Conformal prediction is a model-free machine learning method for creating prediction regions with a guaranteed coverage probability level. However, a data scientist often faces three challenges in practice: (i) the determination of a conformal prediction region is only approximate, jeopardizing the finite-sample validity of prediction, (ii) the computation required could be prohibitively expensive, and (iii) the shape of a conformal prediction region is hard to control. This article offers new insights into the relationship among the monotonicity of the non-conformity measure, the monotonicity of the plausibility function, and the exact determination of a conformal prediction region. Based on these new insights, we propose a simple strategy to alleviate the three challenges simultaneously.

The parameters of a machine learning model are typically learned by minimizing a loss function on a set of training data. However, this can come with the risk of overtraining; in order for the model to generalize well, it is of great importance that we are able to find the optimal parameter for the model on the entire population -- not only on the given training sample. In this paper, we construct valid confidence sets for this optimal parameter of a machine learning model, which can be generated using only the training data without any knowledge of the population. We then show that studying the distribution of this confidence set allows us to assign a notion of confidence to arbitrary regions of the parameter space, and we demonstrate that this distribution can be well-approximated using bootstrapping techniques.

A fundamental problem in statistics and machine learning is that of using observed data to predict future observations. This is particularly challenging for model-based approaches because often the goal is to carry out this prediction with no or minimal model assumptions. For example, the inferential model (IM) approach is attractive because it has certain validity guarantees, but requires specification of a parametric model. Here we show that a new perspective on a recently developed generalized IM approach can be applied to construct an IM for prediction that satisfies the desirable validity guarantees without specification of a model. One important special case of this approach corresponds to the powerful conformal prediction framework and, consequently, the desirable properties of conformal prediction follow immediately from the general IM validity theory. Several numerical examples are presented to illustrate the theory and highlight the method's performance and flexibility.

In this paper, we generalize the basic notions and results of Dempster-Shafer theory from predicates to formal concepts. Results include the representation of conceptual belief functions as inner measures of suitable probability functions, and a Dempster-Shafer rule of combination on belief functions on formal concepts.

We propose a new model for forming beliefs and learning about unknown probabilities (such as the probability of picking a red marble from a bag with an unknown distribution of coloured marbles). The most widespread model for such situations of 'radical uncertainty' is in terms of imprecise probabilities, i.e. representing the agent's knowledge as a set of probability measures. We add to this model a plausibility map, associating to each measure a plausibility number, as a way to go beyond what is known with certainty and represent the agent's beliefs about probability. There are a number of standard examples: Shannon Entropy, Centre of Mass etc. We then consider learning of two types of information: (1) learning by repeated sampling from the unknown distribution (e.g. picking marbles from the bag); and (2) learning higher-order information about the distribution (in the shape of linear inequalities, e.g. we are told there are more red marbles than green marbles). The first changes only the plausibility map (via a 'plausibilistic' version of Bayes' Rule), but leaves the given set of measures unchanged; the second shrinks the set of measures, without changing their plausibility. Beliefs are defined as in Belief Revision Theory, in terms of truth in the most plausible worlds. But our belief change does not comply with standard AGM axioms, since the revision induced by (1) is of a non-AGM type. This is essential, as it allows our agents to learn the true probability: we prove that the beliefs obtained by repeated sampling converge almost surely to the correct belief (in the true probability). We end by sketching the contours of a dynamic doxastic logic for statistical learning.

We consider the problem of rational uncertainty about unproven mathematical statements, remarked on by G\"odel and others. Using Bayesian-inspired arguments we build a normative model of fair bets under deductive uncertainty which draws from both probability and the theory of algorithms. We comment on connections to Zeilberger's notion of "semi-rigorous proofs", particularly that inherent subjectivity would be present. We also discuss a financial view with models of arbitrage where traders have limited computational resources.

We study relations between evidence theory and S-approximation spaces. Both theories have their roots in the analysis of Dempster's multivalued mappings and lower and upper probabilities and have close relations to rough sets. We show that an S-approximation space, satisfying a monotonicity condition, can induce a natural belief structure which is a fundamental block in evidence theory. We also demonstrate that one can induce a natural belief structure on one set, given a belief structure on another set if those sets are related by a partial monotone S-approximation space.

We consider the iterated belief change that occurs following an alternating sequence of actions and observations. At each instant, an agent has beliefs about the actions that have occurred as well as beliefs about the resulting state of the world. We represent such problems by a sequence of ranking functions, so an agent assigns a quantitative plausibility value to every action and every state at each point in time. The resulting formalism is able to represent fallible belief, erroneous perception, exogenous actions, and failed actions. We illustrate that our framework is a generalization of several existing approaches to belief change, and it appropriately captures the non-elementary interaction between belief update and belief revision.

We define a new notion of conditional belief, which plays the same role for Dempster-Shafer belief functions as conditional probability does for probability functions. Our definition is different from the standard definition given by Dempster, and avoids many of the well-known problems of that definition. Just as the conditional probability Pr (lB) is a probability function which is the result of conditioning on B being true, so too our conditional belief function Bel (lB) is a belief function which is the result of conditioning on B being true. We define the conditional belief as the lower envelope (that is, the inf) of a family of conditional probability functions, and provide a closed form expression for it. An alternate way of understanding our definition of conditional belief is provided by considering ideas from an earlier paper [Fagin and Halpern, 1989], where we connect belief functions with inner measures. In particular, we show here how to extend the definition of conditional probability to non measurable sets, in order to get notions of inner and outer conditional probabilities, which can be viewed as best approximations to the true conditional probability, given our lack of information. Our definition of conditional belief turns out to be an exact analogue of our definition of inner conditional probability.

The fundamental updating process in the transferable belief model is related to the concept of specialization and can be described by a specialization matrix. The degree of belief in the truth of a proposition is a degree of justified support. The Principle of Minimal Commitment implies that one should never give more support to the truth of a proposition than justified. We show that Dempster's rule of conditioning corresponds essentially to the least committed specialization, and that Dempster's rule of combination results essentially from commutativity requirements. The concept of generalization, dual to thc concept of specialization, is described.