This short note revisit information metric, underlining that it is a pseudo metric on manifolds of observables (random variables), rather than as usual on probability laws. Geodesics are characterized in terms of their boundaries and conditional independence condition. Pythagorean theorem is given, providing in special case potentially interesting natural integer triplets. This metric is computed for illustration on Diabetes dataset using infotopo package.

This material may be downloaded for personal use only. Any other use requires prior permission of the American Society of Civil Engineers. This material may be found at https://doi.org/10.1061/AJRUA6.0001106 ABSTRACT In data-driven SHM, the signals recorded from systems in operation can be noisy and incomplete. Data corresponding to each of the operational, environmental, and damage states are rarely available a priori; furthermore, labelling to describe the measurements is often unavailable. In consequence, the algorithms used to implement SHM should be robust and adaptive, while accommodating for missing information in the training-data - such that new information can be included if it becomes available. By reviewing novel techniques for statistical learning (introduced in previous work), it is argued that probabilistic algorithms offer a natural solution to the modelling of SHM data in practice. In three case-studies, probabilistic methods are adapted for applications to SHM signals -- including semi-supervised learning, active learning, and multi-task learning. Various machine learning tools have been applied in the literature, for example (Vanik et al. 2000; Sohn et al. 2003; Chatzi and Smyth 2009), and used to infer the health or performance state of the monitored system, either directly or indirectly. Generally, algorithms for regression, classification, density estimation, or clustering learn patterns in the measured signals (available for training), and the associated patterns can be used to infer the state of the system in operation, given future measurements (Worden and Manson 2006). Unsurprisingly, there are numerous ways to apply machine learning to SHM. Notably (and categorised generally), advances have focussed on various probabilistic (e.g. Each approach has its advantages; however, considering certain challenges associated with SHM data (outlined in the next section) the current work focusses on probabilistic (i.e. Additionally, probabilistic methods can lead to predictions under uncertainty (Papoulis 1965) - a significant advantage in risk-based applications.

Sampled from a distribution: Many machine learning algorithms make assumptions that the data is sampled from a frequency. For example, linear regression assumes gaussian distribution and logistic regression assumes that the data is sampled from a Bernoulli distribution.

In statistical modeling, we have to calculate the estimator to determine the equation of your model. The problem is, the estimator itself is difficult to calculate, especially when it involves some distributions like Beta, Gamma, or even Gompertz distribution. Maximum Likelihood Estimator (MLE) is one of many methods to calculate the estimator for those distributions. In this article, I will give you some examples to calculate MLE with the Newton-Raphson method using R. Newton-Raphson method is an iterative procedure to calculate the roots of function f. The goal of this method is to make the approximated result as close as possible with the exact result (that is, the roots of the function).

Hyperspectral imaging, also known as image spectrometry, is a landmark technique in geoscience and remote sensing (RS). In the past decade, enormous efforts have been made to process and analyze these hyperspectral (HS) products mainly by means of seasoned experts. However, with the ever-growing volume of data, the bulk of costs in manpower and material resources poses new challenges on reducing the burden of manual labor and improving efficiency. For this reason, it is, therefore, urgent to develop more intelligent and automatic approaches for various HS RS applications. Machine learning (ML) tools with convex optimization have successfully undertaken the tasks of numerous artificial intelligence (AI)-related applications. However, their ability in handling complex practical problems remains limited, particularly for HS data, due to the effects of various spectral variabilities in the process of HS imaging and the complexity and redundancy of higher dimensional HS signals. Compared to the convex models, non-convex modeling, which is capable of characterizing more complex real scenes and providing the model interpretability technically and theoretically, has been proven to be a feasible solution to reduce the gap between challenging HS vision tasks and currently advanced intelligent data processing models.

BayeSuites is the first web framework for learning, visualizing, and interpreting Bayesian networks (BNs) that can scale to tens of thousands of nodes while providing fast and friendly user experience. All the necessary features that enable this are reviewed in this paper; these features include scalability, extensibility, interoperability, ease of use, and interpretability. Scalability is the key factor in learning and processing massive networks within reasonable time; for a maintainable software open to new functionalities, extensibility and interoperability are necessary. Ease of use and interpretability are fundamental aspects of model interpretation, fairly similar to the case of the recent explainable artificial intelligence trend. We present the capabilities of our proposed framework by highlighting a real example of a BN learned from genomic data obtained from Allen Institute for Brain Science.

Bayesian inference has been long called for Bayesian computation techniques that are scalable to large data sets and applicable in big and complex models with a huge number of unknown parameters to infer. Sampling methods, such as Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC), in their current development do not meet this need. Sampling methods have not been successfully used in some modern areas such as deep neural networks. Even in more traditional areas such as graphical modelling and mixture modelling, it is very challenging to use MCMC and SMC. Variational Bayes (VB) is an optimization-based technique for approximate Bayesian inference, and provides a computationally efficient alternative to sampling methods.

Recently, particle-based variational inference (ParVI) methods have gained interest because they directly minimize the Kullback-Leibler divergence and do not suffer from approximation errors from the evidence-based lower bound. However, many ParVI approaches do not allow arbitrary sampling from the posterior, and the few that do allow such sampling suffer from suboptimality. This work proposes a new method for learning to approximately sample from the posterior distribution. We construct a neural sampler that is trained with the functional gradient of the KL-divergence between the empirical sampling distribution and the target distribution, assuming the gradient resides within a reproducing kernel Hilbert space. Our generative ParVI (GPVI) approach maintains the asymptotic performance of ParVI methods while offering the flexibility of a generative sampler. Through carefully constructed experiments, we show that GPVI outperforms previous generative ParVI methods such as amortized SVGD, and is competitive with ParVI as well as gold-standard approaches like Hamiltonian Monte Carlo for fitting both exactly known and intractable target distributions.

We explore the limitations of and best practices for using black-box variational inference to estimate posterior summaries of the model parameters. By taking an importance sampling perspective, we are able to explain and empirically demonstrate: 1) why the intuitions about the behavior of approximate families and divergences for low-dimensional posteriors fail for higher-dimensional posteriors, 2) how we can diagnose the pre-asymptotic reliability of variational inference in practice by examining the behavior of the density ratios (i.e., importance weights), 3) why the choice of variational objective is not as relevant for higher-dimensional posteriors, and 4) why, although flexible variational families can provide some benefits in higher dimensions, they also introduce additional optimization challenges. Based on these findings, for high-dimensional posteriors we recommend using the exclusive KL divergence that is most stable and easiest to optimize, and then focusing on improving the variational family or using model parameter transformations to make the posterior more similar to the approximating family. Our results also show that in low to moderate dimensions, heavy-tailed variational families and mass-covering divergences can increase the chances that the approximation can be improved by importance sampling.

Existing deterministic variational inference approaches for diffusion processes use simple proposals and target the marginal density of the posterior. We construct the variational process as a controlled version of the prior process and approximate the posterior by a set of moment functions. In combination with moment closure, the smoothing problem is reduced to a deterministic optimal control problem. Exploiting the path-wise Fisher information, we propose an optimization procedure that corresponds to a natural gradient descent in the variational parameters. Our approach allows for richer variational approximations that extend to state-dependent diffusion terms. The classical Gaussian process approximation is recovered as a special case.