Probabilism, entropies and strictly proper scoring rules

Landes, Jürgen

doi:10.1016/j.ijar.2015.05.007

Cited by 16 publications

(11 citation statements)

References 53 publications

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“…The entropy maximization is a well-known method for determination of prior probabilities [31][32][33][34][35][36][37]. This method was developed within statistical physics [38], and it reflects the second law of thermodynamics, i.e., the natural tendency of the entropy to increase in closed systems [38].…”

Section: A General Discussion On Entropy Maximization Versus Its Mini...mentioning

confidence: 99%

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Adaptive decision making via entropy minimization

Allahverdyan

Galstyan²,

Abbas

et al. 2018

International Journal of Approximate Reasoning

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An agent choosing between various actions tends to take the one with the lowest cost. But this choice is arguably too rigid (not adaptive) to be useful in complex situations, e.g., where explorationexploitation trade-off is relevant in creative task solving or when stated preferences differ from revealed ones. Here we study an agent who is willing to sacrifice a fixed amount of expected utility for adaptation. How can/ought our agent choose an optimal (in a technical sense) mixed action? We explore consequences of making this choice via entropy minimization, which is argued to be a specific example of risk-aversion. This recovers the ǫ-greedy probabilities known in reinforcement learning. We show that the entropy minimization leads to rudimentary forms of intelligent behavior: (i) the agent assigns a non-negligible probability to costly events; but (ii) chooses with a sizable probability the action related to less cost (lesser of two evils) when confronted with two actions with comparable costs; (iii) the agent is subject to effects similar to cognitive dissonance and frustration. Neither of these features are shown by entropy maximization. See section II for more details. We stress that we do not mean the delayed reward situation, where the utility is constant, but is discounted by some known factor, because the action is performed now, while its reward will come in future.How to assign prior probabilities to avoid the strictly deterministic (1)? Such probabilities should hold a natural constraint that actions related to higher cost are getting smaller probabilities. Two ad hoc solutions are especially simple: one can take into account only the second-best action, or take all non-best actions with the same (small) probability. In reinforcement learning the latter prior probability is known as the ǫ-greedy [3]. It is preferable to have a regular method of choosing non-deterministic probabilities, which will reflect people's attitudes towards the decision making in an uncertain situation, and which will include the above ad hoc solutions as particular cases.Here we explore the possibility of defining the prior probabilities via risk minimization (or maximization); see [9, 10] for reviews on the notion of risk and its various interpretations. We assume that the agent first decides how much average utility E−min k [ε k ] he invests into exploration by going into nonoptimal-in the sense of not holding (1)-behavior. We employ the notion of risk in a specific context, namely when comparing the behavior of agents having the same utilities for various actions and the same value of E. We argue below that maximizing (minimizing) risk in this specific situation can be done via maximizing (minimizing) the entropy − n k=1 p k ln p k . People demonstrate both risk minimization (aversion) and maximization (seeking) [12,25], though the risk in those situations is a less specific (and more difficult to describe) notion-first because it involves agents having different utilities for same actions, and second because it involves a d...

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Section: A General Discussion On Entropy Maximization Versus Its Mini...mentioning

confidence: 99%

“…Likewise, one shows thatŜ(E) is a concave function of E: d 2Ŝ dE 2 ≤ 0; cf. (36). Now using (41, 39) we find…”

Section: Entropy Maximization For Risk-seeking Agentsmentioning

confidence: 99%

Adaptive decision making via entropy minimization

Allahverdyan

Galstyan²,

Abbas

et al. 2018

International Journal of Approximate Reasoning

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“…However, our analysis is unique in that it focuses on measuring a model's behavior in uncertainty quantification and takes a rigorous, decision-theoretic view of the problem. As a result, it works with a special family of risk functions (i.e., the strictly proper scoring rule) that measure a model's performance in uncertainty calibration, handles the existence of unknown domain via a minimax formulation, and derives the solution by using a generalized version of maximum entropy theorem for Bergman scores [27,42]. The form of the optimal solution we derived in ( 5) takes an intuitive form, and has already been used widely as a training objective in many uncertainty works that leverage adversarial training and generative modeling to detect OOD examples [30,31,46,50,51].…”

Section: Related Workmentioning

confidence: 99%

Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness

Liu¹,

Zi²,

Padhy³

et al. 2020

Preprint

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Bayesian neural networks (BNN) and deep ensembles are principled approaches to estimate the predictive uncertainty of a deep learning model. However their practicality in real-time, industrial-scale applications are limited due to their heavy memory and inference cost. This motivates us to study principled approaches to high-quality uncertainty estimation that require only a single deep neural network (DNN). By formalizing the uncertainty quantification as a minimax learning problem, we first identify input distance awareness, i.e., the model's ability to quantify the distance of a testing example from the training data in the input space, as a necessary condition for a DNN to achieve high-quality (i.e., minimax optimal) uncertainty estimation. We then propose Spectral-normalized Neural Gaussian Process (SNGP), a simple method that improves the distance-awareness ability of modern DNNs, by adding a weight normalization step during training and replacing the output layer. On a suite of vision and language understanding tasks and on modern architectures (Wide-ResNet and BERT), SNGP is competitive with deep ensembles in prediction, calibration and out-of-domain detection, and outperforms the other single-model approaches.

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“…Pettigrew ( 2016b ) favours the quadratic Brier score, though his justification of PI1 considered a class of inaccuracy measures. Different classes of inaccuracy measures have appeared in the literature, often delineated by technical fruitfulness rather than philosophical considerations—e.g., ‘strictly proper’ inaccuracy measures are particularly conducive to proving the required theorems (Landes 2015 ). As yet, we are far from a consensus as to which functions are appropriate as inaccuracy measures.…”

Section: Consequences For Consequentialismmentioning

confidence: 99%

Justifying the principle of indifference

Williamson

2018

Euro Jnl Phil Sci

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This paper presents a new argument for the Principle of Indifference. This argument can be thought of in two ways: as a pragmatic argument, justifying the principle as needing to hold if one is to minimise worst-case expected loss, or as an epistemic argument, justifying the principle as needing to hold in order to minimise worst-case expected inaccuracy. The question arises as to which interpretation is preferable. I show that the epistemic argument contradicts Evidentialism and suggest that the relative plausibility of Evidentialism provides grounds to prefer the pragmatic interpretation. If this is right, it extends to a general preference for pragmatic arguments for the Principle of Indifference, and also to a general preference for pragmatic arguments for other norms of Bayesian epistemology. Keywords Principle of indifference · Bayesianism · Epistemic consequentialism · AccuracyMany Bayesians are committed to some version or other of the Principle of Indifference, which holds that in certain situations one's degrees of belief should be equivocal. Section 1 introduces three such versions in order of increasing strength. In Section 2, I develop a consequentialist argument for the strongest version. This can be thought of as motivating the principle in terms of its pragmatic consequences: if one is to minimise worst-case expected loss, then one should satisfy the Principle of Indifference. As I explain in Section 3, an analogous argument can be constructed

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Probabilism, entropies and strictly proper scoring rules

Cited by 16 publications

References 53 publications

Adaptive decision making via entropy minimization

Adaptive decision making via entropy minimization

Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness

Justifying the principle of indifference

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