Related papers: Solving the Problem of Induction
The Turing machine, as it was presented by Turing himself, models the calculations done by a person. This means that we can compute whatever any Turing machine can compute, and therefore we are Turing complete. The question addressed here…
The subset sum problem is known to be an NP-hard problem in the field of computer science with the fastest known approach having a run-time complexity of $O(2^{0.3113n})$. A modified version of this problem is known as the perfect sum…
The paper investigates the role of data, hypotheses and mathematical methods that can be used in the discovery of a law y=fo(u), relating variables u and y of a physical phenomenon, making use of experimental measurements of such variables.…
Suppose you are told that taking a statin will reduce your risk of a heart attack or stroke by 3% in the next ten years, or that women have better emotional intelligence than men. You may wonder how accurate the 3% is, or how confident we…
We characterize a notion of confidence that arises in learning or updating beliefs: the amount of trust one has in incoming information and its impact on the belief state. This learner's confidence can be used alongside (and is easily…
We introduce a framework uniting algorithmic randomness with exchangeable credences to address foundational questions in philosophy of probability and philosophy of science. To demonstrate its power, we show how one might use the framework…
In this work we introduce a novel approach, based on sampling, for finding assignments that are likely to be solutions to stochastic constraint satisfaction problems and constraint optimisation problems. Our approach reduces the size of the…
In the footsteps of the book \textit{Measure Theory and Integration By and For the Learner} of our series in Probability Theory and Statistics, we intended to devote a special volume of the very probabilistic aspects of the first cited…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
We address the multiple testing problem under the assumption that the true/false hypotheses are driven by a Hidden Markov Model (HMM), which is recognized as a fundamental setting to model multiple testing under dependence since the seminal…
Mathematical proof aims to deliver confident conclusions, but a very similar process of deduction can be used to make uncertain estimates that are open to revision. A key ingredient in such reasoning is the use of a "default" estimate of…
(l) I have enough evidence to render the sentence S probable. (la) So, relative to what I know, it is rational of me to believe S. (2) Now that I have more evidence, S may no longer be probable. (2a) So now, relative to what I know, it is…
When testing multiple hypothesis in a survey --e.g. many different source locations, template waveforms, and so on-- the final result consists in a set of confidence intervals, each one at a desired confidence level. But the probability…
Belief functions are a powerful and popular framework for the mathematical characterisation of uncertainty, in particular in situations in which lack of data renders learning a probability distribution for the problem impractical. The first…
This paper is a review of a particular approach to the method of maximum entropy as a general framework for inference. The discussion emphasizes the pragmatic elements in the derivation. An epistemic notion of information is defined in…
According to the probability ranking principle, the document set with the highest values of probability of relevance optimizes information retrieval effectiveness given the probabilities are estimated as accurately as possible. The key…
Instruction subsets are heuristics that can reduce the size of the inductive programming search space by tens of orders of magnitude. Comprising many overlapping subsets of different sizes, they serve as predictions of the instructions…
An impossibility theorem demonstrates that a particular problem or set of problems cannot be solved as described in the claim. Such theorems put limits on what is possible to do concerning artificial intelligence, especially the…
Inductive reasoning enables humans to infer abstract rules from limited examples and apply them to novel situations. In this work, we compare an LLM-based hypothesis search framework with direct program generation approaches on few-shot…
Here we deconstruct, and then in a reasoned way reconstruct, the concept of "entropy of a system," paying particular attention to where the randomness may be coming from. We start with the core concept of entropy as a COUNT associated with…