scholarly journals Multiscale Convergence of the Inverse Problem for Chemotaxis in the Bayesian Setting

Computation ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 119
Author(s):  
Kathrin Hellmuth ◽  
Christian Klingenberg ◽  
Qin Li ◽  
Min Tang
Keyword(s):  
Inverse Problem ◽  
Kinetic Equation ◽  
Posterior Distribution ◽  
Population Level ◽  
Chemical Stimulus ◽  
Posterior Distributions ◽  
Asymptotic Equivalence ◽  
Time Dynamics ◽  
Multiscale Convergence ◽  
Mesoscopic Level

Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller–Segel equation and a chemotaxis kinetic equation. These two equations describe the organism’s movement at the macro- and mesoscopic level, respectively, and are asymptotically equivalent in the parabolic regime. The way in which the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller–Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms’ population level movement when reacting to certain stimulation. From this, one infers the chemotaxis response, which constitutes an inverse problem. In this paper, we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated with two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought. We prove the asymptotic equivalence of the two posterior distributions.

Metropolis-Hastings Algorithms for DSGE Models

2015 ◽  
Author(s):  
Edward P. Herbst ◽  
Frank Schorfheide
Keyword(s):  
Random Walk ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Large Scale ◽  
Likelihood Function ◽  
Dsge Models ◽  
Posterior Distributions ◽  
Dsge Model ◽  
Alternative Proposal ◽  
Posterior Moments

This chapter talks about the most widely used method to generate draws from posterior distributions of a DSGE model: the random walk MH (RWMH) algorithm. The DSGE model likelihood function in combination with the prior distribution leads to a posterior distribution that has a fairly regular elliptical shape. In turn, the draws from a simple RWMH algorithm can be used to obtain an accurate numerical approximation of posterior moments. However, in many other applications, particularly those involving medium- and large-scale DSGE models, the posterior distributions could be very non-elliptical. Irregularly shaped posterior distributions are often caused by identification problems or misspecification. In lieu of the difficulties caused by irregularly shaped posterior surfaces, the chapter reviews various alternative MH samplers, which use alternative proposal distributions.

scholarly journals Bayesian Derivative Order Estimation for a Fractional Logistic Model

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 109
Author(s):  
Francisco J. Ariza-Hernandez ◽  
Martin P. Arciga-Alejandre ◽  
Jorge Sanchez-Ortiz ◽  
Alberto Fleitas-Imbert
Keyword(s):  
Inverse Problem ◽  
Logistic Model ◽  
Likelihood Function ◽  
Synthetic Data ◽  
Real Data ◽  
Posterior Distributions ◽  
Rate Parameter ◽  
Order Estimation ◽  
Proposed Model ◽  
Derivative Order

In this paper, we consider the inverse problem of derivative order estimation in a fractional logistic model. In order to solve the direct problem, we use the Grünwald-Letnikov fractional derivative, then the inverse problem is tackled within a Bayesian perspective. To construct the likelihood function, we propose an explicit numerical scheme based on the truncated series of the derivative definition. By MCMC samples of the marginal posterior distributions, we estimate the order of the derivative and the growth rate parameter in the dynamic model, as well as the noise in the observations. To evaluate the methodology, a simulation was performed using synthetic data, where the bias and mean square error are calculated, the results give evidence of the effectiveness for the method and the suitable performance of the proposed model. Moreover, an example with real data is presented as evidence of the relevance of using a fractional model.

scholarly journals Multiple asymptotics of kinetic equations with internal states

2020 ◽  
Vol 30 (06) ◽  
pp. 1041-1073
Author(s):  
Benoit Perthame ◽  
Weiran Sun ◽  
Min Tang ◽  
Shugo Yasuda
Keyword(s):  
Kinetic Equation ◽  
Equilibrium Equation ◽  
Internal State ◽  
Kinetic Equations ◽  
Chemical Stimulus ◽  
Adaptation Time ◽  
Macroscopic Models ◽  
Internal States ◽  
New Type ◽  
Run And Tumble

The run and tumble process is well established in order to describe the movement of bacteria in response to a chemical stimulus. However, the relation between the tumbling rate and the internal state of bacteria is poorly understood. This study aims at deriving macroscopic models as limits of the mesoscopic kinetic equation in different regimes. In particular, we are interested in the roles of the stiffness of the response and the adaptation time in the kinetic equation. Depending on the asymptotics chosen both the standard Keller–Segel equation and the flux-limited Keller–Segel (FLKS) equation can appear. An interesting mathematical issue arises with a new type of equilibrium equation leading to solution with singularities.

scholarly journals Reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation

2018 ◽  
Vol 8 (2) ◽  
pp. 145-151 ◽  
Author(s):  
Esra Karatas Akgül
Keyword(s):  
Hilbert Space ◽  
Inverse Problem ◽  
Kinetic Equation ◽  
Inverse Problems ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Approximate Solutions ◽  
Kernel Functions ◽  
Coefficient Inverse Problem ◽  
Space Method

On the basis of a reproducing kernel Hilbert space, reproducing kernel functions for solving the coefficient inverse problem for the kinetic equation are given in this paper. Reproducing kernel functions found in the reproducing kernel Hilbert space imply that they can be considered for solving such inverse problems. We obtain approximate solutions by reproducing kernel functions. We show our results by a table. We prove the eciency of the reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation.

A Crash Course in Bayesian Inference

2015 ◽  
Author(s):  
Edward P. Herbst ◽  
Frank Schorfheide
Keyword(s):  
Decision Making ◽  
Bayesian Inference ◽  
Linear Regression ◽  
Closed Form ◽  
Posterior Distribution ◽  
Ar Model ◽  
Model Parameters ◽  
Posterior Distributions ◽  
Interval Estimates ◽  
Point Estimates

This chapter provides a self-contained review of Bayesian inference and decision making. It begins with a discussion of Bayesian inference for a simple autoregressive (AR) model, which takes the form of a Gaussian linear regression. For this model, the posterior distribution can be characterized analytically and closed-form expressions for its moments are readily available. The chapter also examines how to turn posterior distributions into point estimates, interval estimates, forecasts, and how to solve general decision problems. The chapter shows how in a Bayesian setting, the calculus of probability is used to characterize and update an individual's state of knowledge or degree of beliefs with respect to quantities such as model parameters or future observations.

scholarly journals On the derivation of the wave kinetic equation for NLS

2021 ◽  
Vol 9 ◽  
Author(s):  
Yu Deng ◽  
Zaher Hani
Keyword(s):  
Kinetic Equation ◽  
Time Scales ◽  
Scaling Laws ◽  
Scaling Law ◽  
Turbulence Theory ◽  
Kinetic Description ◽  
Time Dynamics ◽  
Tree Expansion ◽  
Long Time ◽  
Cubic Nonlinear Schrödinger Equation

Abstract A fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a large kinetic time scale $T_{\mathrm {kin}} \gg 1$ and in a limiting regime where the size L of the domain goes to infinity and the strength $\alpha $ of the nonlinearity goes to $0$ (weak nonlinearity). For the cubic nonlinear Schrödinger equation, $T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$ and $\alpha $ is related to the conserved mass $\lambda $ of the solution via $\alpha =\lambda ^2 L^{-d}$ . In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particular scaling law in which the $(\alpha , L)$ limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to be two favourable scaling laws: when $\alpha $ approaches $0$ like $L^{-\varepsilon +}$ or like $L^{-1-\frac {\varepsilon }{2}+}$ (for arbitrary small $\varepsilon $ ), we exhibit the wave kinetic equation up to time scales $O(T_{\mathrm {kin}}L^{-\varepsilon })$ , by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales $T_*\ll T_{\mathrm {kin}}$ and identify specific interactions that become very large for times beyond $T_*$ . In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond $T_*$ toward $T_{\mathrm {kin}}$ for such scaling laws seems to require new methods and ideas.

scholarly journals Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem

2022 ◽  
Author(s):  
Hanne Kekkonen
Keyword(s):  
Inverse Problem ◽  
Convergence Rate ◽  
Gaussian Process ◽  
Reconstruction Error ◽  
Posterior Distributions ◽  
Smooth Functions ◽  
True Parameter ◽  
Linear Inverse Problem ◽  
Point Evaluations ◽  
Truncated Gaussian

Abstract We consider the statistical non-linear inverse problem of recovering the absorption term f>0 in the heat equation with given sufficiently smooth functions describing boundary and initial values respectively. The data consists of N discrete noisy point evaluations of the solution u_f. We study the statistical performance of Bayesian nonparametric procedures based on a large class of Gaussian process priors. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate for the reconstruction error of the associated posterior means. We also consider the optimality of the contraction rates and prove a lower bound for the minimax convergence rate for inferring f from the data, and show that optimal rates can be achieved with truncated Gaussian priors.

scholarly journals Component-wise Approximate Bayesian Computation via Gibbs-like steps

Biometrika ◽  
2020 ◽  
Author(s):  
Grégoire Clarté ◽  
Christian P Robert ◽  
Robin J Ryder ◽  
Julien Stoehr
Keyword(s):  
Markov Chain ◽  
Posterior Distribution ◽  
Approximate Bayesian Computation ◽  
Generative Models ◽  
Bayesian Computation ◽  
Summary Statistics ◽  
Posterior Distributions ◽  
Reduced Dimensions ◽  
Standard Solution ◽  
Approximate Bayesian

Abstract Approximate Bayesian computation methods are useful for generative models with intractable likelihoods. These methods are however sensitive to the dimension of the parameter space, requiring exponentially increasing resources as this dimension grows. To tackle this difficulty, we explore a Gibbs version of the Approximate Bayesian computation approach that runs component-wise approximate Bayesian computation steps aimed at the corresponding conditional posterior distributions, and based on summary statistics of reduced dimensions. While lacking the standard justifications for the Gibbs sampler, the resulting Markov chain is shown to converge in distribution under some partial independence conditions. The associated stationary distribution can further be shown to be close to the true posterior distribution and some hierarchical versions of the proposed mechanism enjoy a closed form limiting distribution. Experiments also demonstrate the gain in efficiency brought by the Gibbs version over the standard solution.

Uncertainty quantification in seismic facies inversion

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. M43-M56
Author(s):  
Erick Costa e Silva Talarico ◽  
Dario Grana ◽  
Leandro Passos de Figueiredo ◽  
Sinesio Pesco
Keyword(s):  
Inverse Problem ◽  
Vertical Distribution ◽  
Uncertainty Quantification ◽  
Posterior Distribution ◽  
Seismic Data ◽  
Transition Matrix ◽  
Transition Probabilities ◽  
Vertical Direction ◽  
Seismic Facies ◽  
Transition Matrices

In seismic reservoir characterization, facies prediction from seismic data often is formulated as an inverse problem. However, the uncertainty in the parameters that control their spatial distributions usually is not investigated. In a probabilistic setting, the vertical distribution of facies often is described by statistical models, such as Markov chains. Assuming that the transition probabilities in the vertical direction are known, the most likely facies sequence and its uncertainty can be obtained by computing the posterior distribution of a Bayesian inverse problem conditioned by seismic data. Generally, the model hyperparameters such as the transition matrix are inferred from seismic data and nearby wells using a Bayesian inference framework. It is assumed that there is a unique set of hyperparameters that optimally fit the measurements. The novelty of the proposed work is to investigate the nonuniqueness of the transition matrix and show the multimodality of their distribution. We then generalize the Bayesian inversion approach based on Markov chain models by assuming that the hyperparameters, the facies prior proportions and transition matrix, are unknown and derive the full posterior distribution. Including all of the possible transition matrices in the inversion improves the uncertainty quantification of the predicted facies conditioned by seismic data. Our method is demonstrated on synthetic and real seismic data sets, and it has high relevance in exploration studies due to the limited number of well data and in geologic environments with rapid lateral variations of the facies vertical distribution.

scholarly journals On Default Priors for Robust Bayesian Estimation with Divergences

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 29
Author(s):  
Tomoyuki Nakagawa ◽  
Shintaro Hashimoto
Keyword(s):  
Bayesian Estimation ◽  
Bayesian Methods ◽  
Posterior Distribution ◽  
Bayesian Framework ◽  
Moment Matching ◽  
Simulation Studies ◽  
Posterior Distributions ◽  
Prior Distributions ◽  
Matching Priors ◽  
Selection Of

This paper presents objective priors for robust Bayesian estimation against outliers based on divergences. The minimum γ-divergence estimator is well-known to work well in estimation against heavy contamination. The robust Bayesian methods by using quasi-posterior distributions based on divergences have been also proposed in recent years. In the objective Bayesian framework, the selection of default prior distributions under such quasi-posterior distributions is an important problem. In this study, we provide some properties of reference and moment matching priors under the quasi-posterior distribution based on the γ-divergence. In particular, we show that the proposed priors are approximately robust under the condition on the contamination distribution without assuming any conditions on the contamination ratio. Some simulation studies are also presented.

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