Belief and Desire: On Information and its Value
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1 Belief and Desire: On Information and its Value Ariel Caticha Department of Physics University at Albany SUNY Info-Metrics Institute 04/26/2013 1
2 Part 1: Belief 2
3 What is information? a) Epistemic: What is conveyed by an informative answer. Everyday usage. Concerned with meaning. b) Probabilistic: Shannon information. Communication theory, Physics, Econometrics... Concerned with amounts... not with meaning. c) Algorithmic: Kolmogorov complexity. Computer science, complexity... Concerned with amount not meaning (arguable...). 3
4 The goal: the prior q(x) the posterior p(x) To update from old beliefs to new beliefs when new information becomes available.?? We seek an epistemic concept of information defined directly in terms of its effects on the beliefs of rational agents. 4
5 An analogy from physics: initial state of motion. final state of motion. Force Force is whatever induces a change of motion: F dp dt 5
6 Inference is dynamics too! old beliefs new beliefs QM! information Information is what induces the change in rational beliefs. 6
7 What is information? Information is what induces the change in rational beliefs. Information is what constrains rational beliefs. Mathematical expression: information = constraints on probabilities Epistemic information does not measure amount of information. 7
8 Question: Answer: How do we select a distribution from among all those that satisfy the constraints? Rank the distributions according to preference. The design of Entropic Inference: Transitive ranking: To each p associate an S[p,q]. Entropies are real functionals designed to be maximized. Select the posterior that maximizes an entropy S[p,q] subject to the available constraints. 8
9 Design Specifications Universality To be useful the method must be of general applicability. We focus on what all inferences have in common. Minimal Updating Parsimony: previous information is valuable. Objectivity: the DS prescribe what not to update. 9
10 DS1: Locality Local information has local effects. If the information does not refer to a domain D, then q(x D) is not updated, p( x D) q( x D). (Bayes rule as a special case.) 10
11 DS2: Coordinate invariance Coordinates carry no information. DS3: Independence Not everything matters. When two systems are believed to be independent and we get information about one, it should not matter whether the other is included in the analysis or not. 11
12 Conclusion: The ranking of universal applicability that implements minimal updating is given by relative entropy, S[ p, q] dx p( x)log p( x). q( x) Other entropies may be useful for other purposes. 12
13 Entropic Inference: Summary prior posterior constraint q p S[ p, q] dx p( x)log p( x) q( x) Pr( dv ) e S [ p, q] dv Includes Bayes, MaxEnt and Large deviations as special cases. 13
14 Part 2: Desire Question: Answer: Why collect information? Why bother? Better beliefs lead to better rational decisions. 14
15 Decision Theory Some states of the world are more desirable than others. To each x associate a u(x). Desirability/value is described by a utility function. Different actions lead to different x s. Rational decision: Choose the action that maximizes utility. 15
16 Complication: The consequences of our actions are uncertain. The uncertainty is described by a probability q( x ). Then the expected utility is ( ) dx q( x ) u( x). The rational decision is: Choose the action that maximizes expected utility, max U q q U q ( ). 16
17 Why should we update? Why bother? If we receive information we can update, q( x ) p( x ) U p ( ) dx p( x ) u( x) The new expected utility is. Now we can make a better decision ( ). max p U p 17
18 The value of information Value is measured by utility. U U q U p U U p ( p) U p( q) q p U dx p( x ) p( x ) u( x). p q 18
19 Conclusions Information is what information does: It affects your beliefs. Information has value: It leads to better decisions. 19
20 Conclusions Epistemic information is the constraints. Minimal updating: previous information is valuable. The tool for updating is (relative) Entropy. Entropy needs no interpretation, and no amount either. MaxEnt, Bayes and Large Deviations are special cases. The value of information is the change in expected utility. 20
21 Thank you! 21
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