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We consider statistical inference in factor analysis for ergodic and non-ergodic diffusion processes from discrete observations. Factor model based on high frequency time series data has been mainly discussed in the field of high dimensional covariance matrix estimation. In this field, the method based on principal component analysis has been mainly used. However, this method is effective only for high dimensional model. On the other hand, there is a method based on the quasi-likelihood. However, since the factor is assumed to be observable, we cannot use this method when the factor is latent. Thus, the existing methods are not effective when the factor is latent and the dimension of the observable variable is not so high. Therefore, we propose an effective method in the situation.

In this paper we analyze a fully discrete scheme for a general Cahn-Hilliard equation coupled with a nonsteady Magneto-hydrodynamics flow, which describes two immiscible, incompressible and electrically conducting fluids with different mobilities, fluid viscosities and magnetic diffusivities. A typical fully discrete scheme, which is comprised of conforming finite element method and the Euler semi-implicit discretization based on a convex splitting of the energy of the equation is considered in detail. We prove that our scheme is unconditionally energy stability and obtain some optimal error estimates for the concentration field, the chemical potential, the velocity field, the magnetic field and the pressure. The results of numerical tests are presented to validate the rates of convergence.

Since the two seminal papers by Fisher (1915, 1921) were published, the test under a fixed value correlation coefficient null hypothesis for the bivariate normal distribution constitutes an important statistical problem. In the framework of asymptotic robust statistics, it remains being a topic of great interest to be investigated. For this and other tests, focused on paired correlated normal random samples, R\'enyi's pseudodistance estimators are proposed, their asymptotic distribution is established and an iterative algorithm is provided for their computation. From them the Wald-type test statistics are constructed for different problems of interest and their influence function is theoretically studied. For testing null correlation in different contexts, an extensive simulation study and two real data based examples support the robust properties of our proposal.

Probabilistic databases (PDBs) model uncertainty in data. The current standard is to view PDBs as finite probability spaces over relational database instances. Since many attributes in typical databases have infinite domains, such as integers, strings, or real numbers, it is often more natural to view PDBs as infinite probability spaces over database instances. In this paper, we lay the mathematical foundations of infinite probabilistic databases. Our focus then is on independence assumptions. Tuple-independent PDBs play a central role in theory and practice of PDBs. Here, we study infinite tuple-independent PDBs as well as related models such as infinite block-independent disjoint PDBs. While the standard model of PDBs focuses on a set-based semantics, we also study tuple-independent PDBs with a bag semantics and independence in PDBs over uncountable fact spaces. We also propose a new approach to PDBs with an open-world assumption, addressing issues raised by Ceylan et al. (Proc. KR 2016) and generalizing their work, which is still rooted in finite tuple-independent PDBs. Moreover, for countable PDBs we propose an approximate query answering algorithm.

Simulation models, in particular agent-based models, are gaining popularity in economics. The considerable flexibility they offer, as well as their capacity to reproduce a variety of empirically observed behaviours of complex systems, give them broad appeal, and the increasing availability of cheap computing power has made their use feasible. Yet a widespread adoption in real-world modelling and decision-making scenarios has been hindered by the difficulty of performing parameter estimation for such models. In general, simulation models lack a tractable likelihood function, which precludes a straightforward application of standard statistical inference techniques. Several recent works have sought to address this problem through the application of likelihood-free inference techniques, in which parameter estimates are determined by performing some form of comparison between the observed data and simulation output. However, these approaches are (a) founded on restrictive assumptions, and/or (b) typically require many hundreds of thousands of simulations. These qualities make them unsuitable for large-scale simulations in economics and can cast doubt on the validity of these inference methods in such scenarios. In this paper, we investigate the efficacy of two classes of black-box approximate Bayesian inference methods that have recently drawn significant attention within the probabilistic machine learning community: neural posterior estimation and neural density ratio estimation. We present benchmarking experiments in which we demonstrate that neural network based black-box methods provide state of the art parameter inference for economic simulation models, and crucially are compatible with generic multivariate time-series data. In addition, we suggest appropriate assessment criteria for future benchmarking of approximate Bayesian inference procedures for economic simulation models.

Spectral clustering algorithms are very popular. Starting from a pairwise similarity matrix, spectral clustering gives a partition of data that approximately minimizes the total similarity scores across clusters. Since there is no need to model how data are distributed within each cluster, such a method enjoys algorithmic simplicity and robustness in clustering non-Gaussian data such as those near manifolds. Nevertheless, several important questions are unaddressed, such as how to estimate the similarity scores and cluster assignment probabilities, as important uncertainty estimates in clustering. In this article, we propose to solve these problems with a discovered generative modeling counterpart. Our clustering model is based on a spanning forest graph that consists of several disjoint spanning trees, with each tree corresponding to a cluster. Taking a Bayesian approach, we assign proper densities on the root and leaf nodes, and we prove that the posterior mode is almost the same as spectral clustering estimates. Further, we show that the associated generative process, named "forest process", is a continuous extension to the classic urn process, hence inheriting many nice properties such as having unbounded support for the number of clusters and being amenable to existing partition probability function; at the same time, we carefully characterize their differences. We demonstrate a novel application in joint clustering of multiple-subject functional magnetic resonance imaging scans of the human brain.

With a variety of local feature attribution methods being proposed in recent years, follow-up work suggested several evaluation strategies. To assess the attribution quality across different attribution techniques, the most popular among these evaluation strategies in the image domain use pixel perturbations. However, recent advances discovered that different evaluation strategies produce conflicting rankings of attribution methods and can be prohibitively expensive to compute. In this work, we present an information-theoretic analysis of evaluation strategies based on pixel perturbations. Our findings reveal that the results output by different evaluation strategies are strongly affected by information leakage through the shape of the removed pixels as opposed to their actual values. Using our theoretical insights, we propose a novel evaluation framework termed Remove and Debias (ROAD) which offers two contributions: First, it mitigates the impact of the confounders, which entails higher consistency among evaluation strategies. Second, ROAD does not require the computationally expensive retraining step and saves up to 99% in computational costs compared to the state-of-the-art. Our source code is available at //github.com/tleemann/road_evaluation.

We prove a formula for the evaluation of averages containing a scalar function of a Gaussian random vector multiplied by a product of the random vector components, each one raised at a power. Some powers could be of zeroth-order, and, for averages containing only one vector component to the first power, the formula reduces to Stein's lemma for the multivariate normal distribution. Also, by setting the said function inside average equal to one, we easily derive Isserlis theorem and its generalizations, regarding higher order moments of a Gaussian random vector. We provide two proofs of the formula, with the first being a rigorous proof via mathematical induction. The second is a formal, constructive derivation based on treating the average not as an integral, but as the action of pseudodifferential operators defined via the moment-generating function of the Gaussian random vector.

Approximate Bayesian Computation (ABC) enables statistical inference in complex models whose likelihoods are difficult to calculate but easy to simulate from. ABC constructs a kernel-type approximation to the posterior distribution through an accept/reject mechanism which compares summary statistics of real and simulated data. To obviate the need for summary statistics, we directly compare empirical distributions with a Kullback-Leibler (KL) divergence estimator obtained via classification. In particular, we blend flexible machine learning classifiers within ABC to automate fake/real data comparisons. We consider the traditional accept/reject kernel as well as an exponential weighting scheme which does not require the ABC acceptance threshold. Our theoretical results show that the rate at which our ABC posterior distributions concentrate around the true parameter depends on the estimation error of the classifier. We derive limiting posterior shape results and find that, with a properly scaled exponential kernel, asymptotic normality holds. We demonstrate the usefulness of our approach on simulated examples as well as real data in the context of stock volatility estimation.

Training datasets for machine learning often have some form of missingness. For example, to learn a model for deciding whom to give a loan, the available training data includes individuals who were given a loan in the past, but not those who were not. This missingness, if ignored, nullifies any fairness guarantee of the training procedure when the model is deployed. Using causal graphs, we characterize the missingness mechanisms in different real-world scenarios. We show conditions under which various distributions, used in popular fairness algorithms, can or can not be recovered from the training data. Our theoretical results imply that many of these algorithms can not guarantee fairness in practice. Modeling missingness also helps to identify correct design principles for fair algorithms. For example, in multi-stage settings where decisions are made in multiple screening rounds, we use our framework to derive the minimal distributions required to design a fair algorithm. Our proposed algorithm decentralizes the decision-making process and still achieves similar performance to the optimal algorithm that requires centralization and non-recoverable distributions.