arXiv subjects

Matthias C. M. Troffaes

Publications and source records attributed to Matthias C. M. Troffaes.

At least 19 recordsLinked to original sources

arXiv2024

Robust Bayesian causal estimation for causal inference in medical diagnosis

Causal effect estimation is a critical task in statistical learning that aims to find the causal effect on subjects by identifying causal links between a number of predictor (or, explanatory) variables and the outcome of a treatment. In a regressional framework, we assign a treatment and outcome model to estimate the average causal effect. Additionally, for high dimensional regression problems, variable selection methods are also used to find a subset of predictor variables that maximises the predictive performance of the underlying model for better estimation of the causal effect. In this paper, we propose a different approach. We focus on the variable selection aspects of high dimensional causal estimation problem. We suggest a cautious Bayesian group LASSO (least absolute shrinkage and selection operator) framework for variable selection using prior sensitivity analysis. We argue that in some cases, abstaining from selecting (or, rejecting) a predictor is beneficial and we should gather more information to obtain a more decisive result. We also show that for problems with very limited information, expert elicited variable selection can give us a more stable causal effect estimation as it avoids overfitting. Lastly, we carry a comparative study with synthetic dataset and show the applicability of our method in real-life situations.

stat.ME↗

arXiv2024

Regret-based budgeted decision rules under severe uncertainty

One way to make decisions under uncertainty is to select an optimal option from a possible range of options, by maximizing the expected utilities derived from a probability model. However, under severe uncertainty, identifying precise probabilities is hard. For this reason, imprecise probability models uncertainty through convex sets of probabilities, and considers decision rules that can return multiple options to reflect insufficient information. Many well-founded decision rules have been studied in the past, but none of those standard rules are able to control the number of returned alternatives. This can be a problem for large decision problems, due to the cognitive burden decision makers have to face when presented with a large number of alternatives. Our contribution proposes regret-based ideas to construct new decision rules which return a bounded number of options, where the limit on the number of options is set in advance by the decision maker as an expression of their cognitive limitation. We also study their consistency and numerical behaviour.

math.ST↗

arXiv2024

Data-driven Estimation of Under Frequency Load Shedding after Outages in Small Power Systems

This paper presents a data-driven methodology for estimating Under Frequency Load Shedding (UFLS) in small power systems. UFLS plays a vital role in maintaining system stability by shedding load when the frequency drops below a specified threshold following loss of generation. Using a dynamic System Frequency Response (SFR) model we generate different values of UFLS (i.e., labels) predicated on a set of carefully selected operating conditions (i.e., features). Machine Learning (ML) algorithms are then applied to learn the relationship between chosen features and the UFLS labels. A novel regression tree and the Tobit model are suggested for this purpose and we show how the resulting non-linear model can be directly incorporated into a Mixed Integer Linear Programming (MILP) problem. The trained model can be used to estimate UFLS in security-constrained operational planning problems, improving frequency response, optimizing reserve allocation, and reducing costs. The methodology is applied to the La Palma island power system, demonstrating its accuracy and effectiveness. The results confirm that the amount of UFLS can be estimated with the Mean Absolute Error (MAE) as small as 0.213 MW for the whole process, with a model that is representable as a MILP for use in scheduling problems such as unit commitment among others.

eess.SY↗

arXiv2022

Iterative importance sampling with Markov chain Monte Carlo sampling in robust Bayesian analysis

Bayesian inference under a set of priors, called robust Bayesian analysis, allows for estimation of parameters within a model and quantification of epistemic uncertainty in quantities of interest by bounded (or imprecise) probability. Iterative importance sampling can be used to estimate bounds on the quantity of interest by optimizing over the set of priors. A method for iterative importance sampling when the robust Bayesian inference rely on Markov chain Monte Carlo (MCMC) sampling is proposed. To accommodate the MCMC sampling in iterative importance sampling, a new expression for the effective sample size of the importance sampling is derived, which accounts for the correlation in the MCMC samples. To illustrate the proposed method for robust Bayesian analysis, iterative importance sampling with MCMC sampling is applied to estimate the lower bound of the overall effect in a previously published meta-analysis with a random effects model. The performance of the method compared to a grid search method and under different degrees of prior-data conflict is also explored.

stat.CO↗

arXiv2022

A robust Bayesian analysis of variable selection under prior ignorance

We propose a cautious Bayesian variable selection routine by investigating the sensitivity of a hierarchical model, where the regression coefficients are specified by spike and slab priors. We exploit the use of latent variables to understand the importance of the co-variates. These latent variables also allow us to obtain the size of the model space which is an important aspect of high dimensional problems. In our approach, instead of fixing a single prior, we adopt a specific type of robust Bayesian analysis, where we consider a set of priors within the same parametric family to specify the selection probabilities of these latent variables. We achieve that by considering a set of expected prior selection probabilities, which allows us to perform a sensitivity analysis to understand the effect of prior elicitation on the variable selection. The sensitivity analysis provides us sets of posteriors for the regression coefficients as well as the selection indicators and we show that the posterior odds of the model selection probabilities are monotone with respect to the prior expectations of the selection probabilities. We also analyse synthetic and real life datasets to illustrate our cautious variable selection method and compare it with other well known methods.

stat.ME↗

arXiv2022

A robust Bayesian bias-adjusted random effects model for consideration of uncertainty about bias terms in evidence synthesis

Meta-analysis is a statistical method used in evidence synthesis for combining, analyzing and summarizing studies that have the same target endpoint and aims to derive a pooled quantitative estimate using fixed and random effects models or network models. Differences among included studies depend on variations in target populations (i.e. heterogeneity) and variations in study quality due to study design and execution (i.e. bias). The risk of bias is usually assessed qualitatively using critical appraisal, and quantitative bias analysis can be used to evaluate the influence of bias on the quantity of interest. We propose a way to consider ignorance or ambiguity in how to quantify bias terms in a bias analysis by characterizing bias with imprecision (as bounds on probability) and use robust Bayesian analysis to estimate the overall effect. Robust Bayesian analysis is here seen as Bayesian updating performed over a set of coherent probability distributions, where the set emerges from a set of bias terms. We show how the set of bias terms can be specified based on judgments on the relative magnitude of biases (i.e., low, unclear and high risk of bias) in one or several domains of the Cochrane's risk of bias table. For illustration, we apply a robust Bayesian bias-adjusted random effects model to an already published meta-analysis on the effect of Rituximab for rheumatoid arthritis from the Cochrane Database of Systematic Reviews.

stat.ME↗

arXiv2021

Improving and benchmarking of algorithms for $Γ$-maximin, $Γ$-maximax and interval dominance

$Γ$-maximin, $Γ$-maximax and inteval dominance are familiar decision criteria for making decisions under severe uncertainty, when probability distributions can only be partially identified. One can apply these three criteria by solving sequences of linear programs. In this study, we present new algorithms for these criteria and compare their performance to existing standard algorithms. Specifically, we use efficient ways, based on previous work, to find common initial feasible points for these algorithms. Exploiting these initial feasible points, we develop early stopping criteria to determine whether gambles are either $Γ$-maximin, $Γ$-maximax or interval dominant. We observe that the primal-dual interior point method benefits considerably from these improvements. In our simulation, we find that our proposed algorithms outperform the standard algorithms when the size of the domain of lower previsions is less or equal to the sizes of decisions and outcomes. However, our proposed algorithms do not outperform the standard algorithms in the case that the size of the domain of lower previsions is much larger than the sizes of decisions and outcomes.

math.OC↗

arXiv2021

Robust decision analysis under severe uncertainty and ambiguous tradeoffs: an invasive species case study

Bayesian decision analysis is a useful method for risk management decisions, but is limited in its ability to consider severe uncertainty in knowledge, and value ambiguity in management objectives. We study the use of robust Bayesian decision analysis to handle problems where one or both of these issues arise. The robust Bayesian approach models severe uncertainty through bounds on probability distributions, and value ambiguity through bounds on utility functions. To incorporate data, standard Bayesian updating is applied on the entire set of distributions. To elicit our expert's utility representing the value of different management objectives, we use a modified version of the swing weighting procedure that can cope with severe value ambiguity. We demonstrate these methods on an environmental management problem to eradicate an alien invasive marmorkrebs recently discovered in Sweden, which needed a rapid response despite substantial knowledge gaps if the species was still present (i.e. severe uncertainty) and the need for difficult tradeoffs and competing interests (i.e. value ambiguity). We identify that the decision alternatives to drain the system and remove individuals in combination with dredging and sieving with or without a degradable biocide, or increasing pH, are consistently bad under the entire range of probability and utility bounds. This case study shows how robust Bayesian decision analysis provides a transparent methodology for integrating information in risk management problems where little data are available and/or where the tradeoffs ambiguous.

stat.AP↗

arXiv2019

Improving and benchmarking of algorithms for decision making with lower previsions

Maximality, interval dominance, and E-admissibility are three well-known criteria for decision making under severe uncertainty using lower previsions. We present a new fast algorithm for finding maximal gambles. We compare its performance to existing algorithms, one proposed by Troffaes and Hable (2014), and one by Jansen, Augustin, and Schollmeyer (2017). To do so, we develop a new method for generating random decision problems with pre-specified ratios of maximal and interval dominant gambles. Based on earlier work, we present efficient ways to find common feasible starting points in these algorithms. We then exploit these feasible starting points to develop early stopping criteria for the primal-dual interior point method, further improving efficiency. We find that the primal-dual interior point method works best. We also investigate the use of interval dominance to eliminate non-maximal gambles. This can make the problem smaller, and we observe that this benefits Jansen et al.'s algorithm, but perhaps surprisingly, not the other two algorithms. We find that our algorithm, without using interval dominance, outperforms all other algorithms in all scenarios in our benchmarking.

math.OC↗

arXiv2019

Evaluating betting odds and free coupons using desirability

In the UK betting market, bookmakers often offer a free coupon to new customers. These free coupons allow the customer to place extra bets, at lower risk, in combination with the usual betting odds. We are interested in whether a customer can exploit these free coupons in order to make a sure gain, and if so, how the customer can achieve this. To answer this question, we evaluate the odds and free coupons as a set of desirable gambles for the bookmaker. We show that we can use the Choquet integral to check whether this set of desirable gambles incurs sure loss for the bookmaker, and hence, results in a sure gain for the customer. In the latter case, we also show how a customer can determine the combination of bets that make the best possible gain, based on complementary slackness. As an illustration, we look at some actual betting odds in the market and find that, without free coupons, the set of desirable gambles derived from those odds avoids sure loss. However, with free coupons, we identify some combinations of bets that customers could place in order to make a guaranteed gain.

q-fin.MF↗

arXiv2018

Improved linear programming methods for checking avoiding sure loss

We review the simplex method and two interior-point methods (the affine scaling and the primal-dual) for solving linear programming problems for checking avoiding sure loss, and propose novel improvements. We exploit the structure of these problems to reduce their size. We also present an extra stopping criterion, and direct ways to calculate feasible starting points in almost all cases. For benchmarking, we present algorithms for generating random sets of desirable gambles that either avoid or do not avoid sure loss. We test our improvements on these linear programming methods by measuring the computational time on these generated sets. We assess the relative performance of the three methods as a function of the number of desirable gambles and the number of outcomes. Overall, the affine scaling and primal-dual methods benefit from the improvements, and they both outperform the simplex method in most scenarios. We conclude that the simplex method is not a good choice for checking avoiding sure loss. If problems are small, then there is no tangible difference in performance between all methods. For large problems, our improved primal-dual method performs at least three times faster than any of the other methods.

math.OC↗

arXiv2018

Decision making under uncertainty using imprecise probabilities

Various ways for decision making with imprecise probabilities (admissibility, maximal expected utility, maximality, E-admissibility, $Γ$-maximax, $Γ$-maximin, all of which are well-known from the literature) are discussed and compared. We generalize a well-known sufficient condition for existence of optimal decisions. A simple numerical example shows how these criteria can work in practice, and demonstrates their differences. Finally, we suggest an efficient approach to calculate optimal decisions under these decision criteria.

math.PR↗

arXiv2018

Imprecise Monte Carlo simulation and iterative importance sampling for the estimation of lower previsions

We develop a theoretical framework for studying numerical estimation of lower previsions, generally applicable to two-level Monte Carlo methods, importance sampling methods, and a wide range of other sampling methods one might devise. We link consistency of these estimators to Glivenko-Cantelli classes, and for the sub-Gaussian case we show how the correlation structure of this process can be used to bound the bias and prove consistency. We also propose a new upper estimator, which can be used along with the standard lower estimator, in order to provide a simple confidence interval. As a case study of this framework, we then discuss how importance sampling can be exploited to provide accurate numerical estimates of lower previsions. We propose an iterative importance sampling method to drastically improve the performance of imprecise importance sampling. We demonstrate our results on the imprecise Dirichlet model.

stat.CO↗

arXiv2014

A geometric and game-theoretic study of the conjunction of possibility measures

In this paper, we study the conjunction of possibility measures when they are interpreted as coherent upper probabilities, that is, as upper bounds for some set of probability measures. We identify conditions under which the minimum of two possibility measures remains a possibility measure. We provide graphical way to check these conditions, by means of a zero-sum game formulation of the problem. This also gives us a nice way to adjust the initial possibility measures so their minimum is guaranteed to be a possibility measure. Finally, we identify conditions under which the minimum of two possibility measures is a coherent upper probability, or in other words, conditions under which the minimum of two possibility measures is an exact upper bound for the intersection of the credal sets of those two possibility measures.

math.PR↗

arXiv2013

A robust Bayesian approach to modelling epistemic uncertainty in common-cause failure models

In a standard Bayesian approach to the alpha-factor model for common-cause failure, a precise Dirichlet prior distribution models epistemic uncertainty in the alpha-factors. This Dirichlet prior is then updated with observed data to obtain a posterior distribution, which forms the basis for further inferences. In this paper, we adapt the imprecise Dirichlet model of Walley to represent epistemic uncertainty in the alpha-factors. In this approach, epistemic uncertainty is expressed more cautiously via lower and upper expectations for each alpha-factor, along with a learning parameter which determines how quickly the model learns from observed data. For this application, we focus on elicitation of the learning parameter, and find that values in the range of 1 to 10 seem reasonable. The approach is compared with Kelly and Atwood's minimally informative Dirichlet prior for the alpha-factor model, which incorporated precise mean values for the alpha-factors, but which was otherwise quite diffuse. Next, we explore the use of a set of Gamma priors to model epistemic uncertainty in the marginal failure rate, expressed via a lower and upper expectation for this rate, again along with a learning parameter. As zero counts are generally less of an issue here, we find that the choice of this learning parameter is less crucial. Finally, we demonstrate how both epistemic uncertainty models can be combined to arrive at lower and upper expectations for all common-cause failure rates. Thereby, we effectively provide a full sensitivity analysis of common-cause failure rates, properly reflecting epistemic uncertainty of the analyst on all levels of the common-cause failure model.

stat.ME↗

arXiv2012

Identifying subtree perfectness in decision trees

In decision problems, often, utilities and probabilities are hard to determine. In such cases, one can resort to so-called choice functions. They provide a means to determine which options in a particular set are optimal, and allow incomparability among any number of options. Applying choice functions in sequential decision problems can be highly non-trivial, as the usual properties of maximising expected utility may no longer be satisfied. In this paper, we study one of these properties: we revisit and reinterpret Selten's concept of subgame perfectness in the context of decision trees, leading us to the concept of subtree perfectness, which basically says that the optimal solution of a decision tree should not depend on any larger tree it may be embedded in. In other words, subtree perfectness excludes counterfactual reasoning, and therefore may be desirable from some philosophical points of view. Subtree perfectness is also desirable from a practical point of view, because it admits efficient algorithms for solving decision trees, such as backward induction. The main contribution of this paper is a very simple non-technical criterion for determining whether any given choice function will satisfy subtree perfectness or not. We demonstrate the theorem and illustrate subtree perfectness, or the lack thereof, through numerous examples, for a wide variety of choice functions, where incomparability amongst strategies can be caused by imprecision in either probabilities or utilities. We find that almost no choice function, except for maximising expected utility, satisfies it in general. We also find that choice functions other than maximising expected utility can satisfy it, provided that we restrict either the structure of the tree, or the structure of the choice function.

math.ST↗

arXiv2012

Normal form backward induction for decision trees with coherent lower previsions

We examine normal form solutions of decision trees under typical choice functions induced by lower previsions. For large trees, finding such solutions is hard as very many strategies must be considered. In an earlier paper, we extended backward induction to arbitrary choice functions, yielding far more efficient solutions, and we identified simple necessary and sufficient conditions for this to work. In this paper, we show that backward induction works for maximality and E-admissibility, but not for interval dominance and Gamma-maximin. We also show that, in some situations, a computationally cheap approximation of a choice function can be used, even if the approximation violates the conditions for backward induction; for instance, interval dominance with backward induction will yield at least all maximal normal form solutions.

math.ST↗

arXiv2012

Finite approximations to coherent choice

This paper studies and bounds the effects of approximating loss functions and credal sets on choice functions, under very weak assumptions. In particular, the credal set is assumed to be neither convex nor closed. The main result is that the effects of approximation can be bounded, although in general, approximation of the credal set may not always be practically possible. In case of pairwise choice, I demonstrate how the situation can be improved by showing that only approximations of the extreme points of the closure of the convex hull of the credal set need to be taken into account, as expected.

stat.CO↗