On Higher-Order Probabilistic Subrecursion

We study the expressive power of subrecursive probabilistic higher-order calculi. More specifically, we show that endowing a very expressive deterministic calculus like G\"odel's $\mathbb{T}$ with various forms of probabilistic choice operators may result in calculi which are not equivalent as for the class of distributions they give rise to, although they all guarantee almost-sure termination. Along the way, we introduce a probabilistic variation of the classic reducibility technique, and we prove that the simplest form of probabilistic choice leaves the expressive power of $\mathbb{T}$ essentially unaltered. The paper ends with some observations about the functional expressive power: expectedly, all the considered calculi capture the functions which $\mathbb{T}$ itself represents, at least when standard notions of observations are considered.

Estimating Lost Sales for Substitutable Products with Uncertain On-Shelf Availability

Problem definition: We address the problem of how to estimate lost sales for substitutable products when there is no reliable on-shelf availability (OSA) information. Academic/practical relevance: We develop a novel approach to estimating lost sales using only sales data, a market share estimate, and an estimate of overall availability. We use the method to illustrate the negative consequences of using potentially inaccurate inventory records as indicators of availability. Methodology: We suggest a partially hidden Markov model of OSA to generate probabilistic choice sets and incorporate these probabilistic choice sets into the estimation of a multinomial logit demand model using a nested expectation-maximization algorithm. We highlight the importance of considering inventory reliability problems first through simulation and then by applying the procedure to a data set from a major U.S. retailer. Results: The simulations show that the method converges in seconds and produces estimates with similar or lower bias than state-of-the-art benchmarks. For the product category under consideration at the retailer, our procedure finds lost sales of around 3.0% compared with 0.2% when relying on the inventory record as an indicator of availability. Managerial implications: The method efficiently computes estimates that can be used to improve inventory management and guide managers on how to use their scarce resources to improve stocking execution. The research also shows that ignoring inventory record inaccuracies when estimating lost sales can produce substantially inaccurate estimates, which leads to incorrect parameters in supply chain planning.

Probabilistic Timed Automata with One Clock and Initialised Clock-Dependent Probabilities

Clock-dependent probabilistic timed automata extend classical timed automata with discrete probabilistic choice, where the probabilities are allowed to depend on the exact values of the clocks. Previous work has shown that the quantitative reachability problem for clock-dependent probabilistic timed automata with at least three clocks is undecidable. In this paper, we consider the subclass of clock-dependent probabilistic timed automata that have one clock, that have clock dependencies described by affine functions, and that satisfy an initialisation condition requiring that, at some point between taking edges with non-trivial clock dependencies, the clock must have an integer value. We present an approach for solving in polynomial time quantitative and qualitative reachability problems of such one-clock initialised clock-dependent probabilistic timed automata. Our results are obtained by a transformation to interval Markov decision processes.

Task Learnability Modulates Surprise but Not Valence Processing for Reinforcement Learning in Probabilistic Choice Tasks

The goal of temporal difference (TD) reinforcement learning is to maximize outcomes and improve future decision-making. It does so by utilizing a prediction error (PE), which quantifies the difference between the expected and the obtained outcome. In gambling tasks, however, decision-making cannot be improved because of the lack of learnability. On the basis of the idea that TD utilizes two independent bits of information from the PE (valence and surprise), we asked which of these aspects is affected when a task is not learnable. We contrasted behavioral data and ERPs in a learning variant and a gambling variant of a simple two-armed bandit task, in which outcome sequences were matched across tasks. Participants were explicitly informed that feedback could be used to improve performance in the learning task but not in the gambling task, and we predicted a corresponding modulation of the aspects of the PE. We used a model-based analysis of ERP data to extract the neural footprints of the valence and surprise information in the two tasks. Our results revealed that task learnability modulates reinforcement learning via the suppression of surprise processing but leaves the processing of valence unaffected. On the basis of our model and the data, we propose that task learnability can selectively suppress TD learning as well as alter behavioral adaptation based on a flexible cost–benefit arbitration.

On continuation-passing transformations and expected cost analysis

We define a continuation-passing style (CPS) translation for a typed λ-calculus with probabilistic choice, unbounded recursion, and a tick operator — for modeling cost. The target language is a (non-probabilistic) λ-calculus, enriched with a type of extended positive reals and a fixpoint operator. We then show that applying the CPS transform of an expression M to the continuation λ v . 0 yields the expected cost of M . We also introduce a formal system for higher-order logic, called EHOL, prove it sound, and show it can derive tight upper bounds on the expected cost of classic examples, including Coupon Collector and Random Walk. Moreover, we relate our translation to Kaminski et al.’s ert-calculus, showing that the latter can be recovered by applying our CPS translation to (a generalization of) the classic embedding of imperative programs into λ-calculus. Finally, we prove that the CPS transform of an expression can also be used to compute pre-expectations and to reason about almost sure termination.

Falsification of Hybrid Systems Using Adaptive Probabilistic Search

We present and analyse an algorithm that quickly finds falsifying inputs for hybrid systems. Our method is based on a probabilistically directed tree search, whose distribution adapts to consider an increasingly fine-grained discretization of the input space. In experiments with standard benchmarks, our algorithm shows comparable or better performance to existing techniques, yet it does not build an explicit model of a system. Instead, at each decision point within a single trial, it makes an uninformed probabilistic choice between simple strategies to extend the input signal by means of exploration or exploitation. Key to our approach is the way input signal space is decomposed into levels, such that coarse segments are more probable than fine segments. We perform experiments to demonstrate how and why our approach works, finding that a fully randomized exploration strategy performs as well as our original algorithm that exploits robustness. We propose this strategy as a new baseline for falsification and conclude that more discriminative benchmarks are required.

Combinatorial Assortment Optimization

Assortment optimization refers to the problem of designing a slate of products to offer potential customers, such as stocking the shelves in a convenience store. The price of each product is fixed in advance, and a probabilistic choice function describes which product a customer will choose from any given subset. We introduce the combinatorial assortment problem, where each customer may select a bundle of products. We consider a choice model in which each consumer selects a utility-maximizing bundle subject to a private valuation function, and study the complexity of the resulting optimization problem. Our main result is an exact algorithm for additive k -demand valuations, under a model of vertical differentiation in which customers agree on the relative value of each pair of items but differ in their absolute willingness to pay. For valuations that are vertically differentiated but not necessarily additive k -demand, we show how to obtain constant approximations under a “well-priced” condition, where each product’s price is sufficiently high. We further show that even for a single customer with known valuation, any sub-polynomial approximation to the problem requires exponentially many demand queries when the valuation function is XOS and that no FPTAS exists even when the valuation is succinctly representable.

A trustful monad for axiomatic reasoning with probability and nondeterminism

The algebraic properties of the combination of probabilistic choice and nondeterministic choice have long been a research topic in program semantics. This paper explains a formalization in the Coq proof assistant of a monad equipped with both choices: the geometrically convex monad. This formalization has an immediate application: it provides a model for a monad that implements a nontrivial interface, which allows for proofs by equational reasoning using probabilistic and nondeterministic effects. We explain the technical choices we made to go from the literature to a complete Coq formalization, from which we identify reusable theories about mathematical structures such as convex spaces and concrete categories, and that we integrate in a framework for monadic equational reasoning.

Ensembled Machine Learning Model for Aviation Incident Risk Prediction

With the fabulous development of air traffic request expected throughout the following two decades, the security of the air transportation framework is of expanding concern. In this paper, we encourage the "proactive security" worldview to expand framework wellbeing with an emphasis on anticipating the seriousness of strange flight occasions as far as their hazard levels. To achieve this objective, a prescient model should be created to look at a wide assortment of potential cases and measure the hazard related with the conceivable result. By using the episode reports accessible in the Aviation Safety Reporting System (ASRS), we construct a half breed model comprising of help vector machine and K-closest neighbor calculation to evaluate the hazard related with the result of each perilous reason. The proposed system is created in four stages. Initially, we classify all the occasions, in view of the degree of hazard related with the occasion result, into five gatherings: high hazard, decently high hazard, medium hazard, respectably medium hazard, and okay. Furthermore, a help vector machine model is utilized to find the connections between the occasion outline in text configuration and occasion result. In this application K-closest neighbors (KNN) and bolster vector machines (SVM) are applied to group the everyday nearby climate types In equal, knn calculation is utilized to highlights and occasion results subsequently improving the forecast. At long last, the forecast on hazard level order is stretched out to occasion level results through a probabilistic choice tree