Linear logic in normed cones: probabilistic coherence spaces and beyond

Ehrhard et al. (2018. Proceedings of the ACM on Programming Languages, POPL 2, Article 59.) proposed a model of probabilistic functional programming in a category of normed positive cones and stable measurable cone maps, which can be seen as a coordinate-free generalization of probabilistic coherence spaces (PCSs). However, unlike the case of PCSs, it remained unclear if the model could be refined to a model of classical linear logic. In this work, we consider a somewhat similar category which gives indeed a coordinate-free model of full propositional linear logic with nondegenerate interpretation of additives and sound interpretation of exponentials. Objects are dual pairs of normed cones satisfying certain specific completeness properties, such as existence of norm-bounded monotone weak limits, and morphisms are bounded (adjointable) positive maps. Norms allow us a distinct interpretation of dual additive connectives as product and coproduct. Exponential connectives are modeled using real analytic functions and distributions that have representations as power series with positive coefficients. Unlike the familiar case of PCSs, there is no reference or need for a preferred basis; in this sense the model is invariant. PCSs form a full subcategory, whose objects, seen as posets, are lattices. Thus, we get a model fitting in the tradition of interpreting linear logic in a linear algebraic setting, which arguably is free from the drawbacks of its predecessors.

Learning the probability of immediate interactional cues: Children’s acquisition of the discourse marker un in Japanese

How do children learn to use discourse markers in conversational interactions? This study focused on a Japanese discourse marker un, typically used as a positive response for yes-no questions and as a backchannel, and tested our prediction that children first learn to use un to respond to questions and then use it as backchannels after interlocutors signal the continuation of their discourse. To this end, we built generalised linear models on the longitudinal conversation data from seven children aged between 1 and 5 years and their caregivers. Our model revealed that children not only increase the general probability of un to reach adults’ rates, but also learn to use un in response to yes-no questions as we predicted. Children also tend to produce un as a backchannel after the interlocutor’s final modal particle ne, which is typically used to set a common ground. Our results show that children gradually learn different interactional contexts for the use of un from local probabilistic coherence between turns in conversations.

Improving probabilistic infectious disease forecasting through coherence

With an estimated $10.4 billion in medical costs and 31.4 million outpatient visits each year, influenza poses a serious burden of disease in the United States. To provide insights and advance warning into the spread of influenza, the U.S. Centers for Disease Control and Prevention (CDC) runs a challenge for forecasting weighted influenza-like illness (wILI) at the national and regional level. Many models produce independent forecasts for each geographical unit, ignoring the constraint that the national wILI is a weighted sum of regional wILI, where the weights correspond to the population size of the region. We propose a novel algorithm that transforms a set of independent forecast distributions to obey this constraint, which we refer to as probabilistically coherent. Enforcing probabilistic coherence led to an increase in forecast skill for 79% of the models we tested over multiple flu seasons, highlighting the importance of respecting the forecasting system’s geographical hierarchy.

One example of epistemological idealization

In different branches of science we find idealizations. In physics we find frictionless surfaces, point-particles of molecules that don?t exert force on one another, in game theory we find fully rational agents who are aware of all of their preferences, and in biology we find infinite populations which do not migrate, and which do not suffer any sort of evolutionary pressure. It seems to us that similar idealizations can be found in epistemology. To be more specific, we mean the models of probabilistic coherentism. Exactly like scientific idealizations which disregard certain very real factors, because their goal is to shed light on the connections between important scientific concepts, this model of probabilistic coherence idealizes concepts of reliability of information sources and independence to show the connection between coherence and reliability.

Why Approximate Coherence?

Chapter 5 continues to answer the question of how Bayesians can justify the claim that approximating probabilistic coherence is beneficial for non-ideal thinkers. Another popular argument for why coherence is rationally required is the accuracy dominance argument for probabilism. If we use an appropriate measure of distance to coherence, reducing incoherence leads to improved accuracy in every possible world. We can show, moreover, that for any incoherent credence function, it is always possible to measure distance from coherence in such a way that there is a series of less incoherent credence functions that are both more accurate in every possible world and less Dutch book-vulnerable.

Why Approximate Coherence?

Chapter 4 begins to answer the question of how Bayesians can justify the claim that approximating probabilistic coherence is beneficial for non-ideal thinkers. Dutch book arguments are often put forth to argue that ideal rationality requires being coherent. I show that we can justify that it is better to be less incoherent by showing that decreased incoherence is associated with decreased losses from Dutch books. While incoherent thinkers can never be immune from Dutch book losses, the amount they stand to lose, given that we standardize bet sizes, is greater the more incoherent their credences are.

One example of formal coherence

The aim of this paper is to present one case of probabilistic formalization of our intuitive notion of coherence. To that end, we will have to provide answers for the questions, what are all relevant relations between beliefs, as far as coherence is concerned, and of course, what is intuitive coherence. After we settle those questions, we will try to show how, by applying certain probabilsitic theories of confirmation to those relations, we can arrive at a basic probabilistic theory of coherence. We will point out certain problems of that theory. At the end of the paper, we will sum up the differences between intuitive and probabilistic coherence, and we will try to provide reasons why the successful formalization of this relation, should be a desired result in epistemology.