Bayesian Change-Point Detection via Context-Tree Weighting

One of the most common types of models that helps us to understand neuron behavior is based on the Hodgkin–Huxley ion channel formulation (HH model). A major challenge with inferring parameters in HH models is non-uniqueness: many different sets of ion channel parameter values produce similar outputs for the same input stimulus. Such phenomena result in an objective function that exhibits multiple modes (i.e., multiple local minima). This non-uniqueness of local optimality poses challenges for parameter estimation with many algorithmic optimization techniques. HH models additionally have severe non-linearities resulting in further challenges for inferring parameters in an algorithmic fashion. To address these challenges with a tractable method in high-dimensional parameter spaces, we propose using a particular Markov chain Monte Carlo (MCMC) algorithm, which has the advantage of inferring parameters in a Bayesian framework. The Bayesian approach is designed to be suitable for multimodal solutions to inverse problems. We introduce and demonstrate the method using a three-channel HH model. We then focus on the inference of nine parameters in an eight-channel HH model, which we analyze in detail. We explore how the MCMC algorithm can uncover complex relationships between inferred parameters using five injected current levels. The MCMC method provides as a result a nine-dimensional posterior distribution, which we analyze visually with solution maps or landscapes of the possible parameter sets. The visualized solution maps show new complex structures of the multimodal posteriors, and they allow for selection of locally and globally optimal value sets, and they visually expose parameter sensitivities and regions of higher model robustness. We envision these solution maps as enabling experimentalists to improve the design of future experiments, increase scientific productivity and improve on model structure and ideation when the MCMC algorithm is applied to experimental data.

ABSTRACTWe invert prestack seismic amplitude data to find rock properties of a vertical profile of the earth. In particular we focus on lithology, porosity and fluid. Our model includes vertical dependencies of the rock properties. This allows us to compute quantities valid for the full profile such as the probability that the vertical profile contains hydrocarbons and volume distributions of hydrocarbons. In a standard point wise approach, these quantities can not be assessed. We formulate the problem in a Bayesian framework, and model the vertical dependency using spatial statistics. The relation between rock properties and elastic parameters is established through a stochastic rock model, and a convolutional model links the reflectivity to the seismic. A Markov chain Monte Carlo (MCMC) algorithm is used to generate multiple realizations that honours both the seismic data and the prior beliefs and respects the additional constraints imposed by the vertical dependencies. Convergence plots are used to provide quality check of the algorithm and to compare it with a similar method. The implementation has been tested on three different data sets offshore Norway, among these one profile has well control. For all test cases the MCMC algorithm provides reliable estimates with uncertainty quantification within three hours. The inversion result is consistent with the observed well data. In the case example we show that the seismic amplitudes make a significant impact on the inversion result even if the data have a moderate well tie, and that this is due to the vertical dependency imposed on the lithology fluid classes in our model. The vertical correlation in elastic parameters mainly influences the upside potential of the volume distribution.The approach is best suited to evaluate a few selected vertical profiles since the MCMC algorithm is computer demanding.

In this paper, we derive a Markov chain Monte Carlo (MCMC) algorithm supported by a neural network. In particular, we use the neural network to substitute derivative calculations made during a Metropolis adjusted Langevin algorithm (MALA) step with inexpensive neural network evaluations. Using a complex, high-dimensional blood coagulation model and a set of measurements, we define a likelihood function on which we evaluate the new MCMC algorithm. The blood coagulation model is a dynamic model, where derivative calculations are expensive and hence limit the efficiency of derivative-based MCMC algorithms. The MALA adaptation greatly reduces the time per iteration, while only slightly affecting the sample quality. We also test the new algorithm on a 2-dimensional example with a non-convex shape, a case where the MALA algorithm has a clear advantage over other state of the art MCMC algorithms. To assess the impact of the new algorithm, we compare the results to previously generated results of the MALA and the random walk Metropolis Hastings (RWMH).

There is an increasing use of Markov chain Monte Carlo (MCMC) algorithms for fitting statistical models in psychometrics, especially in situations where the traditional estimation techniques are very difficult to apply. One of the disadvantages of using an MCMC algorithm is that it is not straightforward to determine the convergence of the algorithm. Using the output of an MCMC algorithm that has not converged may lead to incorrect inferences on the problem at hand. The convergence is not one to a point, but that of the distribution of a sequence of generated values to another distribution, and hence is not easy to assess; there is no guaranteed diagnostic tool to determine convergence of an MCMC algorithm in general. This article examines the convergence of MCMC algorithms using a number of convergence diagnostics for two real data examples from psychometrics. Findings from this research have the potential to be useful to researchers using the algorithms. For both the examples, the number of iterations required (suggested by the diagnostics) to be reasonably confident that the MCMC algorithm has converged may be larger than what many practitioners consider to be safe.

Bayesian model updating provides a rigorous framework to account for uncertainty induced by lack of knowledge about engineering systems in their respective mathematical models through updates of the joint probability density function (PDF), the so-called posterior PDF, of the unknown model parameters. The Markov chain Monte Carlo (MCMC) methods are currently the most popular approaches for generating samples from the posterior PDF. However, these methods often found wanting when sampling from difficult distributions (e.g., high-dimensional PDFs, PDFs with flat manifolds, multimodal PDFs, and very peaked PDFs). This paper introduces a new multi-level sampling approach for Bayesian model updating, called Sequential Gauss-Newton algorithm, which is inspired by the Transitional Markov chain Monte Carlo (TMCMC) algorithm. The Sequential Gauss-Newton algorithm improves two aspects of TMCMC to make an efficient and effective MCMC algorithm for drawing samples from difficult posterior PDFs. First, the statistical efficiency of the algorithm is enhanced by use of the systematic resampling scheme. Second, a new MCMC algorithm, called Gauss-Newton MCMC algorithm, is proposed which is essentially an M-H algorithm with a Gaussian proposal PDF tailored to the posterior PDF using the gradient and Hessian information of the negative log posterior. The effectiveness of the proposed algorithm for solving the Bayesian model updating problem is illustrated using three examples with irregularly shaped posterior PDFs.

In an effort to build seismic models that are the most consistent with multiple data sets we have applied a new probabilistic inverse technique. This method uses a Markov chain Monte Carlo (MCMC) algorithm to sample models from a prior distribution and test them against multiple data types to generate a posterior distribution. While computationally expensive, this approach has several advantages over deterministic models, notably the seamless reconciliation of different data types that constrain the model, the proper handling of both data and model uncertainties, and the ability to easily incorporate a variety of prior information, all in a straightforward, natural fashion. A real advantage of the technique is that it provides a more complete picture of the solution space. By mapping out the posterior probability density function, we can avoid simplistic assumptions about the model space and allow alternative solutions to be identified, compared, and ranked. Here we use this method to determine the crust and upper mantle structure of the Yellow Sea and Korean Peninsula region. The model is parameterized as a series of seven layers in a regular latitude‐longitude grid, each of which is characterized by thickness and seismic parameters (Vp, Vs, and density). We use surface wave dispersion and body wave traveltime data to drive the model. We find that when properly tuned (i.e., the Markov chains have had adequate time to fully sample the model space and the inversion has converged), the technique behaves as expected. The posterior model reflects the prior information at the edge of the model where there is little or no data to constrain adjustments, but the range of acceptable models is significantly reduced in data‐rich regions, producing values of sediment thickness, crustal thickness, and upper mantle velocities consistent with expectations based on knowledge of the regional tectonic setting.

Monitoring the spread of infectious diseases during an epidemic or pandemic presents a great challenge to public health systems worldwide. To facilitate informed decision-making, concerned parties usually rely on short-term daily or weekly projections generated via predictive modeling. Traditional epidemiological models, particularly growth models, are frequently employed with the assumption that epidemics or pandemics exhibit a single-peaked pattern in case numbers or mortality rates. However, this may not hold true for all diseases, as evidenced by the prolonged and multi-peaked nature of infectious diseases like COVID-19. Additionally, government interventions have led to time-varying disease transmission rates, making the observed data multi-modal. In response to these complexities, this paper presents BAGELS, a generalized Bayesian stochastic growth model augmented by a change-point detection mechanism to account for multiple peaks. A trans-dimensional reversible jump Markov chain Monte Carlo algorithm is used to sample the posterior distributions of interpretable epidemiological parameters while estimating the number and locations of change-points. We demonstrate the effectiveness of BAGELS through extensive evaluations using both simulated data and COVID-19 data from major U.S. states. Not only does this work offer a robust tool for longitudinal epidemiological studies, but it also advances the Bayesian change-point detection methodology for growth curves.

Using Bayesian inference to calibrate constitutive model parameters has recently seen a rise in interest. The Markov chain Monte Carlo (MCMC) algorithm is one of the most commonly used methods to sample from the posterior. However, the choice of which MCMC algorithm to apply is typically pragmatic and based on considerations such as software availability and experience. We compare three commonly used MCMC algorithms: Metropolis-Hastings (MH), Affine Invariant Stretch Move (AISM) and No-U-Turn sampler (NUTS). For the comparison, we use the Kullback-Leibler (KL) divergence as a convergence criterion, which measures the statistical distance between the sampled and the ‘true’ posterior. We apply the Bayesian framework to a Newtonian squeeze flow problem, for which there exists an analytical model. Furthermore, we have collected experimental data using a tailored setup. The ground truth for the posterior is obtained by evaluating it on a uniform reference grid. We conclude that, for the same number of samples, the NUTS results in the lowest KL divergence, followed by the AISM sampler and last the MH sampler.

The identification of useful temporal dependence structure in discrete time series data is an important component of algorithms applied to many tasks in statistical inference and machine learning, and used in a wide variety of problems across the spectrum of biological studies. Most of the early statistical approaches were ineffective in practice, because the amount of data required for reliable modelling grew exponentially with memory length. On the other hand, many of the more modern methodological approaches that make use of more flexible and parsimonious models result in algorithms that do not scale well and are computationally ineffective for larger data sets. In this paper we describe a class of novel methodological tools for effective Bayesian inference for general discrete time series, motivated primarily by questions regarding data originating from studies in genetics and neuroscience. Our starting point is the development of a rich class of Bayesian hierarchical models for variable-memory Markov chains. The particular prior structure we adopt makes it possible to design effective, linear-time algorithms that can compute most of the important features of the relevant posterior and predictive distributions without resorting to Markov chain Monte Carlo simulation. The origin of some of these algorithms can be traced to the family of Context Tree Weighting (CTW) algorithms developed for data compression since the mid-1990s. We have used the resulting methodological tools in numerous application-specific tasks (including prediction, segmentation, classification, anomaly detection, entropy estimation, and causality testing) on data from different areas of application. The results obtained compare quite favourably with those obtained using earlier approaches, such as Probabilistic Suffix Trees (PST), Variable-Length Markov Chains (VLMC), and the class of Markov Transition Distributions (MTD).

Markov Chain Monte Carlo (MCMC) is widely applied to fit complex, practical models in various disciplines, such as engineering, physics, agriculture, etc. Indeed, when the target density $f$ is not available in closed form, a Markov chain is constructed with stationary density $f$ for forming Monte Carlo estimators of means with respect to $f$. Although a Harris ergodic Markov chain will converge to the target distribution as the Monte Carlo sample size tends to infinity, and the population means can be consistently estimated by the corresponding MCMC estimators, in practice, it is important to estimate the standard errors associated with these Monte Carlo estimates due to the availability of only a finite sample. Therefore, the geometric ergodicity of a Markov Chain is highly desired as it is the most standard method for guaranteeing a central limit theorem (CLT) for MCMC estimators and finding its standard errors. Among the different MCMC algorithms, Metropolis-Hastings (MH) algorithms are predominant. This thesis establishes conditions under which MH algorithms with a position-dependent proposal covariance matrix will or will not have the geometric convergence rate. Indeed, we provide these results for MH algorithms where the normal proposal density has a general mean function $c(x)$ and a covariance matrix $G(x)$ depending on the current position $x$ of the Markov chain. For modern variants of the popular Metropolis adjusted Langevin algorithms (MALA), like the manifold MALA (MMALA) that adapt to the geometry of the target distributions, the proposal covariance matrix changes in every iteration. Thus, we provide conditions for the geometric ergodicity of different variations of the Langevin algorithms, e.g., MMALA, position-dependent MALA (PMALA), and the pre-conditioned MALA (PCMALA). We also provide conditions guaranteeing geometric ergodicity of the pre-conditioned unadjusted Langevin algorithms (PCULA). Generalized linear mixed models (GLMMs), where the response variable is allowed to follow a non-Gaussian distribution, are popular statistical models, although their likelihood functions are not available in closed form. GLMMs can handle the over-dispersion often present in real-world data, and MCMC methods are generally used to fit frequentist and Bayesian GLMMs. Since the likelihood function for GLMMs is not available in closed form, Monte Carlo EM or Monte Carlo maximum likelihood methods are often used for making inferences from GLMMs. Simulation from the conditional (on the data) distribution of the random effects is, in turn, needed for implementing the Monte Carlo EM or Monte Carlo maximum likelihood methods. In this thesis, our general conditions for geometric ergodicity of variants of Langevin algorithms are also verified in the context of conditional simulation from the two most popular GLMMs, namely the binomial GLMM with the logit link and the Poisson GLMM with the log link. These results have important practical implications as users can now confidently fit these models by attaching asymptotically valid standard errors with their Markov chain-based Monte Carlo estimates. Spatial generalized linear mixed models (SGLMMs) are GLMMs where the random effects form a spatial process and are useful for modeling spatially correlated data. We compare the performance of different variants of MALA mentioned above as well as other popular MH algorithms like Random Walk Metropolis in the context of conditional simulation from SGLMMs. Next, we consider MCMC algorithms for full Bayesian analysis of GLMMs. Blocking and reparameterization are techniques used to improve the efficiency and convergence properties of MCMC algorithms. We combine these strategies with pre-conditioning to develop efficient Langevin algorithms for Bayesian GLMMs and SGLMMs. We illustrate our algorithms using extensive real and simulated data analyses.

Left-censored linear regression models are quite popular models and have been deeply considered in the last three decades. In this paper, we consider a completely Bayesian approach for making a new Markov chain Monte Carlo (MCMC) algorithm with tractable full posteriors. Simulated consequences and real data analyses depict that the new Markov chain Monte Carlo algorithm has excellent mixing property and carry out very well than the present methods based on prediction accuracy.

Application of the transitional Markov chain Monte Carlo algorithm to probabilistic site characterization

Maximum likelihood estimation techniques are widely used in twin and family studies, but soon reach computational boundaries when applied to highly complex models (e.g., models including gene-by-environment interaction and gene-environment correlation, item response theory measurement models, repeated measures, longitudinal structures, extended pedigrees). Markov Chain Monte Carlo (MCMC) algorithms are very well suited to fit complex models with hierarchically structured data. This article introduces the key concepts of Bayesian inference and MCMC parameter estimation and provides a number of scripts describing relatively simple models to be estimated by the freely obtainable BUGS software. In addition, inference using BUGS is illustrated using a data set on follicle-stimulating hormone and luteinizing hormone levels with repeated measures. The examples provided can serve as stepping stones for more complicated models, tailored to the specific needs of the individual researcher.

An integrated Markov chain Monte Carlo algorithm for upscaling hydrological and geochemical parameters from column to field scale

Spatial Poisson models for areal count data use nonstationary “intrinsic autoregressions,” also often referred to as “conditionally autoregressive” (CAR) models. Bayesian inference for these models has generally involved using single parameter updating Markov chain Monte Carlo algorithms, which often exhibit slow mixing (i.e., poor convergence) properties. These spatial models are richly parameterized and lend themselves to the structured Markov chain Monte Carlo (SMCMC) algorithms. SMCMC provides a simple, general, and flexible framework for accelerating convergence in an MCMC sampler by providing a systematic way to block groups of similar parameters while taking full advantage of the posterior correlation structure induced by the model and data. Among the SMCMC strategies considered here are blocking using different size blocks (grouping by geographical region), reparameterization, updating jointly with and without model hyperparameters, “oversampling” some of the model parameters, and “pilot adaptation” versus continuous tuning techniques for the proposal density. We apply the techniques presented here to datasets on cancer mortality and late detection in the state of Minnesota. We find that, compared to univariate sampling procedures, our techniques will typically lead to more accurate posterior estimates, and they are sometimes also far more efficient in terms of the number of effective samples generated per second.