This document discusses agent-based models for disease transmission and sequential Monte Carlo algorithms for statistical inference of these models. It begins with an overview of agent-based models and their use in epidemiology. It then describes an agent-based SIS model where each agent can be susceptible or infected. Observations are the number of reported infections over time. The likelihood of the model involves a sum over all possible state sequences, which is intractable for large populations. The document proposes using sequential Monte Carlo methods to approximate the likelihood, including the bootstrap particle filter and auxiliary particle filter.
Statistical inference for agent-based SIS and SIR modelsJeremyHeng10
The document discusses statistical inference methods for agent-based models of epidemics. It presents an agent-based SIS model where each agent's state is modeled over time. Likelihood-based inference for these models is computationally challenging due to their complexity. The document proposes using sequential Monte Carlo methods like the auxiliary particle filter to approximate the likelihood. It additionally introduces a controlled sequential Monte Carlo method that uses dimensionality reduction to more efficiently construct proposal distributions.
This document summarizes a presentation on controlled sequential Monte Carlo. It discusses state space models, sequential Monte Carlo, and particle marginal Metropolis-Hastings for parameter inference. Controlled sequential Monte Carlo is proposed to lower the variance of the marginal likelihood estimator compared to standard sequential Monte Carlo, improving the performance of parameter inference methods. The method is illustrated on a neuroscience example where it reduces variance for different particle sizes.
Sequential Monte Carlo algorithms for agent-based models of disease transmissionJeremyHeng10
This document discusses sequential Monte Carlo algorithms for statistical inference in agent-based models of disease transmission. It begins with an overview of agent-based models and their use in epidemiology. It then describes an agent-based SIS model where each agent's state and transitions depend on covariates. The likelihood involves marginalizing over the latent states of all agents. Sequential Monte Carlo methods like particle filters are proposed to approximate this intractable likelihood. The document outlines the bootstrap particle filter and auxiliary particle filter approaches.
1) The document discusses unbiased Markov chain Monte Carlo (MCMC) methods for estimating expectations with respect to a target distribution.
2) It proposes running two coupled Markov chains that meet at a random termination time to provide an unbiased estimator. Averaging over multiple copies of this estimator then yields a consistent estimate.
3) Key aspects that enable unbiasedness are using a maximal coupling to propose moves that allow the chains to meet, and a bias correction term that accounts for the initial distributions not matching the target.
This document discusses unbiased estimation using coupled Markov chains. It introduces several unbiased estimators that require simulating coupled chains X and Y that meet at some point. Various Markov kernels are considered for coupling, including random walk Metropolis-Hastings, Gibbs samplers, and Hamiltonian Monte Carlo. Maximal coupling is discussed as a key tool for simulating chains that meet, and its algorithm is presented. Examples of coupling random walk Metropolis-Hastings, Gibbs samplers, and Hamiltonian Monte Carlo are provided.
Seminar at IEEE Computational Intelligence Society, Singapore Chapter at School of Electrical and Electronic Engineering, NTU, Singapore, 20 February 2019
short course at CIRM, Bayesian Masterclass, October 2018Christian Robert
Markov Chain Monte Carlo (MCMC) methods generate dependent samples from a target distribution using a Markov chain. The Metropolis-Hastings algorithm constructs a Markov chain with a desired stationary distribution by proposing moves to new states and accepting or rejecting them probabilistically. The algorithm is used to approximate integrals that are difficult to compute directly. It has been shown to converge to the target distribution as the number of iterations increases.
Statistical inference for agent-based SIS and SIR modelsJeremyHeng10
The document discusses statistical inference methods for agent-based models of epidemics. It presents an agent-based SIS model where each agent's state is modeled over time. Likelihood-based inference for these models is computationally challenging due to their complexity. The document proposes using sequential Monte Carlo methods like the auxiliary particle filter to approximate the likelihood. It additionally introduces a controlled sequential Monte Carlo method that uses dimensionality reduction to more efficiently construct proposal distributions.
This document summarizes a presentation on controlled sequential Monte Carlo. It discusses state space models, sequential Monte Carlo, and particle marginal Metropolis-Hastings for parameter inference. Controlled sequential Monte Carlo is proposed to lower the variance of the marginal likelihood estimator compared to standard sequential Monte Carlo, improving the performance of parameter inference methods. The method is illustrated on a neuroscience example where it reduces variance for different particle sizes.
Sequential Monte Carlo algorithms for agent-based models of disease transmissionJeremyHeng10
This document discusses sequential Monte Carlo algorithms for statistical inference in agent-based models of disease transmission. It begins with an overview of agent-based models and their use in epidemiology. It then describes an agent-based SIS model where each agent's state and transitions depend on covariates. The likelihood involves marginalizing over the latent states of all agents. Sequential Monte Carlo methods like particle filters are proposed to approximate this intractable likelihood. The document outlines the bootstrap particle filter and auxiliary particle filter approaches.
1) The document discusses unbiased Markov chain Monte Carlo (MCMC) methods for estimating expectations with respect to a target distribution.
2) It proposes running two coupled Markov chains that meet at a random termination time to provide an unbiased estimator. Averaging over multiple copies of this estimator then yields a consistent estimate.
3) Key aspects that enable unbiasedness are using a maximal coupling to propose moves that allow the chains to meet, and a bias correction term that accounts for the initial distributions not matching the target.
This document discusses unbiased estimation using coupled Markov chains. It introduces several unbiased estimators that require simulating coupled chains X and Y that meet at some point. Various Markov kernels are considered for coupling, including random walk Metropolis-Hastings, Gibbs samplers, and Hamiltonian Monte Carlo. Maximal coupling is discussed as a key tool for simulating chains that meet, and its algorithm is presented. Examples of coupling random walk Metropolis-Hastings, Gibbs samplers, and Hamiltonian Monte Carlo are provided.
Seminar at IEEE Computational Intelligence Society, Singapore Chapter at School of Electrical and Electronic Engineering, NTU, Singapore, 20 February 2019
short course at CIRM, Bayesian Masterclass, October 2018Christian Robert
Markov Chain Monte Carlo (MCMC) methods generate dependent samples from a target distribution using a Markov chain. The Metropolis-Hastings algorithm constructs a Markov chain with a desired stationary distribution by proposing moves to new states and accepting or rejecting them probabilistically. The algorithm is used to approximate integrals that are difficult to compute directly. It has been shown to converge to the target distribution as the number of iterations increases.
This document discusses computational issues that arise in Bayesian statistics. It provides examples of latent variable models like mixture models that make computation difficult due to the large number of terms that must be calculated. It also discusses time series models like the AR(p) and MA(q) models, noting that they have complex parameter spaces due to stationarity constraints. The document outlines the Metropolis-Hastings algorithm, Gibbs sampler, and other methods like Population Monte Carlo and Approximate Bayesian Computation that can help address these computational challenges.
Estimation of the score vector and observed information matrix in intractable...Pierre Jacob
This document discusses methods for estimating the score vector and observed information matrix for intractable models. It begins with an overview of using derivatives in sampling algorithms. It then discusses iterated filtering, a method for estimating derivatives in hidden Markov models when the likelihood is not available in closed form. Iterated filtering introduces a perturbed model and relates the posterior mean to the score and posterior variance to the observed information matrix. The document outlines proofs that show this relationship as the prior concentration increases.
This document summarizes controlled sequential Monte Carlo, which aims to efficiently estimate intractable likelihoods p(y|θ) in state space models. It does this by defining a target path measure P(dx0:T) and proposal Markov chain Q(dx0:T) to approximate P(dx0:T). Standard sequential Monte Carlo (SMC) methods provide unbiased estimation but can have inadequate performance for practical particle sizes N due to discrepancy between P and Q. The document proposes using twisted path measures that depend on observations to better match P and Q, by defining proposal transitions P(dxt|xt-1,yt:T) that incorporate backward information filters ψ*t(xt)=P(yt
1. The document discusses approximate Bayesian computation (ABC), a technique used when the likelihood function is intractable. ABC works by simulating parameters from the prior and simulating data, rejecting simulations that are not close to the observed data based on a tolerance level.
2. Random forests can be used in ABC to select informative summary statistics from a large set of possibilities and estimate parameters. The random forests classify simulations as accepted or rejected based on the summaries, implicitly selecting important summaries.
3. Calibrating the tolerance level in ABC is important but difficult, as it determines how close simulations must be to the observed data. Methods discussed include using quantiles of prior predictive simulations or asymptotic convergence properties.
This document summarizes a talk given by Heiko Strathmann on using partial posterior paths to estimate expectations from large datasets without full posterior simulation. The key ideas are:
1. Construct a path of "partial posteriors" by sequentially adding mini-batches of data and computing expectations over these posteriors.
2. "Debias" the path of expectations to obtain an unbiased estimator of the true posterior expectation using a technique from stochastic optimization literature.
3. This approach allows estimating posterior expectations with sub-linear computational cost in the number of data points, without requiring full posterior simulation or imposing restrictions on the likelihood.
Experiments on synthetic and real-world examples demonstrate competitive performance versus standard M
The document summarizes a talk given by Mark Girolami on manifold Monte Carlo methods. It discusses using stochastic diffusions and geometric concepts to improve MCMC methods. Specifically, it proposes using discretized Langevin and Hamiltonian diffusions across a Riemann manifold as an adaptive proposal mechanism. This is founded on deterministic geodesic flows on the manifold. Examples presented include a warped bivariate Gaussian, Gaussian mixture model, and log-Gaussian Cox process.
This document discusses various importance sampling methods for approximating Bayes factors, which are used for Bayesian model selection. It compares regular importance sampling, bridge sampling, harmonic means, mixtures to bridge sampling, and Chib's solution. An example application to probit modeling of diabetes in Pima Indian women is presented to illustrate regular importance sampling. Markov chain Monte Carlo methods like the Metropolis-Hastings algorithm and Gibbs sampling can be used to sample from the probit models.
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 3.
More info at http://summerschool.ssa.org.ua
The document discusses particle filtering and state-space processes. It provides an overview of two commonly used particle filters: the bootstrap filter and auxiliary particle filter. It also presents an example of applying particle filtering to a stochastic volatility model.
Rao-Blackwellisation schemes for accelerating Metropolis-Hastings algorithmsChristian Robert
Aggregate of three different papers on Rao-Blackwellisation, from Casella & Robert (1996), to Douc & Robert (2010), to Banterle et al. (2015), presented during an OxWaSP workshop on MCMC methods, Warwick, Nov 20, 2015
Those are the slides for my Master course on Monte Carlo Statistical Methods given in conjunction with the Monte Carlo Statistical Methods book with George Casella.
This document discusses Markov chain Monte Carlo (MCMC) methods. It begins with an outline of the Metropolis-Hastings algorithm, which is a generic MCMC method for obtaining a sequence of random samples from a probability distribution when direct sampling is difficult. The document then provides details on the Metropolis-Hastings algorithm, including its convergence properties. It also discusses the independent Metropolis-Hastings algorithm as a special case and provides an example to illustrate it.
Approximate Bayesian Computation with Quasi-LikelihoodsStefano Cabras
This document describes ABC-MCMC algorithms that use quasi-likelihoods as proposals. It introduces quasi-likelihoods as approximations to true likelihoods that can be estimated from pilot runs. The ABCql algorithm uses a quasi-likelihood estimated from a pilot run as the proposal in an ABC-MCMC algorithm. Examples applying ABCql to mixture of normals, coalescent, and gamma models are provided to demonstrate its effectiveness compared to standard ABC-MCMC.
This document provides an overview of Markov chain Monte Carlo (MCMC) methods. It begins with motivations for using MCMC, such as computational difficulties that arise in models with latent variables like mixture models. It then discusses likelihood-based and Bayesian approaches, noting limitations of maximum likelihood methods. Conjugate priors are described that allow tractable Bayesian inference for some simple models. However, conjugate priors are not available for more complex models, motivating the use of MCMC methods which can approximate integrals and distributions of interest for more complex models.
The document discusses Approximate Bayesian Computation (ABC). ABC allows inference for statistical models where the likelihood function is not available in closed form. ABC works by simulating data under different parameter values and comparing simulated to observed data. ABC has been used for model choice by comparing evidence for different models. Consistency of ABC for model choice depends on the criterion used and asymptotic identifiability of the parameters.
Bayesian hybrid variable selection under generalized linear modelsCaleb (Shiqiang) Jin
This document presents a method for Bayesian variable selection under generalized linear models. It begins by introducing the model setting and Bayesian model selection framework. It then discusses three algorithms for model search: deterministic search, stochastic search, and a hybrid search method. The key contribution is a method to simultaneously evaluate the marginal likelihoods of all neighbor models, without parallel computing. This is achieved by decomposing the coefficient vectors and estimating additional coefficients conditioned on the current model's coefficients. Newton-Raphson iterations are used to solve the system of equations and obtain the maximum a posteriori estimates for all neighbor models simultaneously in a single computation. This allows for a fast, inexpensive search of the model space.
Sampling strategies for Sequential Monte Carlo (SMC) methodsStephane Senecal
Sequential Monte Carlo methods use importance sampling and resampling to estimate distributions in state space models recursively over time. This document discusses strategies for sampling in sequential Monte Carlo methods, including:
- Using the optimal proposal distribution of the one-step ahead predictive distribution to minimize weight variance.
- Approximating the predictive distribution using mixtures, expansions, auxiliary variables, or Markov chain Monte Carlo methods.
- Considering blocks of variables over time rather than individual time steps to better diffuse particles, such as using a lagged block, reweighting particles before resampling, or sampling an extended block with an augmented state space.
Sequential Monte Carlo Algorithms for Agent-based Models of Disease TransmissionJeremyHeng10
1) The document presents sequential Monte Carlo algorithms for statistical inference in agent-based models of disease transmission. Agent-based models simulate disease spread through a population of interacting agents.
2) Exact likelihood-based inference is computationally challenging for these models due to their complexity. The document proposes using auxiliary particle filtering and controlled sequential Monte Carlo to approximate the likelihood.
3) These methods aim to improve on the bootstrap particle filter by incorporating more information from the observations through dimension reduction approximations of the agent-based model. This allows efficient proposal distributions to be constructed for sequential Monte Carlo.
Allele Frequencies as Stochastic Processes: Mathematical & Statistical Approa...Gota Morota
The document discusses modeling allele frequency changes over time as stochastic processes. It describes allele frequencies changing as random walks or Brownian motion. It presents the Fokker-Planck equation for describing the probability distribution of allele frequencies over time under various evolutionary forces like genetic drift, selection, and mutation. The steady state distribution of allele frequencies and solutions to the Fokker-Planck equation are discussed for different evolutionary scenarios. Time series analysis methods are introduced for modeling allele frequency change as a discrete time process. An example application to cattle genotype data is shown.
This document discusses computational issues that arise in Bayesian statistics. It provides examples of latent variable models like mixture models that make computation difficult due to the large number of terms that must be calculated. It also discusses time series models like the AR(p) and MA(q) models, noting that they have complex parameter spaces due to stationarity constraints. The document outlines the Metropolis-Hastings algorithm, Gibbs sampler, and other methods like Population Monte Carlo and Approximate Bayesian Computation that can help address these computational challenges.
Estimation of the score vector and observed information matrix in intractable...Pierre Jacob
This document discusses methods for estimating the score vector and observed information matrix for intractable models. It begins with an overview of using derivatives in sampling algorithms. It then discusses iterated filtering, a method for estimating derivatives in hidden Markov models when the likelihood is not available in closed form. Iterated filtering introduces a perturbed model and relates the posterior mean to the score and posterior variance to the observed information matrix. The document outlines proofs that show this relationship as the prior concentration increases.
This document summarizes controlled sequential Monte Carlo, which aims to efficiently estimate intractable likelihoods p(y|θ) in state space models. It does this by defining a target path measure P(dx0:T) and proposal Markov chain Q(dx0:T) to approximate P(dx0:T). Standard sequential Monte Carlo (SMC) methods provide unbiased estimation but can have inadequate performance for practical particle sizes N due to discrepancy between P and Q. The document proposes using twisted path measures that depend on observations to better match P and Q, by defining proposal transitions P(dxt|xt-1,yt:T) that incorporate backward information filters ψ*t(xt)=P(yt
1. The document discusses approximate Bayesian computation (ABC), a technique used when the likelihood function is intractable. ABC works by simulating parameters from the prior and simulating data, rejecting simulations that are not close to the observed data based on a tolerance level.
2. Random forests can be used in ABC to select informative summary statistics from a large set of possibilities and estimate parameters. The random forests classify simulations as accepted or rejected based on the summaries, implicitly selecting important summaries.
3. Calibrating the tolerance level in ABC is important but difficult, as it determines how close simulations must be to the observed data. Methods discussed include using quantiles of prior predictive simulations or asymptotic convergence properties.
This document summarizes a talk given by Heiko Strathmann on using partial posterior paths to estimate expectations from large datasets without full posterior simulation. The key ideas are:
1. Construct a path of "partial posteriors" by sequentially adding mini-batches of data and computing expectations over these posteriors.
2. "Debias" the path of expectations to obtain an unbiased estimator of the true posterior expectation using a technique from stochastic optimization literature.
3. This approach allows estimating posterior expectations with sub-linear computational cost in the number of data points, without requiring full posterior simulation or imposing restrictions on the likelihood.
Experiments on synthetic and real-world examples demonstrate competitive performance versus standard M
The document summarizes a talk given by Mark Girolami on manifold Monte Carlo methods. It discusses using stochastic diffusions and geometric concepts to improve MCMC methods. Specifically, it proposes using discretized Langevin and Hamiltonian diffusions across a Riemann manifold as an adaptive proposal mechanism. This is founded on deterministic geodesic flows on the manifold. Examples presented include a warped bivariate Gaussian, Gaussian mixture model, and log-Gaussian Cox process.
This document discusses various importance sampling methods for approximating Bayes factors, which are used for Bayesian model selection. It compares regular importance sampling, bridge sampling, harmonic means, mixtures to bridge sampling, and Chib's solution. An example application to probit modeling of diabetes in Pima Indian women is presented to illustrate regular importance sampling. Markov chain Monte Carlo methods like the Metropolis-Hastings algorithm and Gibbs sampling can be used to sample from the probit models.
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 3.
More info at http://summerschool.ssa.org.ua
The document discusses particle filtering and state-space processes. It provides an overview of two commonly used particle filters: the bootstrap filter and auxiliary particle filter. It also presents an example of applying particle filtering to a stochastic volatility model.
Rao-Blackwellisation schemes for accelerating Metropolis-Hastings algorithmsChristian Robert
Aggregate of three different papers on Rao-Blackwellisation, from Casella & Robert (1996), to Douc & Robert (2010), to Banterle et al. (2015), presented during an OxWaSP workshop on MCMC methods, Warwick, Nov 20, 2015
Those are the slides for my Master course on Monte Carlo Statistical Methods given in conjunction with the Monte Carlo Statistical Methods book with George Casella.
This document discusses Markov chain Monte Carlo (MCMC) methods. It begins with an outline of the Metropolis-Hastings algorithm, which is a generic MCMC method for obtaining a sequence of random samples from a probability distribution when direct sampling is difficult. The document then provides details on the Metropolis-Hastings algorithm, including its convergence properties. It also discusses the independent Metropolis-Hastings algorithm as a special case and provides an example to illustrate it.
Approximate Bayesian Computation with Quasi-LikelihoodsStefano Cabras
This document describes ABC-MCMC algorithms that use quasi-likelihoods as proposals. It introduces quasi-likelihoods as approximations to true likelihoods that can be estimated from pilot runs. The ABCql algorithm uses a quasi-likelihood estimated from a pilot run as the proposal in an ABC-MCMC algorithm. Examples applying ABCql to mixture of normals, coalescent, and gamma models are provided to demonstrate its effectiveness compared to standard ABC-MCMC.
This document provides an overview of Markov chain Monte Carlo (MCMC) methods. It begins with motivations for using MCMC, such as computational difficulties that arise in models with latent variables like mixture models. It then discusses likelihood-based and Bayesian approaches, noting limitations of maximum likelihood methods. Conjugate priors are described that allow tractable Bayesian inference for some simple models. However, conjugate priors are not available for more complex models, motivating the use of MCMC methods which can approximate integrals and distributions of interest for more complex models.
The document discusses Approximate Bayesian Computation (ABC). ABC allows inference for statistical models where the likelihood function is not available in closed form. ABC works by simulating data under different parameter values and comparing simulated to observed data. ABC has been used for model choice by comparing evidence for different models. Consistency of ABC for model choice depends on the criterion used and asymptotic identifiability of the parameters.
Bayesian hybrid variable selection under generalized linear modelsCaleb (Shiqiang) Jin
This document presents a method for Bayesian variable selection under generalized linear models. It begins by introducing the model setting and Bayesian model selection framework. It then discusses three algorithms for model search: deterministic search, stochastic search, and a hybrid search method. The key contribution is a method to simultaneously evaluate the marginal likelihoods of all neighbor models, without parallel computing. This is achieved by decomposing the coefficient vectors and estimating additional coefficients conditioned on the current model's coefficients. Newton-Raphson iterations are used to solve the system of equations and obtain the maximum a posteriori estimates for all neighbor models simultaneously in a single computation. This allows for a fast, inexpensive search of the model space.
Sampling strategies for Sequential Monte Carlo (SMC) methodsStephane Senecal
Sequential Monte Carlo methods use importance sampling and resampling to estimate distributions in state space models recursively over time. This document discusses strategies for sampling in sequential Monte Carlo methods, including:
- Using the optimal proposal distribution of the one-step ahead predictive distribution to minimize weight variance.
- Approximating the predictive distribution using mixtures, expansions, auxiliary variables, or Markov chain Monte Carlo methods.
- Considering blocks of variables over time rather than individual time steps to better diffuse particles, such as using a lagged block, reweighting particles before resampling, or sampling an extended block with an augmented state space.
Sequential Monte Carlo Algorithms for Agent-based Models of Disease TransmissionJeremyHeng10
1) The document presents sequential Monte Carlo algorithms for statistical inference in agent-based models of disease transmission. Agent-based models simulate disease spread through a population of interacting agents.
2) Exact likelihood-based inference is computationally challenging for these models due to their complexity. The document proposes using auxiliary particle filtering and controlled sequential Monte Carlo to approximate the likelihood.
3) These methods aim to improve on the bootstrap particle filter by incorporating more information from the observations through dimension reduction approximations of the agent-based model. This allows efficient proposal distributions to be constructed for sequential Monte Carlo.
Allele Frequencies as Stochastic Processes: Mathematical & Statistical Approa...Gota Morota
The document discusses modeling allele frequency changes over time as stochastic processes. It describes allele frequencies changing as random walks or Brownian motion. It presents the Fokker-Planck equation for describing the probability distribution of allele frequencies over time under various evolutionary forces like genetic drift, selection, and mutation. The steady state distribution of allele frequencies and solutions to the Fokker-Planck equation are discussed for different evolutionary scenarios. Time series analysis methods are introduced for modeling allele frequency change as a discrete time process. An example application to cattle genotype data is shown.
Data fusion is the process of combining data from different sources to enhance the utility of the combined product. In remote sensing, input data sources are typically massive, noisy, and have different spatial supports and sampling characteristics. We take an inferential approach to this data fusion problem: we seek to infer a true but not directly observed spatial (or spatio-temporal) field from heterogeneous inputs. We use a statistical model to make these inferences, but like all models it is at least somewhat uncertain. In this talk, we will discuss our experiences with the impacts of these uncertainties and some potential ways addressing them.
Bayesian inference for mixed-effects models driven by SDEs and other stochast...Umberto Picchini
An important, and well studied, class of stochastic models is given by stochastic differential equations (SDEs). In this talk, we consider Bayesian inference based on measurements from several individuals, to provide inference at the "population level" using mixed-effects modelling. We consider the case where dynamics are expressed via SDEs or other stochastic (Markovian) models. Stochastic differential equation mixed-effects models (SDEMEMs) are flexible hierarchical models that account for (i) the intrinsic random variability in the latent states dynamics, as well as (ii) the variability between individuals, and also (iii) account for measurement error. This flexibility gives rise to methodological and computational difficulties.
Fully Bayesian inference for nonlinear SDEMEMs is complicated by the typical intractability of the observed data likelihood which motivates the use of sampling-based approaches such as Markov chain Monte Carlo. A Gibbs sampler is proposed to target the marginal posterior of all parameters of interest. The algorithm is made computationally efficient through careful use of blocking strategies, particle filters (sequential Monte Carlo) and correlated pseudo-marginal approaches. The resulting methodology is is flexible, general and is able to deal with a large class of nonlinear SDEMEMs [1]. In a more recent work [2], we also explored ways to make inference even more scalable to an increasing number of individuals, while also dealing with state-space models driven by other stochastic dynamic models than SDEs, eg Markov jump processes and nonlinear solvers typically used in systems biology.
[1] S. Wiqvist, A. Golightly, AT McLean, U. Picchini (2020). Efficient inference for stochastic differential mixed-effects models using correlated particle pseudo-marginal algorithms, CSDA, https://doi.org/10.1016/j.csda.2020.107151
[2] S. Persson, N. Welkenhuysen, S. Shashkova, S. Wiqvist, P. Reith, G. W. Schmidt, U. Picchini, M. Cvijovic (2021). PEPSDI: Scalable and flexible inference framework for stochastic dynamic single-cell models, bioRxiv doi:10.1101/2021.07.01.450748.
Stochastic reaction networks (SRNs) are a particular class of continuous-time Markov chains used to model a wide range of phenomena, including biological/chemical reactions, epidemics, risk theory, queuing, and supply chain/social/multi-agents networks. In this context, we explore the efficient estimation of statistical quantities, particularly rare event probabilities, and propose two alternative importance sampling (IS) approaches [1,2] to improve the Monte Carlo (MC) estimator efficiency. The key challenge in the IS framework is to choose an appropriate change of probability measure to achieve substantial variance reduction, which often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection between finding optimal IS parameters and solving a variance minimization problem via a stochastic optimal control formulation. We pursue two alternative approaches to mitigate the curse of dimensionality when solving the resulting dynamic programming problem. In the first approach [1], we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. As an alternative, we present in [2] a dimension reduction method, based on mapping the problem to a significantly lower dimensional space via the Markovian projection (MP) idea. The output of this model reduction technique is a low dimensional SRN (potentially one dimension) that preserves the marginal distribution of the original high-dimensional SRN system. The dynamics of the projected process are obtained via a discrete $L^2$ regression. By solving a resulting projected Hamilton-Jacobi-Bellman (HJB) equation for the reduced-dimensional SRN, we get projected IS parameters, which are then mapped back to the original full-dimensional SRN system, and result in an efficient IS-MC estimator of the full-dimensional SRN. Our analysis and numerical experiments verify that both proposed IS (learning based and MP-HJB-IS) approaches substantially reduce the MC estimator’s variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators. [1] Ben Hammouda, C., Ben Rached, N., and Tempone, R., and Wiechert, S. Learning-based importance sampling via stochastic optimal control for stochastic reaction net-works. Statistics and Computing 33, no. 3 (2023): 58. [2] Ben Hammouda, C., Ben Rached, N., and Tempone, R., and Wiechert, S. (2023). Automated Importance Sampling via Optimal Control for Stochastic Reaction Networks: A Markovian Projection-based Approach. To appear soon.
Dependent processes in Bayesian NonparametricsJulyan Arbel
This document summarizes dependent processes in Bayesian nonparametrics. It motivates the need for dependent random probability measures to accommodate temporal dependence structures beyond the exchangeability assumption. It describes modeling collections of random probability measures indexed by time as either discrete-time or continuous-time processes. The diffusive Dirichlet process is introduced as a dependent Dirichlet process with Dirichlet marginal distributions at each time point and continuous sample paths. Simulation and estimation methods are discussed for this model.
Looking Inside Mechanistic Models of CarcinogenesisSascha Zöllner
This talk discusses the basic mathematical approaches and motivations underlying mechanistic models of carcinogenesis, specifically multi-stage models. After discussing simple ODE-based deterministic models, stochastic cancer models are introduced. On the simplest examples of the 1-stage (Poisson) process and a minimal 2-stage model, the basic features of such models are laid out. We then proceed to treat the widely used two-stage model with clonal expansion (TSCE), and its application to calculating risks due to external agents, such as radiation.
Noise is unwanted sound considered unpleasant, loud, or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arises when the brain receives and perceives a sound.
Monte Carlo methods use random sampling to solve quantitative problems. They were first used by Stanislaw Ulam and Nicholas Metropolis to solve non-random problems by transforming them into random forms. Monte Carlo simulations play a major role in experimental physics by designing experiments, evaluating potential outputs and risks, and validating results. Random numbers are generated using pseudorandom number generators or by transforming uniform random variables using probability distribution functions. The accuracy of Monte Carlo simulations improves as the number of samples increases, with the standard error declining proportionally with the square root of the number of samples.
Spatial Point Processes and Their Applications in EpidemiologyLilac Liu Xu
Spatial statistics can be used in epidemiology to analyze spatial point patterns of disease cases and controls. Common models include homogeneous and inhomogeneous Poisson processes, which describe patterns of complete spatial randomness and non-random clustering or dispersion. Descriptive statistics like the first-order intensity function λ(s) and second-order K-function can quantify clustering in a point pattern. A case-control study compares these statistics between case and control patterns to test for non-random spatial variations in disease risk. Monte Carlo simulations are used to calculate p-values when testing hypotheses about relative risk and clustering.
Xi Zhang presented their Ph.D. dissertation which analyzed functional regression models and their application to high-frequency financial data. The presentation included:
1. An introduction to functional data analysis and the use of intraday cumulative return curves from stock price data.
2. A simulation study comparing predictive methods in functional autoregressive models, finding the estimated kernel method performed well.
3. An application of functional extensions of the Capital Asset Pricing Model to predict intraday return curves, finding simpler models with intercepts had better predictive performance than more complex models.
This document provides a course calendar and lecture plans for topics related to Bayesian estimation methods. The course calendar lists 12 class dates from September to December covering topics like Bayes estimation, Kalman filters, particle filters, hidden Markov models, supervised learning, and clustering algorithms. One lecture plan provides details on the hidden Markov model, including the introduction, definition of HMMs, and problems of evaluation, decoding, and learning. Another lecture plan covers particle filters, including the sequential importance sampling algorithm, choice of proposal density, and the particle filter algorithm of sampling, weight update, resampling, and state estimation.
Monte Carlo methods rely on repeated random sampling to compute results. They generate random samples from a population according to a probability distribution and use them to obtain numerical results. The founders of the Monte Carlo method were J. von Neumann and S. Ulam during the Manhattan Project in the 1940s. Monte Carlo methods can be used to solve multidimensional integrals and have better convergence than classical numerical integration methods for dimensions greater than 4. The variance of Monte Carlo estimates decreases as 1/N, where N is the number of samples, resulting in slow convergence. Variance reduction techniques can improve the convergence rate.
Quantitatively describe inherently random quantities using the methods of probability and statistics
Plot random data using a relative frequency histogram, cumulative frequency histogram and probability plot
Use the method of moments to fit a probability distribution.
Use frequency analysis to calculate flood flows of a given return period
Calculate the probability and risk associated with hydrologic events.
This document discusses dynamics of structures with uncertainties. It begins with an introduction to stochastic single degree of freedom systems and how natural frequency variability can be modeled using probability distributions. It then discusses how to extend this approach to stochastic multi degree of freedom systems using stochastic finite element formulations and modal projections. Key challenges with statistical overlap of eigenvalues are noted. The document provides mathematical models of equivalent damping in stochastic systems and examples of stochastic frequency response functions.
The tau-leap method for simulating stochastic kinetic modelsColin Gillespie
This document discusses approximate methods for simulating chemically reacting systems stochastically in a more computationally efficient manner than the direct method. It introduces the τ-leap method, where reactions are simulated in fixed time intervals (τ) by assuming reaction rates are constant over τ. It describes how τ can be chosen to satisfy a "leap condition" and minimize errors. The midpoint estimation technique is introduced to further reduce errors by estimating propensities at the midpoint of each τ interval. Examples applying these methods to a Lotka-Volterra system are provided to illustrate the techniques.
A brief introduction to mutual information and its applicationHyun-hwan Jeong
Mutual information is introduced as a measure for dependence between random variables that is based on entropy. It is defined as the difference between the joint entropy of two variables and the sum of their individual entropies. Mutual information has various applications including association measures between genomic features and outcomes, using mutual information for distance measures in clustering to detect epistatic interactions, and constructing outcome-guided mutual information networks for prediction. Challenges with mutual information include handling noise in continuous data and assessing statistical significance while accounting for multiple testing.
Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time...SYRTO Project
Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time Series Models. Andre Lucas. Amsterdam - June, 25 2015. European Financial Management Association 2015 Annual Meetings.
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Sequential Monte Carlo algorithms for agent-based models of disease transmission
1. Sequential Monte Carlo algorithms for
agent-based models of disease transmission
Jeremy Heng
ESSEC Business School
Joint work with Phyllis Ju (Purdue) and Pierre Jacob (ESSEC)
KAUST
1 September 2021
JH Agent-based models 1/ 42
4. Agent-based models
• Agent-based models specify how a population of agents
interact and evolve over time
• Flexible, interpretable and widely employed in many fields
(e.g. ecology, epidemiology, transportation)
• Can render realistic macroscopic phenomena from simple
microscopic rules
Figure: SimCity by Electronic Arts
JH Agent-based models 3/ 42
5. A brief history of agent-based models
• Dates back to work by von Neumann and Ulam in 1940s on
cellular automata
• Popularized in many disciplines during the 1990s for various
reasons
• Growing computational power made it possible to simulate
such models
• Low levels of mathematical sophistication required to build
such models
JH Agent-based models 4/ 42
7. Calibration of agent-based models
• These models are typically calibrated by matching key
features of simulated and actual data
• Can be computationally intensive and difficult to calibrate
126 CHAPTER 5
Figure 5.3. Simulated and historical settlement patterns, in red, for Long House
Valley in A.D. 1125. North is to the top of the page.
of the 1270–1450 period could have supported a reduced but substantial
population in small settlements dispersed across suitable farming habitats
located primarily in areas of high potential crop production in the
Figure: Simulated and historical settlement patterns in long house valley
JH Agent-based models 6/ 42
8. Statistical inference for agent-based models
• Given occasional noisy measurements of the population, we
could consider statistical inference for such models
• Few works have addressed this important topic as
likelihood-based inference is computationally challenging
• We propose sequential Monte Carlo algorithms for some
classical agent-based models in epidemiology
• The general principle is to ‘open the black box’ nature of
these models and exploit its inherent structure
JH Agent-based models 7/ 42
9. Compartmental models in epidemiology
• A population-level approach assigns the population to
compartments and models the number of people in each
compartment over time
SIR model
Susceptible
Infected
Recovered
λ
γ
SIS model
Susceptible
Infected
λ
γ
JH Agent-based models 8/ 42
10. Agent-based models in epidemiology
• The agent-based approach assumes agents can take these
states and models the state of each agent n over time
SIR model
Susceptible
Infected
Recovered
λn
γn
SIS model
Susceptible
Infected
λn
γn
JH Agent-based models 9/ 42
11. Why agent-based models?
• May be unrealistic to assume agents are interchangeable
0.0
0.1
0.2
0.3
0.4
0.5
0 10 20 30
Incubation period
Density
(days)
Gender Men Women
0.0
0.1
0.2
0.3
0 10 20 30
Incubation period
Density
(days)
Age <50 >=50
Figure: Gender-specific (left) and age-specific (right) distributions of
COVID-19 incubation period (Zhao et al. 2020, AoAS)
JH Agent-based models 10/ 42
12. Why agent-based models?
• May be unrealistic to assume all agents interact
1
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4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
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17
18
19
20
Figure: Fully connected network versus small world network
JH Agent-based models 11/ 42
14. Agent-based SIS model
• We consider the agent-based SIS model and encode
Susceptible = 0 and Infected = 1
• Let Xt = (Xn
t )n∈[1:N] ∈ {0, 1}N denote the state of a closed
population of N agents at time t ∈ [0 : T]
• Initialization X0 ∼ µθ given by
Xn
0 ∼ Ber(αn
0), independently for n ∈ [1 : N]
• Markov transition Xt ∼ fθ(·|Xt−1) at time t ∈ [1 : T] is
given by
Xn
t ∼ Ber(αn
(Xt−1)), independently for n ∈ [1 : N]
JH Agent-based models 12/ 42
15. Agent-based SIS model
• Transition probability specified as
αn
(Xt−1) =
(
λnD(n)−1
P
m∈N(n) Xm
t−1, if Xn
t−1 = 0
1 − γn, if Xn
t−1 = 1
• Interactions specified by an undirected network: D(n) and
N(n) denote the degree and neighbours of agent n
• Infection and recovery rates are modelled using
agent-specific attributes
λn
= (1 + exp(−β>
λ wn
))−1
, γn
= (1 + exp(−β>
γ wn
))−1
,
where βλ, βγ ∈ Rd are parameters and wn ∈ Rd are the
covariates of agent n (similarly αn
0 depends on β0)
JH Agent-based models 13/ 42
16. Agent-based SIS model
• If the network is fully connected D(n) = N, N(n) = [1 : N]
and the agents are homogeneous λn = λ, γn = γ
• We recover the classical SIS model of Kermack and
McKendrick (1927), which has a deterministic limit as
N → ∞
• These simpler models offer dimension reduction which
facilitates inference
• However, one cannot incorporate network information and
agent attributes
• We will use these simplifications to construct efficient SMC
proposal distributions for the agent-based model
JH Agent-based models 14/ 42
17. Agent-based SIS model
• Observations (Yt)t∈[0:T] are the number of infections
reported over time
• Modelled as conditionally independent given (Xt)t∈[0:T], and
Yt ∼ gθ(·|Xt) = Bin(I(Xt), ρ)
• I(Xt) =
PN
n=1 Xn
t is the number of infections and ρ ∈ (0, 1) is
the reporting rate
• Parameters to be inferred θ = (β0, βλ, βγ, ρ)
JH Agent-based models 15/ 42
19. Likelihood of agent-based SIS model
• We have a standard hidden Markov model
pθ(x0:T , y0:T ) = µθ(x0)
T
Y
t=1
fθ(xt|xt−1)
T
Y
t=0
gθ(yt|xt)
• The marginal likelihood is
pθ(y0:T ) =
X
x0:T ∈{0,1}N×(T+1)
pθ(x0:T , y0:T ),
• Maximum likelihood estimation computes
arg max
θ
pθ(y0:T )
• Bayesian inference samples from
p(θ|y0:T ) ∝ p(θ)pθ(y0:T )
JH Agent-based models 17/ 42
20. Likelihood of agent-based SIS model
• We have a hidden Markov model on a discrete state-space
• We can employ forward algorithm to compute the marginal
likelihood exactly
• The cost is of order
(no. of states)2
× (no. of observations) = O(22N
T)
• For large N, we have to rely on sequential Monte Carlo
(SMC) methods to approximate the marginal likelihood
JH Agent-based models 18/ 42
22. Sequential Monte Carlo
• Sequential Monte Carlo (SMC) methods, aka particle
filters, are now quite advanced and well-understood since its
introduction in the 90s
• The idea is to recursively simulate an interacting particle
system of size P
• For time t ∈ [0 : T], we have P states and ancestor indexes
(X
(1)
t , . . . , X
(P)
t ), (A
(1)
t , . . . , A
(P)
t )
JH Agent-based models 19/ 42
23. Sequential Monte Carlo
…
X0 X1 XT
Y0 Y1 YT
…
✓
For time t = 0 and particle p ∈ [1 : P]
sample X
(p)
0 ∼ q0(x0|θ)
JH Agent-based models 20/ 42
24. Sequential Monte Carlo
…
X0 X1 XT
Y0 Y1 YT
…
✓
For time t = 0 and particle p ∈ [1 : P]
weight W
(p)
0 ∝ w0(X
(p)
0 )
JH Agent-based models 20/ 42
25. Sequential Monte Carlo
…
X0 X1 XT
Y0 Y1 YT
…
✓
X
X
X
For time t = 0 and particle p ∈ [1 : P]
sample ancestor A
(p)
0 ∼ R
W
(1)
0 , . . . , W
(P)
0
, resampled particle: X
A
(p)
0
0
JH Agent-based models 20/ 42
26. Sequential Monte Carlo
…
X0 X1 XT
Y0 Y1 YT
…
✓
X
X
X
For time t = 1 and particle p ∈ [1 : P]
sample X
(p)
1 ∼ q1(x1|X
A
(p)
0
0 , θ)
JH Agent-based models 20/ 42
27. Sequential Monte Carlo
…
X0 X1 XT
Y0 Y1 YT
…
✓
X
X
X
For time t = 1 and particle p ∈ [1 : P]
weight W n
1 ∝ w1(X
A
(p)
0
0 , X
(p)
1 )
JH Agent-based models 20/ 42
28. Sequential Monte Carlo
…
X0 X1 XT
Y0 Y1 YT
…
✓
X
X
X X
For time t = 1 and particle p ∈ [1 : P]
sample ancestor A
(p)
1 ∼ R
W
(1)
1 , . . . , W
(P)
1
, resampled particle: X
A
(p)
1
1
JH Agent-based models 20/ 42
29. Sequential Monte Carlo
…
X0 X1 XT
Y0 Y1 YT
…
✓
X
X
X X
X
X
X
Repeat for time t ∈ [2 : T].
JH Agent-based models 20/ 42
30. Sequential Monte Carlo
…
X0 X1 XT
Y0 Y1 YT
…
✓
X
X
X X
X
X
X
Repeat for time t ∈ [2 : T]. Note this is for a given θ!
JH Agent-based models 20/ 42
31. Likelihood estimation
• Weight functions (wt)t∈[0:T] and proposals distributions
(qt)t∈[0:T] have to satisfy
w0(x0)
T
Y
t=1
wt(xt−1, xt) =
pθ(x0:T , y0:T )
q(x0:T |θ)
where q(x0:T |θ) = q0(x0|θ)
QT
t=1 qt(xt|xt−1, θ)
• We can compute a marginal likelihood estimator
p̂θ(y0:T ) =
1
P
P
X
p=1
w0(X
(p)
0 )
T
Y
t=1
1
P
P
X
p=1
wt(X
(A
(p)
t−1)
t−1 , X
(p)
t )
• Unbiasedness and consistency as P → ∞ follow from Del
Moral (2004)
JH Agent-based models 21/ 42
32. Bootstrap particle filter
• The bootstrap particle filter (BPF) of Gordon et al. (1993)
employs the proposal distributions
q0(x0|θ) = µθ(x0), qt(xt|xt−1, θ) = fθ(xt|xt−1)
and weight functions
wt(xt) = gθ(yt|xt)
• BPF can be readily implemented as simulating the latent
process is straightforward
• However, it suffers from curse of dimensionality for large N
– need large P to control variance of p̂θ(y0:T )
– p̂θ(y0:T ) can collapse to zero
JH Agent-based models 22/ 42
33. Likelihood estimation
• Efficiency of SMC crucially relies on the choice of proposal
distributions
• Poor performance of BPF is not surprising, as it does not use
any information from the observations
• We show how to implement the fully adapted auxiliary
particle filter that accounts for the next observation
• We propose a novel controlled SMC method that takes the
entire observation sequence into account
JH Agent-based models 23/ 42
34. Auxiliary particle filter
• The auxiliary particle filter (APF) was introduced in Pitt
and Shephard (1999) and Carpenter et al. (1999)
• It employs the proposal distributions
q0(x0|θ) = pθ(x0|y0), qt(xt|xt−1, θ) = pθ(xt|xt−1, yt)
and weight functions
wt(xt−1) = pθ(yt|xt−1)
• Sampling from these proposals and evaluating these weights
are not always tractable
JH Agent-based models 24/ 42
35. Auxiliary particle filter
• The predictive likelihood is
pθ(yt|xt−1) =
X
xt ∈{0,1}N
fθ(xt|xt−1)gθ(yt|xt)
JH Agent-based models 25/ 42
36. Auxiliary particle filter
• The predictive likelihood is
pθ(yt|xt−1) =
X
xt ∈{0,1}N
fθ(xt|xt−1)gθ(yt|xt)
=
X
xt ∈{0,1}N
N
Y
n=1
Ber(xn
t ; αn
(xt−1))Bin(yt; I(xt), ρ)
JH Agent-based models 25/ 42
37. Auxiliary particle filter
• The predictive likelihood is
pθ(yt|xt−1) =
X
xt ∈{0,1}N
fθ(xt|xt−1)gθ(yt|xt)
=
X
xt ∈{0,1}N
N
Y
n=1
Ber(xn
t ; αn
(xt−1))Bin(yt; I(xt), ρ)
=
N
X
it =yt
PoiBin(it; αn
(xt−1))Bin(yt; it, ρ)
since the sum of independent Bernoulli with non-identical
success probabilities follows a Poisson binomial distribution
JH Agent-based models 25/ 42
38. Auxiliary particle filter
• The predictive likelihood is
pθ(yt|xt−1) =
X
xt ∈{0,1}N
fθ(xt|xt−1)gθ(yt|xt)
=
X
xt ∈{0,1}N
N
Y
n=1
Ber(xn
t ; αn
(xt−1))Bin(yt; I(xt), ρ)
=
N
X
it =yt
PoiBin(it; αn
(xt−1))Bin(yt; it, ρ)
since the sum of independent Bernoulli with non-identical
success probabilities follows a Poisson binomial distribution
• Poisson binomial PMF costs O(N2) to compute (Chen and
Liu, 1997)
JH Agent-based models 25/ 42
39. Auxiliary particle filter
• To sample from pθ(xt|xt−1, yt), we augment It = I(Xt) as an
auxiliary variable
pθ(xt, it|xt−1, yt) = pθ(it|xt−1, yt)pθ(xt|xt−1, it)
• Conditional distribution of the number of infections is
pθ(it|xt−1, yt) =
PoiBin(it; αn(xt−1))Bin(yt; it, ρ)
pθ(yt|xt−1)
• Distribution of agent states conditioned on their sum is a
conditioned Bernoulli
pθ(xt|xt−1, it) = CondBer(xt; α(xt−1), it),
which costs O(N2) to sample (Chen and Liu, 1997)
JH Agent-based models 26/ 42
40. Auxiliary particle filter
• Hence the overall cost of APF is O(N2TP)
• We can reduce the cost to O(N log(N)TP) using two ideas
• Reduce cost of Poisson binomial PMF evaluation to O(N)
using translated Poisson approximation at a bias of
O(N−1/2) (Barbour and Ćekanavićius, 2002)
• Reduce cost of conditioned Bernoulli sampling to
O(N log(N)) using Markov chain Monte Carlo (Heng,
Jacob and Ju, 2020)
JH Agent-based models 27/ 42
41. Controlled sequential Monte Carlo
• We introduce a novel implementation of the controlled SMC
(cSMC) proposed by Heng et al. (2020)
• The optimal proposal that gives a zero variance marginal
likelihood estimator is the smoothing distribution
pθ(x0:T |y0:T ) = pθ(x0|y0:T )
T
Y
t=1
pθ(xt|xt−1, yt:T )
• At time t ∈ [1 : T], the transition is
pθ(xt|xt−1, yt:T ) =
fθ(xt|xt−1)ψ?
t (xt)
fθ(ψ?
t |xt−1)
• ψ?
t (xt) = p(yt:T |xt) is the backward information filter (BIF)
and fθ(ψ?
t |xt−1) =
P
xt ∈{0,1}N fθ(xt|xt−1)ψ?
t (xt)
JH Agent-based models 28/ 42
42. Controlled sequential Monte Carlo
• BIF satisfies the backward recursion ψ?
T (xT ) = gθ(yT |xT ),
ψ?
t (xt) = gθ(yt|xt)fθ(ψ?
t+1|xt), t ∈ [0 : T − 1]
• This costs O(22NT) to compute, so approximations are
necessary when N is large
• Our approach is based on dimensionality reduction by
coarse-graining the agent-based model
• We approximate the model with heterogenous agents by a
model with homogenous agents whose individual infection and
recovery rates given by their population averages, i.e.
λn ≈ λ̄ = N−1
PN
n=1 λn and γn ≈ γ̄ = N−1
PN
n=1 γn
JH Agent-based models 29/ 42
43. Controlled sequential Monte Carlo
• BIF of the approximate model ψt(I(xt)) can be computed
exactly in O(N3T) cost, and approximately in O(N2T)
• We then define the SMC proposal transition as
qt(xt|xt−1, θ) =
fθ(xt|xt−1)ψt(I(xt))
fθ(ψt|xt−1)
,
and employ the weight function
wt(xt) =
gθ(yt|xt)fθ(ψt+1|xt)
ψt(I(xt))
• Sampling and weighting can be done in a similar manner as
APF
JH Agent-based models 30/ 42
44. Controlled sequential Monte Carlo
• Quality of proposals depend on the coarse-graining
approximation
• We establish a bound on the Kullback–Leibler divergence from
q(x0:T |θ) and pθ(x0:T |y0:T ) (Chatterjee and Diaconis, 2018)
• Finer-grained approximations can be obtained using clustering
of the infection and recovery rates, at the expense of
increased cost
JH Agent-based models 31/ 42
51. Influenza outbreak in a boarding school
• SMC methods can be readily deployed within particle
MCMC for parameter and state inference (Andrieu, Doucet
and Holenstein, 2010)
JH Agent-based models 38/ 42
52. Influenza outbreak in a boarding school
• SMC methods can be readily deployed within particle
MCMC for parameter and state inference (Andrieu, Doucet
and Holenstein, 2010)
• Choice of network can impact inference results
0.0
0.2
0.4
0.6
0 2 4 6 8
λ γ
Density
Network Erdos−Renyi Full Small−world D=4 Small−world D=2
JH Agent-based models 38/ 42
54. Smallpox outbreak in a church community
• We considered APF and cSMC for the agent-based SIR model
• We can analyze vaccine efficacy in this framework
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0
10
20
30
0 30 60 89
date
removal
0.000
0.025
0.050
0.075
0.100
0 1 2 3 4 5 6
R0
proportion
vaccinated
Yes
No
JH Agent-based models 39/ 42
55. Concluding remarks
• A general alternative to SMC methods is MCMC algorithms
to sample from the smoothing distribution
• Preprint https://arxiv.org/abs/2101.12156
• R package https://github.com/nianqiaoju/agents
• Slides https://sites.google.com/view/jeremyheng/
JH Agent-based models 40/ 42
56. References
C. Andrieu, A. Doucet, and R. Holenstein. Particle Markov chain Monte Carlo
methods. Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 72(3):269–342, 2010.
A. Barbour and V. Ćekanavićius. Total variation asymptotics for sums of
independent integer random variables. The Annals of Probability,
30(2):509–545, 2002.
J. Carpenter, P. Clifford, and P. Fearnhead. Improved particle filter for nonlinear
problems. IEE Proceedings-Radar, Sonar and Navigation, 146(1):2–7, 1999.
S. Chatterjee and P. Diaconis. The sample size required in importance
sampling. The Annals of Applied Probability, 28(2):1099–1135, 2018.
S. Chen and J. Liu. Statistical applications of the Poisson-Binomial and
conditional Bernoulli distributions. Statistica Sinica, 875–892, 1997.
P. Del Moral. Feynman-kac formulae: Genealogical and Interacting Particle
Systems with Applications. Springer-Verlag New York, 2004.
JH Agent-based models 41/ 42
57. References
N. Gordon, D. Salmond, and A. Smith. Novel approach to
nonlinear/non-gaussian Bayesian state estimation. In IEE proceedings F (radar
and signal processing), volume 140, pages 107–113. IET, 1993.
J. Heng, A. Bishop, G. Deligiannidis, and A. Doucet. Controlled sequential
Monte Carlo. Annals of Statistics, 48(5):2904–2929, 2020.
J. Heng, P. Jacob, and N. Ju. A simple Markov chain for independent Bernoulli
variables conditioned on their sum. arXiv preprint arXiv:2012.03103, 2020.
M. Pitt and N. Shephard. Filtering via simulation: Auxiliary particle filters.
Journal of the American statistical association, 94(446):590–599, 1999.
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