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Reasons for (prior) belief in Bayesian epistemology

Reasons for (prior) belief in Bayesian epistemology. Christian List ( joint work with Franz Dietrich) http://personal.lse.ac.uk/LIST/ Paper forthcoming in Synthese LSE, November 2012. Introduction. Bayesian epistemology

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Reasons for (prior) belief in Bayesian epistemology

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  1. Reasons for (prior) belief in Bayesian epistemology Christian List (joint work with Franz Dietrich) http://personal.lse.ac.uk/LIST/ Paper forthcoming in Synthese LSE, November 2012

  2. Introduction • Bayesian epistemology • tells us how to move from prior to posterior beliefs in light of new evidence or information, • but says little about where our prior beliefs come from. • It offers few resources to describe some prior beliefs as rational or well-justified, and others as irrational or unreasonable.

  3. Rational choice theory • A different strand of epistemology takes the central epistemological question to be • not how to change one’s beliefs in light of new evidence (though this obviously remains important), • but what reasons justify a given set of beliefs in the first place. • We offer an account of rational belief formation that closes some of the gap between Bayesianism and its reason-based alternative. • We formalize the idea that an agent can have reasons for his or her (prior) beliefs, as distinct from evidence/information in the Bayesian sense. • This is part of a larger programme of research on the role of reasons in rational agency (Dietrich and List 2012a,b).

  4. FIXED/GIVEN CHANGEABLE Prior beliefs Evidence/information Posterior beliefs

  5. FIXED/GIVEN CHANGEABLE Credibility relation Doxastic reasons ALSO CHANGEABLE As before (Prior) beliefs Evidence/information Posterior beliefs

  6. Overview of this talk • Beliefs • Reasons for belief • An example • An axiomatic characterization result • The cardinal case

  7. Overview of this talk • Beliefs • Reasons for belief • An example • An axiomatic characterization result • The cardinal case

  8. The objects of belief • We want to model how an agent forms his or her prior beliefs over some set X of basic objects of beliefs. • The elements of X could be, e.g., possible worlds, states of the world, or rival hypotheses. • We call them epistemic possibilities. • We only assume that the alternatives in X are mutually exclusive and jointly exhaustive of the relevant space of possibilities.

  9. An agent’s beliefs • In Bayesian epistemology, the agent’s beliefs are usually represented by a credence function (subjective probability function) on X, which assigns to each possibility in X a real number between 0 and 1, with a sum-total of 1. • However, we here begin by representing the agent’s beliefs by a credence order ≿ on X (a complete and transitive binary relation). x≿y means that the agent believes x at least as strongly as y. (≻ and  denote the induced strict and indifference relations.)

  10. Beliefs and belief formation • Bayesian epistemology gives an account of how an agent’s beliefs should rationally change in response to evidence or information. • If the agent receives evidence that rules out some possibilities in X, he or she must change the credence order so as to rank any possibilities ruled out below (or weakly below) any possibilities not ruled out, while not changing other rankings (Bayesian updating). • Here, however, we focus on the problem of belief formation: • How does the agent arrive at his or her beliefs over X in the first place, before receiving any evidence?

  11. Beliefs and belief formation • We can look at this problem from both positive and normative perspectives, i.e., we can ask eitherhow an agent actually forms his or her beliefs, orhow he or she ought rationally to do so. • We develop a formal framework that can be used to investigate both questions. This is where reasons come into play.

  12. Overview of this talk • Beliefs • Reasons for belief • An example • An axiomatic characterization result • The cardinal case

  13. Reasons in general • Reasons can be conceptualized in a number of ways. • Scanlon, e.g., defines a reason as “a consideration that counts in favor of some judgment-sensitive attitude [e.g., belief or desire]” (What we owe to each other, p. 67). • We adopt a more general definition, preserving the “counting” part but not the “in favor” part of Scanlon’s definition. • We think of reasons as propositions that play a special role (that somehow “count” or “matter”) in the agent’s attitude formation – in the present context, in his/her belief formation.

  14. Doxastic reasons • A proposition (in general) is a subset of X. • It is true of the possibilities contained in it, and false of all others. • More generally, propositions could be represented by sentences from a suitable language. • We can also think of each proposition as capturing a particular property of the epistemic possibilities. • Now suppose that there is some set of propositions, D, that the agent focuses on in his/her belief formation process; we call these the agent’s doxastic reasons.

  15. Doxastic reasons • When a proposition is in D, • this does not mean that the agent believes it; • it only means that, in forming his/her belief about each epistemic possibility, the agent considers whether or not the proposition is true of that possibility. • So, the propositions in D stand for questions that the agent asks him/herself in the process of belief formation. • We further define D to be the set of all possible such sets D. • D could simply be the set of all possible sets of propositions (or smaller – I set technicalities aside).

  16. The focal doxastic reasons • To indicate that the agent’s credence order ≿ depends on his or her set D, we append the subscript D to the symbol ≿. ≿D= the agent’s credence order when D is focal • A full model of an agent’s beliefs requires the ascription of a family (≿D)D  D of credence orders to the agent, one ≿D for each D  D. • So, how exactly does the credence order ≿D depend on D?

  17. Overview of this talk • Beliefs • Reasons for belief • An example • An axiomatic characterization result • The cardinal case

  18. An example: meeting in DC • Suppose I have agreed to meet Alexandru somewhere in Washington DC at 12 noon tomorrow. • We have not agreed on a place, and we have no way to communicate. • I have no evidence in the standard sense as to where Alexandru is likely to expect me. • Here are some possibilities: • Union Station • Lincoln Memorial • White House • Hilary Clinton’s apartment • How do I form my prior beliefs over where Alexandru might expect me?

  19. An example: meeting in DC • The following are some possible considerations that might be relevant: A : The place in question is where one arrives in Washington ({Union}). F : The place in question is world-famous ({Lincoln, WH}). R : The place in question has restricted access ({WH, Hilary C.’s apt.}). • My credence orderings across variations in D might look like this: D={A,F,R}  Union ≻D Lincoln ≻D WH ≻D Hilary Clinton’s apt. D={A,F}  Union ≻DLincoln DWH ≻DHilary Clinton’s apt. D={A,R}  Union ≻DLincoln ≻D WH D Hilary Clinton’s apt. D={F,R}  Lincoln ≻D WH ≻D Union ≻DHilary Clinton’s apt. D={A}  Union ≻D Lincoln D WH DHilary Clinton’s apt. D={F}  WH DLincoln ≻D Union DHilary Clinton’s apt. D={R}  Union DLincoln ≻D WH DHilary Clinton’s apt.   Union DLincoln D WH DHilary Clinton’s apt.

  20. An example: meeting in DC Can we say something systematic about these beliefs? • They are what we call reason-based.

  21. Reason-based beliefs • The agent’s family of credence orders is reason-based if there exists a binary relation  over sets of reasons (a credibility relation) such that: for any D  D and any x, y  X: The agent believes x more than y when focusing on the reasons in D if and only if the set of reasons in D that are true of x is ranked above the set of reasons in D that are true of y; formally, x ≿Dy {RD : xR}  {RD : yD}.

  22. The example again… • The following are a few possible considerations that might be relevant: A: The place in question is where one arrives in Washington ({Union}). F : The place in question is world-famous ({Lincoln, WH}). R: The place in question has restricted access ({WH, Hilary C.’s apt.}). • My credence orderings across variations in Dmight look like this: D={A,F,R}  Union ≻D Lincoln ≻D WH ≻D Hilary Clinton’s apt. D={A,F}  Union ≻DLincoln DWH ≻DHilary Clinton’s apt. D={A,R}  Union ≻DLincoln ≻D WH D Hilary Clinton’s apt. D={F,R}  Lincoln ≻D WH ≻D Union ≻DHilary Clinton’s apt. D={A}  Union ≻D Lincoln D WH DHilary Clinton’s apt. D={F}  WH DLincoln ≻D Union DHilary Clinton’s apt. D={R}  Union DLincoln ≻D WH DHilary Clinton’s apt.   Union DLincoln D WH DHilary Clinton’s apt. • These beliefs are reason-based, w.r.t. the following credibility relation: {A} > {F} > {F,R} >  > {R}

  23. Overview of this talk • Beliefs • Reasons for belief • An example • An axiomatic characterization result • The cardinal case

  24. Two axioms on the relationship between reasons and beliefs • Axiom 1.‘Principle of insufficient reason.’ For any x, y  X and any D  D, if {R  D : xR} = {R  D : y R}, thenxDy. • Axiom 2. ‘Invariance of relative likelihoods under the addition of irrelevant reasons.’ For any x, y X and any D,D’ D with D  D’, if for all R D’\D, x  R and y  R, thenx ≿Dyx ≿D’y.

  25. The basic representation theorem • The agent’s family of credence orders satisfies Axioms 1 and 2 if and only if (2) it is reason-based. That is, there exists a credibility relation  over sets of reasons such that, for any D  D and x, y  X, x≿Dy {RD : xR}  {RD : yR}.

  26. Overview of this talk • Beliefs • Reasons for belief • An example • An axiomatic characterization result • The cardinal case

  27. Beliefs: the cardinal case • We now go beyond considering credence orders and represent an agent’s beliefs in the more standard way by credence functions (subjective probability functions). • A credence function is a function Pr : X [0,1] such that  xX Pr(x) = 1. (Probabilities of propositions are defined in the usual way.) • We write PrD as the agent’s credence function when D is his or her set of doxastic reasons in relation to the possibilities in X. • We are interested in how PrD depends on D.

  28. Two axioms on the relationship between reasons and beliefs (the cardinal case) • Axiom 1.‘Principle of insufficient reason’ For any x, y  X and any D D, if {RD : R is true of x} = {RD : R is true of y}, thenPrD(x) = PrD(y). • Axiom 2. ‘Invariance of likelihood ratios under the addition of irrelevant reasons’ For any x, y X and any D, D’D with D  D’, if no R in D’\D is true of x or y, thenPrD(x)/PrD(y) = PrD’(x)/PrD’(y).

  29. Theorem 2 • The agent’s family of credence functions PrD across all D  D satisfies Axioms 1 and 2 if and only if • there is a credibility function p from possible combinations of doxastic reasons into the real numbers such that, for all D  D and all x  X, p({RD : R is true of x}) PrD(x) = .  x’X p({RD : R is true of x’})

  30. Remarks • The (prior) probability of a given possibility is proportional to the credibility of the set of doxastic reasons that are true of that possibility. • The factor of proportionality, 1 / x’X p({RD : R is true of x’}), depends on D and ensures that probabilities add up to 1. • Interpretationally, • just as practical reasons are good-making features of actions, • so doxastic reasons are plausible-making features of epistemic possibilities.

  31. FIXED/GIVEN CHANGEABLE Credibility relation Doxastic reasons ALSO CHANGEABLE As before (Prior) beliefs Evidence/information Posterior beliefs

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