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Markov Chain Monte Carlo Methods and Applications in Neuroscience
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics. KTH, Centres, Science for Life Laboratory, SciLifeLab.ORCID iD: 0000-0003-3635-8760
2023 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

An important task in brain modeling is that of estimating model parameters and quantifying their uncertainty. In this thesis we tackle this problem from a Bayesian perspective: we use experimental data to update the prior information about model parameters, in order to obtain their posterior distribution. Uncertainty quantification via a direct computation of the posterior has a prohibitive computational cost in high dimensions. An alternative to a direct computation is offered by Markov chain Monte Carlo (MCMC) methods.

The aim of this project is to analyse some of the methods within this class and improve their convergence. In this thesis we describe the following MCMC methods: Metropolis-Hastings (MH) algorithm, Metropolis adjusted Langevin algorithm (MALA), simplified manifold MALA (smMALA) and Approximate Bayesian Computation MCMC (ABCMCMC).

SmMALA is further analysed in Paper A, where we propose an algorithm to approximate a key component of this algorithm (the Fisher Information) when applied to ODE models, with the purpose of reducing the computational cost of the method.

A theoretical analysis of MCMC methods is carried out in Paper B and relies on tools from the theory of large deviations. In particular, we analyse the convergence of the MH algorithm by stating and proving a large deviation principle (LDP) for the empirical measures produced by the algorithm.

Some of the methods analysed in this thesis are implemented in an R package, available on GitHub as “icpm-kth/uqsa” and presented in Paper C, and are applied to subcellular pathway models within neurons in the context of uncertainty quantification of the model parameters.

Abstract [sv]

En viktig uppgift inom hjärnmodellering är att uppskatta parametrar i modellen och kvantifiera deras osäkerhet. I denna avhandling hanterar vi detta problem från ett Bayesianskt perspektiv: vi använder experimentell data för att uppdatera a priori kunskap av modellparametrar, för att erhålla deras posteriori-fördelning. Osäkerhetskvantifiering (UQ) via direkt beräkning av posteriorfördelningen har en hög beräkningskostnad vid höga dimensioner. Ett alternativ till direkt beräkning ges av Markov chain Monte Carlo (MCMC) metoder.

Syftet med det här projektet är att analysera några MCMC metoder och förbättra deras konvergens. I denna avhandling beskriver vi följande MCMC algoritmer: “Metropolis-Hastings” (MH), “Metropolis adjusted Langevin” (MALA), “Simplified Manifold MALA” (smMALA) och “Approximate Bayesian Computation MCMC” (ABCMCMC).

SmMALA analyseras i artikel A. Där presenterar vi en algoritm för att approximera en nyckelkomponent av denna algoritm (Fisher informationen) när den tillämpas på ODE modeller i syfte att minska metodens beräkningskostnad.

En teoretisk analys av MCMC metoder behandlas i artikel B och bygger på verktyg från teorin av stora avvikelser. Mer specifikt, vi analyserar MH algoritmens konvergens genom att formulera och bevisa en stora avvikelser princip (LDP) för de empiriska mått som produceras av algoritmen.

Några av metoderna analyserade i den här avhandlingen har implementerats i ett R paket som finns på GitHub som “icpm-kth/uqsa” och presenteras i artikel C. Metoderna tillämpas på subcellulära vägmodeller inom neuroner i sammanhanget av osäkerhetskvantifieringen av modellparametrar.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2023. , p. 121
Series
TRITA-SCI-FOU ; 2023:29
Keywords [en]
Markov chain Monte Carlo, Large deviations, Subcellular pathway models
Keywords [sv]
Markov chain Monte Carlo, Stora avvikelser, Subcellular pathway models
National Category
Probability Theory and Statistics
Research subject
Applied and Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-334400ISBN: 978-91-8040-652-9 (print)OAI: oai:DiVA.org:kth-334400DiVA, id: diva2:1789375
Presentation
2023-09-14, 3721, Lindstedtsvägen 25, Kungliga Tekniska Högskolan, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 2023-08-21

Available from: 2023-08-21 Created: 2023-08-18 Last updated: 2024-01-02Bibliographically approved
List of papers
1. Sensitivity Approximation by the Peano-Baker Series
Open this publication in new window or tab >>Sensitivity Approximation by the Peano-Baker Series
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we develop a new method for numerically approximating sensitivitiesin parameter-dependent ordinary differential equations (ODEs). Our approach,intended for situations where the standard forward and adjoint sensitivity analysisbecome too computationally costly for practical purposes, is based on the PeanoBaker series from control theory. We give a representation, using this series, for thesensitivity matrix S of an ODE system and use the representation to construct anumerical method for approximating S. We prove that, under standard regularityassumptions, the error of our method scales as O(∆t2max), where ∆tmax is the largesttime step used when numerically solving the ODE. We illustrate the performanceof the method in several numerical experiments, taken from both the systemsbiology setting and more classical dynamical systems. The experiments show thesought-after improvement in running time of our method compared to the forwardsensitivity approach. For example, in experiments involving a random linear system,the forward approach requires roughly √n longer computational time, where n isthe dimension of the parameter space, than our proposed method.

Keywords
Sensitivity analysis, Peano-Baker series, ordinary differential equations, error analysis
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-334388 (URN)10.48550/arxiv.2109.00067 (DOI)
Note

QC 20230824

Available from: 2023-08-18 Created: 2023-08-18 Last updated: 2023-08-24Bibliographically approved
2. A large deviation principle for the empirical measures of Metropolis-Hastings chains
Open this publication in new window or tab >>A large deviation principle for the empirical measures of Metropolis-Hastings chains
(English)Manuscript (preprint) (Other academic)
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-334393 (URN)10.48550/arXiv.2304.02775 (DOI)
Note

QC 20230823

Available from: 2023-08-18 Created: 2023-08-18 Last updated: 2023-08-23Bibliographically approved
3. UQSA - An R-Package for Uncertainty Quantification and Sensitivity Analysis for Biochemical Reaction Network Models
Open this publication in new window or tab >>UQSA - An R-Package for Uncertainty Quantification and Sensitivity Analysis for Biochemical Reaction Network Models
Show others...
(English)Manuscript (preprint) (Other academic)
National Category
Bioinformatics (Computational Biology) Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-334395 (URN)10.48550/arXiv.2308.05527 (DOI)
Note

QC 20230823

Available from: 2023-08-18 Created: 2023-08-18 Last updated: 2023-08-23Bibliographically approved

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