Optimal quantization for infinite nonhomogeneous distributions on the real line

Quantization for probability distributions concerns the best approximation of a $d$-dimensional probability distribution $P$ by a discrete probability with a given number $n$ of supporting points. In this paper, an infinitely generated nonhomogeneous Borel probability measure $P$ is considered on $\mathbb R$. For such a probability measure $P$, an induction formula to determine the optimal sets of $n$-means and the $n$th quantization error for every natural number $n$ is given. In addition, using the induction formula we give some results and observations about the optimal sets of $n$-means for all $n\geq 2$.Comment: arXiv admin note: text overlap with arXiv:1512.0037

Induction without Probabilities

A simple indeterministic system is displayed and it is urged that we cannot responsibly infer inductively over it if we presume that the probability calculus is the appropriate logic of induction. The example illustrates the general thesis of a material theory of induction, that the logic appropriate to a particular domain is determined by the facts that prevail there

There are no universal rules for induction

In a material theory of induction, inductive inferences are warranted by facts that prevail locally. This approach, it is urged, is preferable to formal theories of induction in which the good inductive inferences are delineated as those conforming to universal schemas. An inductive inference problem concerning indeterministic, nonprobabilistic systems in physics is posed, and it is argued that Bayesians cannot responsibly analyze it, thereby demonstrating that the probability calculus is not the universal logic of induction. Copyright 2010 by the Philosophy of Science Association.All right reserved

Ultimate Intelligence Part I: Physical Completeness and Objectivity of Induction

We propose that Solomonoff induction is complete in the physical sense via several strong physical arguments. We also argue that Solomonoff induction is fully applicable to quantum mechanics. We show how to choose an objective reference machine for universal induction by defining a physical message complexity and physical message probability, and argue that this choice dissolves some well-known objections to universal induction. We also introduce many more variants of physical message complexity based on energy and action, and discuss the ramifications of our proposals.Comment: Under review at AGI-2015 conference. An early draft was submitted to ALT-2014. This paper is now being split into two papers, one philosophical, and one more technical. We intend that all installments of the paper series will be on the arxi

Probabilities and health risks: a qualitative approach

Health risks, defined in terms of the probability that an individual will suffer a particular type of adverse health event within a given time period, can be understood as referencing either natural entities or complex patterns of belief which incorporate the observer's values and knowledge, the position adopted in the present paper. The subjectivity inherent in judgements about adversity and time frames can be easily recognised, but social scientists have tended to accept uncritically the objectivity of probability. Most commonly in health risk analysis, the term probability refers to rates established by induction, and so requires the definition of a numerator and denominator. Depending upon their specification, many probabilities may be reasonably postulated for the same event, and individuals may change their risks by deciding to seek or avoid information. These apparent absurdities can be understood if probability is conceptualised as the projection of expectation onto the external world. Probabilities based on induction from observed frequencies provide glimpses of the future at the price of acceptance of the simplifying heuristic that statistics derived from aggregate groups can be validly attributed to individuals within them. The paper illustrates four implications of this conceptualisation of probability with qualitative data from a variety of sources, particularly a study of genetic counselling for pregnant women in a U.K. hospital. Firstly, the official selection of a specific probability heuristic reflects organisational constraints and values as well as predictive optimisation. Secondly, professionals and service users must work to maintain the facticity of an established heuristic in the face of alternatives. Thirdly, individuals, both lay and professional, manage probabilistic information in ways which support their strategic objectives. Fourthly, predictively sub-optimum schema, for example the idea of AIDS as a gay plague, may be selected because they match prevailing social value systems

QUANTIZATION FOR A PROBABILITY DISTRIBUTION GENERATED BY AN INFINITE ITERATED FUNCTION SYSTEM

Quantization for probability distributions concerns the best approximation of a d-dimensional probability distribution P by a discrete probability with a given number n of supporting points. In this paper, we have considered a probability measure generated by an infinite iterated function system associated with a probability vector on ℝ. For such a probability measure P, an induction formula to determine the optimal sets of n-means and the nth quantization error for every natural number n is given. In addition, using the induction formula we give some results and observations about the optimal sets of n-means for all n ≥ 2