Bayesian synthetic likelihood (BSL) is a popular method for estimating the parameter posterior distribution for complex statistical models and stochastic processes that possess a computationally intractable likelihood function. Instead of evaluating the likelihood, BSL approximates the likelihood of a judiciously chosen summary statistic of the data via model simulation and density estimation. Compared to alternative methods such as approximate Bayesian computation (ABC), BSL requires little tuning and requires less model simulations than ABC when the chosen summary statistic is high-dimensional. The original synthetic likelihood relies on a multivariate normal approximation of the intractable likelihood, where the mean and covariance are estimated by simulation. An extension of BSL considers replacing the sample covariance with a penalised covariance estimator to reduce the number of required model simulations. Further, a semi-parametric approach has been developed to relax the normality assumption. In this paper, we present an R package called BSL that amalgamates the aforementioned methods and more into a single, easy-to-use and coherent piece of software. The R package also includes several examples to illustrate how to use the package and demonstrate the utility of the methods.

Numerous engineering problems of interest to the industry are often characterized by expensive black-box objective experiments or computer simulations. Obtaining insight into the problem or performing subsequent optimizations requires hundreds of thousands of evaluations of the objective function which is most often a practically unachievable task. Gaussian Process (GP) surrogate modeling replaces the expensive function with a cheap-to-evaluate data-driven probabilistic model. While the GP does not assume a functional form of the problem, it is defined by a set of parameters, called hyperparameters. The hyperparameters define the characteristics of the objective function, such as smoothness, magnitude, periodicity, etc. Accurately estimating these hyperparameters is a key ingredient in developing a reliable and generalizable surrogate model. Markov chain Monte Carlo (MCMC) is a ubiquitously used Bayesian method to estimate these hyperparameters. At the GE Global Research Center, a customized industry-strength Bayesian hybrid modeling framework utilizing the GP, called GEBHM, has been employed and validated over many years. GEBHM is very effective on problems of small and medium size, typically less than 1000 training points. However, the GP does not scale well in time with a growing dataset and problem dimensionality which can be a major impediment in such problems. In this work, we extend and implement in GEBHM an Adaptive Sequential Monte Carlo (ASMC) methodology for training the GP enabling the modeling of large-scale industry problems. This implementation saves computational time (especially for large-scale problems) while not sacrificing predictability over the current MCMC implementation. We demonstrate the effectiveness and accuracy of GEBHM with ASMC on four mathematical problems and on two challenging industry applications of varying complexity.

We develop a stochastic whole-brain and body simulator of the nematode roundworm Caenorhabditis elegans (C. elegans) and show that it is sufficiently regularizing to allow imputation of latent membrane potentials from partial calcium fluorescence imaging observations. This is the first attempt we know of to "complete the circle," where an anatomically grounded whole-connectome simulator is used to impute a time-varying "brain" state at single-cell fidelity from covariates that are measurable in practice. The sequential Monte Carlo (SMC) method we employ not only enables imputation of said latent states but also presents a strategy for learning simulator parameters via variational optimization of the noisy model evidence approximation provided by SMC. Our imputation and parameter estimation experiments were conducted on distributed systems using novel implementations of the aforementioned techniques applied to synthetic data of dimension and type representative of that which are measured in laboratories currently.

Matrix Factorization (MF) is a common method for generating recommendations, where the proximity of entities like users or items in the embedded space indicates their similarity to one another. Though almost all applications implicitly use a Euclidean embedding space to represent two entity types, recent work has suggested that a hyperbolic Poincar\'e ball may be more well suited to representing multiple entity types, and in particular, hierarchies. We describe a novel method to embed a hierarchy of related music entities in hyperbolic space. We also describe how a parametric empirical Bayes approach can be used to estimate link reliability between entities in the hierarchy. Applying these methods together to build personalized playlists for users in a digital music service yielded a large and statistically significant increase in performance during an A/B test, as compared to the Euclidean model.

Learning robot controllers by minimizing a black-box objective cost using Bayesian optimization (BO) can be time-consuming and challenging. It is very often the case that some roll-outs result in failure behaviors, causing premature experiment detention. In such cases, the designer is forced to decide on heuristic cost penalties because the acquired data is often scarce, or not comparable with that of the stable policies. To overcome this, we propose a Bayesian model that captures exactly what we know about the cost of unstable controllers prior to data collection: Nothing, except that it should be a somewhat large number. The resulting Bayesian model, approximated with a Gaussian process, predicts high cost values in regions where failures are likely to occur. In this way, the model guides the BO exploration toward regions of stability. We demonstrate the benefits of the proposed model in several illustrative and statistical synthetic benchmarks, and also in experiments on a real robotic platform. In addition, we propose and experimentally validate a new BO method to account for unknown constraints. Such method is an extension of Max-Value Entropy Search, a recent information-theoretic method, to solve unconstrained global optimization problems.

Accurately predicting future behaviors of surrounding vehicles is an essential capability for autonomous vehicles in order to plan safe and feasible trajectories. The behaviors of others, however, are full of uncertainties. Both rational and irrational behaviors exist, and the autonomous vehicles need to be aware of this in their prediction module. The prediction module is also expected to generate reasonable results in the presence of unseen and corner scenarios. Two types of prediction models are typically used to solve the prediction problem: learning-based model and planning-based model. Learning-based model utilizes real driving data to model the human behaviors. Depending on the structure of the data, learning-based models can predict both rational and irrational behaviors. But the balance between them cannot be customized, which creates challenges in generalizing the prediction results. Planning-based model, on the other hand, usually assumes human as a rational agent, i.e., it anticipates only rational behavior of human drivers. In this paper, a generic prediction architecture is proposed to address various rationalities in human behavior. We leverage the advantages from both learning-based and planning-based prediction models. The proposed approach is able to predict continuous trajectories that well-reflect possible future situations of other drivers. Moreover, the prediction performance remains stable under various unseen driving scenarios. A case study under a real-world roundabout scenario is provided to demonstrate the performance and capability of the proposed prediction architecture.

I'm following the book Pattern recognition and machine learning by Bishop on Bernoulli mixture model, and trying to code it. But I don't understand how to calculate the conditional probability (page 446 of the first edition) So in the E-step I'm supposed to calculate this. But it is said that we should use the log of the probability, so as to avoid numerical underflow. So how do i apply it here? I can't see any way to do it.

The derivation of physical laws is a dominant topic in scientific research. We propose a new method capable of discovering the physical laws from data to tackle four challenges in the previous methods. The four challenges are: (1) large noise in the data, (2) outliers in the data, (3) integrating the data collected from different experiments, and (4) extrapolating the solutions to the areas that have no available data. To resolve these four challenges, we try to discover the governing differential equations and develop a model-discovering method based on sparse Bayesian inference and subsampling. The subsampling technique is used for improving the accuracy of the Bayesian learning algorithm here, while it is usually employed for estimating statistics or speeding up algorithms elsewhere. The optimal subsampling size is moderate, neither too small nor too big. Another merit of our method is that it can work with limited data by the virtue of Bayesian inference. We demonstrate how to use our method to tackle the four aforementioned challenges step by step through numerical examples: (1) predator-prey model with noise, (2) shallow water equations with outliers, (3) heat diffusion with random initial and boundary conditions, and (4) fish-harvesting problem with bifurcations. Numerical results show that the robustness and accuracy of our new method is significantly better than the other model-discovering methods and traditional regression methods.

Real-world optimization problems typically involve multiple objectives. This is particularly true in the design of engineering systems, where multiple performance outcomes are balanced against budgetary constraints. Among the complexities of optimizing over multiple objectives is the effect of uncertainties in the problem. Design is guided by models known to be imperfect, systems are built using materials with uncertainty regarding their properties, variations occur in the construction of designed systems, and so on. These imperfections, uncertainties and errors cause uncertainty also in the solution to a design problem. In traditional engineering design, one designs a system after choosing a material with appropriate properties for the project from a database of known materials. As a result, the design of the system is constrained by the initial material selection. By coupling material discovery and engineering system design, we can combine these two traditionally separate processes under the umbrella of a unified multiple objective optimization problem. In this paper, we cast the engineering design problem in the framework of computer model calibration.

Bayesian inference is used extensively to infer and to quantify the uncertainty in a field of interest from a measurement of a related field when the two are linked by a physical model. Despite its many applications, Bayesian inference faces challenges when inferring fields that have discrete representations of large dimension, and/or have prior distributions that are difficult to represent mathematically. In this manuscript we consider the use of Generative Adversarial Networks (GANs) in addressing these challenges. A GAN is a type of deep neural network equipped with the ability to learn the distribution implied by multiple samples of a given field. Once trained on these samples, the generator component of a GAN maps the iid components of a low-dimensional latent vector to an approximation of the distribution of the field of interest. In this work we demonstrate how this approximate distribution may be used as a prior in a Bayesian update, and how it addresses the challenges associated with characterizing complex prior distributions and the large dimension of the inferred field. We demonstrate the efficacy of this approach by applying it to the problem of inferring and quantifying uncertainty in the initial temperature field in a heat conduction problem from a noisy measurement of the temperature at later time.