355,979 research outputs found
Probabilities on Sentences in an Expressive Logic
Automated reasoning about uncertain knowledge has many applications. One
difficulty when developing such systems is the lack of a completely
satisfactory integration of logic and probability. We address this problem
directly. Expressive languages like higher-order logic are ideally suited for
representing and reasoning about structured knowledge. Uncertain knowledge can
be modeled by using graded probabilities rather than binary truth-values. The
main technical problem studied in this paper is the following: Given a set of
sentences, each having some probability of being true, what probability should
be ascribed to other (query) sentences? A natural wish-list, among others, is
that the probability distribution (i) is consistent with the knowledge base,
(ii) allows for a consistent inference procedure and in particular (iii)
reduces to deductive logic in the limit of probabilities being 0 and 1, (iv)
allows (Bayesian) inductive reasoning and (v) learning in the limit and in
particular (vi) allows confirmation of universally quantified
hypotheses/sentences. We translate this wish-list into technical requirements
for a prior probability and show that probabilities satisfying all our criteria
exist. We also give explicit constructions and several general
characterizations of probabilities that satisfy some or all of the criteria and
various (counter) examples. We also derive necessary and sufficient conditions
for extending beliefs about finitely many sentences to suitable probabilities
over all sentences, and in particular least dogmatic or least biased ones. We
conclude with a brief outlook on how the developed theory might be used and
approximated in autonomous reasoning agents. Our theory is a step towards a
globally consistent and empirically satisfactory unification of probability and
logic.Comment: 52 LaTeX pages, 64 definiton/theorems/etc, presented at conference
Progic 2011 in New Yor
A Puzzle about Knowing Conditionals
We present a puzzle about knowledge, probability and conditionals. We show that in certain cases some basic and plausible principles governing our reasoning come into conflict. In particular, we show that there is a simple argument that a person may be in a position to know a conditional the consequent of which has a low probability conditional on its antecedent, contra Adams’ Thesis. We suggest that the puzzle motivates a very strong restriction on the inference of a conditional from a disjunction
Sleeping Beauty Reconsidered: Conditioning and Reflection in Asynchronous Systems
A careful analysis of conditioning in the Sleeping Beauty problem is done,
using the formal model for reasoning about knowledge and probability developed
by Halpern and Tuttle. While the Sleeping Beauty problem has been viewed as
revealing problems with conditioning in the presence of imperfect recall, the
analysis done here reveals that the problems are not so much due to imperfect
recall as to asynchrony. The implications of this analysis for van Fraassen's
Reflection Principle and Savage's Sure-Thing Principle are considered.Comment: A preliminary version of this paper appears in Principles of
Knowledge Representation and Reasoning: Proceedings of the Ninth
International Conference (KR 2004). This version will appear in Oxford
Studies in Epistemolog
Joint Probability Trees
We introduce Joint Probability Trees (JPT), a novel approach that makes
learning of and reasoning about joint probability distributions tractable for
practical applications. JPTs support both symbolic and subsymbolic variables in
a single hybrid model, and they do not rely on prior knowledge about variable
dependencies or families of distributions. JPT representations build on tree
structures that partition the problem space into relevant subregions that are
elicited from the training data instead of postulating a rigid dependency model
prior to learning. Learning and reasoning scale linearly in JPTs, and the tree
structure allows white-box reasoning about any posterior probability ,
such that interpretable explanations can be provided for any inference result.
Our experiments showcase the practical applicability of JPTs in
high-dimensional heterogeneous probability spaces with millions of training
samples, making it a promising alternative to classic probabilistic graphical
models
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