The paper discusses shrinkage priors which impose increasing shrinkage in a sequence of parameters. We review the cumulative shrinkage process (CUSP) prior of Legramanti et al. (2020), which is a spike-and-slab shrinkage prior where the spike probability is stochastically increasing and constructed from the stick-breaking representation of a Dirichlet process prior. As a first contribution, this CUSP prior is extended by involving arbitrary stick-breaking representations arising from beta distributions. As a second contribution, we prove that exchangeable spike-and-slab priors, which are popular and widely used in sparse Bayesian factor analysis, can be represented as a finite generalized CUSP prior, which is easily obtained from the decreasing order statistics of the slab probabilities. Hence, exchangeable spike-and-slab shrinkage priors imply increasing shrinkage as the column index in the loading matrix increases, without imposing explicit order constraints on the slab probabilities. An application to sparse Bayesian factor analysis illustrates the usefulness of the findings of this paper. A new exchangeable spike-and-slab shrinkage prior based on the triple gamma prior of Cadonna et al. (2020) is introduced and shown to be helpful for estimating the unknown number of factors in a simulation study.

Clustering is commonly performed as an initial analysis step for uncovering structure in 'omics datasets, e.g. to discover molecular subtypes of disease. The high-throughput, high-dimensional nature of these datasets means that they provide information on a diverse array of different biomolecular processes and pathways. Different groups of variables (e.g. genes or proteins) will be implicated in different biomolecular processes, and hence undertaking analyses that are limited to identifying just a single clustering partition of the whole dataset is therefore liable to conflate the multiple clustering structures that may arise from these distinct processes. To address this, we propose a multi-view Bayesian mixture model that identifies groups of variables (``views"), each of which defines a distinct clustering structure. We consider applications in stratified medicine, for which our principal goal is to identify clusters of patients that define distinct, clinically actionable disease subtypes. We adopt the semi-supervised, outcome-guided mixture modelling approach of Bayesian profile regression that makes use of a response variable in order to guide inference toward the clusterings that are most relevant in a stratified medicine context. We present the model, together with illustrative simulation examples, and examples from pan-cancer proteomics. We demonstrate how the approach can be used to perform integrative clustering, and consider an example in which different 'omics datasets are integrated in the context of breast cancer subtyping.

OWL ontologies are a quite popular way to describe structured knowledge in terms of classes, relations among classes and class instances. In this paper, given a target class T of an OWL ontology, with a focus on ontologies with real- and boolean-valued data properties, we address the problem of learning graded fuzzy concept inclusion axioms with the aim of describing enough conditions for being an individual classified as instance of the class T. To do so, we present PN-OWL that is a two-stage learning algorithm made of a P-stage and an N-stage. Roughly, in the P-stage the algorithm tries to cover as many positive examples as possible (increase recall), without compromising too much precision, while in the N-stage, the algorithm tries to rule out as many false positives, covered by the P-stage, as possible. PN-OWL then aggregates the fuzzy inclusion axioms learnt at the P-stage and the N-stage by combining them via aggregation functions to allow for a final decision whether an individual is instance of T or not. We also illustrate its effectiveness by means of an experimentation. An interesting feature is that fuzzy datatypes are built automatically, the learnt fuzzy concept inclusions can be represented directly into Fuzzy OWL 2 and, thus, any Fuzzy OWL 2 reasoner can then be used to automatically determine/classify (and to which degree) whether an individual belongs to the target class T or not.

TECHNICAL NOTE - TN-2023-02-28, FEBRUARY 2023. 1 Abstract This short technical note points out an erroneous claim about a new rule of combination of basic belief assignments presented recently by Kenn et al. in [1], referred as Kenn's rule of combination (or just as KRC for short). We prove thanks a very simple counter-example that Kenn's rule is not associative. Consequently, the results of the method proposed by Kenn et al. highly depends on the ad-hoc sequential order chosen for the fusion process as proposed by the authors. This serious problem casts in doubt the interest of this method and its real ability to provide trustful results and to make good decisions to help for precise breast cancer therapy. Recently a paper devoted to the Breast Cancer Precision Therapy by Kenn et al. [1] attracted our attention for two main reasons: 1) this application of information fusion is very interesting and important; 2) Kenn's et al. method is based on a new rule of combination of basic belief assignments (BBAs).

In this literature review, we provided a survey of interpretable and explainable machine learning methods (see Tables 1 and 2 for the summary of the techniques), described commonest goals and desiderata for these techniques, motivated their relevance in several fields of application, and discussed their quantitative evaluation. Interpretability and explainability still remain an active area of research, especially, in the face of recent rapid progress in designing highly performant predictive models and inevitable infusion of machine learning into other domains, where decisions have far-reaching consequences. For years the field has been challenged by a lack of clear definitions for interpretability or explainability, these terms being often wielded "in a quasi-mathematical way"[6,122]. For many techniques, there still exist no satisfactory functionally-grounded evaluation criteria and universally accepted benchmarks, hindering reproducibility and model comparison. Moreover, meaningful adaptations of these methods to'real-world' machine learning systems and data analysis problems largely remain a matter for the future. It has been argued that, for successful and widespread use of interpretable and explainable machine learning models, stakeholders need to be involved in the discussion[4, 122]. A meaningful and equal collaboration between machine learning researchers and stakeholders from various domains, such as medicine, natural sciences, and law, is a logical next step within the evolution of interpretable and explainable ML.

Offline reinforcement learning (RL) aims at learning an optimal strategy using a pre-collected dataset without further interactions with the environment. While various algorithms have been proposed for offline RL in the previous literature, the minimax optimality has only been (nearly) established for tabular Markov decision processes (MDPs). In this paper, we focus on offline RL with linear function approximation and propose a new pessimism-based algorithm for offline linear MDP. At the core of our algorithm is the uncertainty decomposition via a reference function, which is new in the literature of offline RL under linear function approximation. Theoretical analysis demonstrates that our algorithm can match the performance lower bound up to logarithmic factors. We also extend our techniques to the two-player zero-sum Markov games (MGs), and establish a new performance lower bound for MGs, which tightens the existing result, and verifies the nearly minimax optimality of the proposed algorithm. To the best of our knowledge, these are the first computationally efficient and nearly minimax optimal algorithms for offline single-agent MDPs and MGs with linear function approximation.

We propose an approach based on function evaluations and Bayesian inference to extract higher-order differential information of objective functions {from a given ensemble of particles}. Pointwise evaluation $\{V(x^i)\}_i$ of some potential $V$ in an ensemble $\{x^i\}_i$ contains implicit information about first or higher order derivatives, which can be made explicit with little computational effort (ensemble-based gradient inference -- EGI). We suggest to use this information for the improvement of established ensemble-based numerical methods for optimization and sampling such as Consensus-based optimization and Langevin-based samplers. Numerical studies indicate that the augmented algorithms are often superior to their gradient-free variants, in particular the augmented methods help the ensembles to escape their initial domain, to explore multimodal, non-Gaussian settings and to speed up the collapse at the end of optimization dynamics.} The code for the numerical examples in this manuscript can be found in the paper's Github repository (https://github.com/MercuryBench/ensemble-based-gradient.git).

Ensemble sampling serves as a practical approximation to Thompson sampling when maintaining an exact posterior distribution over model parameters is computationally intractable. In this paper, we establish a regret bound that ensures desirable behavior when ensemble sampling is applied to the linear bandit problem. This represents the first rigorous regret analysis of ensemble sampling and is made possible by leveraging information-theoretic concepts and novel analytic techniques that may prove useful beyond the scope of this paper.

When estimating unknown parameters in a dynamic model the optimum solution to the parameter estimation problem may not remain constant. Specifically, the optimal values of the model parameters may change through time because of the evolution of the underlying process: finding them is, in general, not straightforward. A survey of basic techniques for tracking the time-varying dynamics of a system is provided in [Ljung and Gunnarsson, 1990] where recursive algorithms in non-stationary stochastic optimization are analysed under different assumptions about the true system's variations, see also [Simonetto et al., 2020] for a review in a purely deterministic setting. In [Delyon and Juditsky, 1995] the problem of tracking the random drifting parameters of a linear regression system is tackled, and [Zhu and Spall, 2016] builds a computable tracking error bound for how a stochastic approximation with constant gain keeps up with a non-stationary target. Successively, [Wilson et al., 2019] introduces a framework for sequentially solving convex stochastic minimization problems, where the distance between successive minimizers is bounded. The minimization problems are then solved by sequentially applying an optimization algorithm, such as stochastic gradient descent (SGD). In a similar setting, [Cao et al., 2019] establishes an upper bound on the regret of a projected SGD algorithm with respect to the drift of the dynamic optima, while [Cutler et al., 2021] provides novel non-asymptotic convergence guarantees for stochastic algorithms with iterate averaging.

Nash Q-learning may be considered one of the first and most known algorithms in multi-agent reinforcement learning (MARL) for learning policies that constitute a Nash equilibrium of an underlying general-sum Markov game. Its original proof provided asymptotic guarantees and was for the tabular case. Recently, finite-sample guarantees have been provided using more modern RL techniques for the tabular case. Our work analyzes Nash Q-learning using linear function approximation -- a representation regime introduced when the state space is large or continuous -- and provides finite-sample guarantees that indicate its sample efficiency. We find that the obtained performance nearly matches an existing efficient result for single-agent RL under the same representation and has a polynomial gap when compared to the best-known result for the tabular case.