSMs using both Bayesian and maximum likelihood inference. More specifically, we propose a new proposal for the particle Metropolis-Hastings algorithm, which includes gradient and Hessian information about the target distribution. We demonstrate that the use of this proposal can reduce the length of the burn-in phase and improve the mixing of the Markov chain.

Furthermore, we develop a novel parameter inference method based on the combination of BO and SMC. We demonstrate that this method requires a relatively small amount of samples from the analytically intractable likelihood, which are computationally costly to obtain. Therefore, it could be a good alternative to other optimisation based parameter inference methods. The proposed BO and SMC combination is also extended for parameter inference in nonlinear SSMs with intractable likelihoods using approximate Bayesian computations. This method is used for parameter inference in a stochastic volatility model with -stable returns using real-world financial data.

Finally, we develop a novel method for input design in nonlinear SSMs which makes use of SMC methods to estimate the expected information matrix. This information is used in combination with graph theory and convex optimisation to estimate optimal inputs with amplitude constraints. We also consider parameter estimation in ARX models with Student-t innovations and unknown model orders. Two different algorithms are used for this inference: reversible Jump Markov chain Monte Carlo and Gibbs sampling with sparseness priors. These methods are used to model real-world EEG data with promising results.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2014. 118 p.
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1652
National Category
Control Engineering
urn:nbn:se:liu:diva-106752 (URN)10.3384/lic.diva-106752 (DOI)LIU-TEK-LIC-2014:85 (Local ID)978-91-7519-369-4 (print) (ISBN)LIU-TEK-LIC-2014:85 (Archive number)LIU-TEK-LIC-2014:85 (OAI)
2014-05-28, Visionen, B-building, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Available from: 2014-05-21 Created: 2014-05-21 Last updated: 2016-05-04Bibliographically approved
2. Accelerating Monte Carlo methods for Bayesian inference in dynamical models
Open this publication in new window or tab >>Accelerating Monte Carlo methods for Bayesian inference in dynamical models
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Making decisions and predictions from noisy observations are two important and challenging problems in many areas of society. Some examples of applications are recommendation systems for online shopping and streaming services, connecting genes with certain diseases and modelling climate change. In this thesis, we make use of Bayesian statistics to construct probabilistic models given prior information and historical data, which can be used for decision support and predictions. The main obstacle with this approach is that it often results in mathematical problems lacking analytical solutions. To cope with this, we make use of statistical simulation algorithms known as Monte Carlo methods to approximate the intractable solution. These methods enjoy well-understood statistical properties but are often computational prohibitive to employ.

The main contribution of this thesis is the exploration of different strategies for accelerating inference methods based on sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC). That is, strategies for reducing the computational effort while keeping or improving the accuracy. A major part of the thesis is devoted to proposing such strategies for the MCMC method known as the particle Metropolis-Hastings (PMH) algorithm. We investigate two strategies: (i) introducing estimates of the gradient and Hessian of the target to better tailor the algorithm to the problem and (ii) introducing a positive correlation between the point-wise estimates of the target.

Furthermore, we propose an algorithm based on the combination of SMC and Gaussian process optimisation, which can provide reasonable estimates of the posterior but with a significant decrease in computational effort compared with PMH. Moreover, we explore the use of sparseness priors for approximate inference in over-parametrised mixed effects models and autoregressive processes. This can potentially be a practical strategy for inference in the big data era. Finally, we propose a general method for increasing the accuracy of the parameter estimates in non-linear state space models by applying a designed input signal.

Abstract [sv]

Borde Riksbanken höja eller sänka reporäntan vid sitt nästa möte för att nå inflationsmålet? Vilka gener är förknippade med en viss sjukdom? Hur kan Netflix och Spotify veta vilka filmer och vilken musik som jag vill lyssna på härnäst?

Dessa tre problem är exempel på frågor där statistiska modeller kan vara användbara för att ge hjälp och underlag för beslut. Statistiska modeller kombinerar teoretisk kunskap om exempelvis det svenska ekonomiska systemet med historisk data för att ge prognoser av framtida skeenden. Dessa prognoser kan sedan användas för att utvärdera exempelvis vad som skulle hända med inflationen i Sverige om arbetslösheten sjunker eller hur värdet på mitt pensionssparande förändras när Stockholmsbörsen rasar. Tillämpningar som dessa och många andra gör statistiska modeller viktiga för många delar av samhället.

Ett sätt att ta fram statistiska modeller bygger på att kontinuerligt uppdatera en modell allteftersom mer information samlas in. Detta angreppssätt kallas för Bayesiansk statistik och är särskilt användbart när man sedan tidigare har bra insikter i modellen eller tillgång till endast lite historisk data för att bygga modellen. En nackdel med Bayesiansk statistik är att de beräkningar som krävs för att uppdatera modellen med den nya informationen ofta är mycket komplicerade. I sådana situationer kan man istället simulera utfallet från miljontals varianter av modellen och sedan jämföra dessa mot de historiska observationerna som finns till hands. Man kan sedan medelvärdesbilda över de varianter som gav bäst resultat för att på så sätt ta fram en slutlig modell. Det kan därför ibland ta dagar eller veckor för att ta fram en modell. Problemet blir särskilt stort när man använder mer avancerade modeller som skulle kunna ge bättre prognoser men som tar för lång tid för att bygga.

I denna avhandling använder vi ett antal olika strategier för att underlätta eller förbättra dessa simuleringar. Vi föreslår exempelvis att ta hänsyn till fler insikter om systemet och därmed minska antalet varianter av modellen som behöver undersökas. Vi kan således redan utesluta vissa modeller eftersom vi har en bra uppfattning om ungefär hur en bra modell ska se ut. Vi kan också förändra simuleringen så att den enklare rör sig mellan olika typer av modeller. På detta sätt utforskas rymden av alla möjliga modeller på ett mer effektivt sätt. Vi föreslår ett antal olika kombinationer och förändringar av befintliga metoder för att snabba upp anpassningen av modellen till observationerna. Vi visar att beräkningstiden i vissa fall kan minska ifrån några dagar till någon timme. Förhoppningsvis kommer detta i framtiden leda till att man i praktiken kan använda mer avancerade modeller som i sin tur resulterar i bättre prognoser och beslut.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1754
Computational statistics, Monte Carlo, Markov chains, Particle filters, Machine learning, Bayesian optimisation, Approximate Bayesian Computations, Gaussian processes, Particle Metropolis-Hastings, Approximate inference, Pseudo-marginal methods
National Category
Probability Theory and Statistics Control Engineering Computational Mathematics
urn:nbn:se:liu:diva-125992 (URN)10.3384/diss.diva-125992 (DOI)978-91-7685-797-7 (Print) (ISBN)
Public defence
2016-05-04, Visionen, B-building, Campus Valla, Linköping, 10:15 (English)
Swedish Research Council, 621-2013-5524Swedish Research Council, 637-2014-466Swedish Foundation for Strategic Research , IIS11-0081
Available from: 2016-03-22 Created: 2016-03-11 Last updated: 2016-04-01Bibliographically approved

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