Conditional Probability in the Light of Qualitative Belief Change. David Makinson LSE Pontignano November 2009

Transcription

1 Conditional Probability in the Light of Qualitative Belief Change David Makinson LSE Pontignano November

2 Contents Explore ways in which qualitative belief change in AGM tradition throws light on options in the treatment of conditional probability. Why we sometimes need to go beyond ratio rule defining conditional from one-place probability. Clarify criteria for choosing between the various axiomatizations for the two-place functions. Suggest novel forms of conditional probability. 2

3 Why Go Beyond the Ratio Rule? Kolmogorov s postulates for one-place probability functions are simple, natural easy to work with Ratio definition of conditional probability is convenient They have become standard So why go beyond them? 3

4 Background One-Place Probability Functions Proper Kolmogorov functions p: L [0,1] such that: p(x) p(y) whenever x l- y p(x y) = p(x)+p(y) whenever x l- y p(x) = 1 for some formula x Improper one The unit function: p(x) = 1 for all formulae x 4

5 Background Ratio Definition Ratio Definition Given 1-place p( ), define 2-place p(, ) p(x a) = p(a x) / p(a) when p(a) >0, otherwise undefined Thus partial function, on Lx{a L: p(a) > 0} Two Problems Should we extend 2nd argument to whole of L? If so, how best to do it? 5

6 First Problem: Should We Extend? Metaphysical Reasons de Finetti, Rényi, Hájek, De Finetti 1974: Every evaluation of probability is conditional; not only on the mentality or psychology of the individual involved, at the time in question, but also, and especially, on the state of information in which he finds himself at that moment. 6

7 Not Very Convincing Substantive: infinite regress, unless you take the universe as your domain Historical: compare with old view universal quantification really conditional, should be over the universe Methodological: Even if conditional, should perhaps not be put in the theory itself. If conditions held fixed, leave them to stage of application. 7

8 Pragmatic Reason Limited expressive capacity Even when p(a) = 0, might like to consider what happens under the supposition that a Example of Borel: p(western hemisphere, equator) 8

9 Critical Zone Zone of difficulty for the second argument a (condition) of a 2-place prob. function p(x,a) {a L: a consistent but p(a) = 0}, {a L: a consistent but p(a,t) = 0} (resp.1-plac /2-place primitive) OK above critical zone: p(a) > 0 - use ratio rule OK below: a inconsistent put p(x,a)= 1 (left projection from a is unit function) 9

10 Degenerate Completions Traditional In critical zone (a consistent but p(a,t) = 0), again put p(x,a)= 1 Left projection from critical a is unit function Carnap Abolish critical zone Axiom: critical zone is empty: when p(a,t) = 0 then a is inconsistent 10

11 Work around the Problem (Rényi) Define 2-place from 1-place functions When p(a) > 0, use ratio rule, when a inconsistent,put p(x,a) = 1 In critical zone, take p(x,a) to be limit of values of p(x a ) for a suitable infinite sequence of noncritical approximations a to a Works well in some cases (eg Borel example) But only when in structure based on reals satisfying suitable conditions 11

12 A Leaf from the AGM Book Compare with situation in qualitative belief change A theory: set of propositions close under classical consequence Expand theory K: add, close: K+a = Cn(K {a}) When a inconsistent, gives blow-out When a in critical zone (consistent but inconsistent with K) also gives blow-out Same problem - how is it dealt with? 12

13 Qualitative Revision When a is in critical zone (consistent but inconsistent with K), first contract a from K, then add a and close under consequence K a = Cn((K a) {a}) where is suitable contraction operation forms a subset of K that is consistent with a (when a itself is consistent) and satisfying certain regularity conditions (AGM postulates) 13

14 Different Behaviour of Expansion and Revision Belief preservation: K K+a Expansion yes, revision no Blow-out: when a in critical zone (consistent but inconsistent with K) then K+a =L Expansion yes, revision no Consistency (of input) preservation: when a is consistent, so is K a Revision yes, expansion no 14

15 Similarly for Conditional Probability (ratio/unit like expansion) Belief preservation: p(x,a) = 1 when p(a,t) = 1 Ratio/unit yes, revisionary should not Blow-out: when a in critical zone (a consistent but p(a,t) = 0) then p(x,a) = 1 for all x Ratio/unit yes, revisionary should not Consistency (of input) preservation: when a is in critical zone, then p(x,a) 1 for some x (eg falsum) Ratio/unit no, revisionary should (?) 15

16 But More than one Kind of Essentially Two main candidates Conditional probability Hosiasson-Lindenbaum 1940 vs Popper 1959 (Rényi 1955 in effect gives a scheme with a parameter, which can be instantiated to either) What s the difference? Can be confusing Qualitative belief revision helps understand what is involved 16

17 Background: Modular Presentation Common postulates for p: L2 [0,1] (van Fraassen 1976, 1995) (vf1) p(x,a) = p(x,b) whenever a b (vf2) p a is a 1-place Kolmogorov fn with p a (a) = 1 (vf3) p(x y,a) = p(x,a) p(y,a x) for all a,x,y (vf2) does not tell us when p a is proper or improper (unit). Axioms consistent with every p a is unit function 17

18 van Fraassen Axioms Imply Write for {a: p a is unit function}. Then is Non-empty When a inconsistent then a Closed downwards When a, b - a then b Closed under disjunction When a,b then a b In summary: is non-empty ideal 18

19 Above the Critical Zone (RP) When p(a,t) > 0, then p a is a proper oneplace Kolmogorov function This gives us Popper system 1959 In it, we know what happens above critical zone (proper) and below (unit) but not within Could all be proper; could all be unit; could have some proper, some unit, depending on choice of a in critical zone 19

20 The Two Degenerate Systems Add to Popper respectively: (Carnap) Critical zone empty,i.e. when a is consistent then p(a,t) > 0 (Unit) When a is consistent but p(a,t) = 0, then p a is the unit function 20

21 Hosiasson-Lindenbaum 1940 Add to the van Fraassen axioms (or equivalently to Popper) (HL) When a is consistent but p(a,t) = 0, then p a is a proper one-place Kolmogorov function 21

22 Conceptual Differences Hosiasson-Lindenbaum radically revisionary Satisfies counterpart of consistency preservation i.e. for every consistent a, p a is a proper Kolmogorov function (no blow-out) Popper partly revisionary, partly expansionary Covers all the HL functions Also the Unit functions Many mixed functions 22

23 Hasse Diagram (mostly Leblanc, Roeper 1989, 1999) vf = Popper {1(, )} Popper Unit HL Unit HL Carnap = Unit HL 23

24 Choosing between Systems Not question of right system and wrong ones Rather: How revisionary do we want to be, in a given context? Completely revisionary: HL Leave degree of revision/expansion to the user: use Popper Not at all revisionary: use Unit system or ratio/unit definition) Different applications, different choices 24

25 Choice of Consequence Operation In a sense, the different systems do not reflect different conceptions of probability, but different choices of the underlying consequence relation = {a L: p a is unit function} is non-empty ideal = {a L: p a is unit function} is non-empty filter Determines a supraclassical consequence operation Cn (A) = Cn(A ) 25

26 Translation Theorem Any Popper function (modulo classical Cn) is itself a Hosiasson-Lindenbaum function modulo a suitably defined supraclassical consequence operation Cn, with certain elements of the critical zone becoming Cn -inconsistent (In field of sets mode, take quotient structure via the ideal, same result) 26

27 Screened Belief Revision Existing work on qualitative belief revision can lead us to further perspectives and ideas for conditional probability Screened revision (Theoria 1997) Basic idea: Two steps: a pre-processing step, possibly followed by an AGM revision 27

28 The Two Step Dance The pre-processor decides whether to revise Check whether proposed input is consistent with a central part of the belief set under consideration, i.e. a privileged subset If answer negative, theory remains unchanged. If it is positive, we apply an AGM revision in a manner that protects the privileged material 28

29 Properties of Screened Revision AGM Success postulate fails Can have a K a When a inconsistent K a = K Other changes also needed see Theoria

30 Probabilistic Analogue? Idea Conditionalize when appropriate, otherwise do nothing Like RP or HL except For values of a where left projection p a would be the unit function, we now put p a = p T 30

31 Properties? Stronger p a always proper Kolmogorov function Weaker Analogue of success is p(a,a) = 1 Fails: when we do nothing: p(a,a) = p a (a) = p T (a), which can be 1. Also forces restriction of product axiom Adequate axiomatization? Open question 31

32 Light in Reverse Direction We have been using AGM belief revision to help throw light on conditional probability. Also insight looking in other direction Natural map from two-place functions satisfying Hosiasson-Lindenbaum (HL) postulates into the family of AGM belief revision operations modulo classical consequence In fact onto. Not injective (essentially because distinct 2-place prob fns can have same top Omit details 32

33 General Moral of the Story Conceptual links between AGM qualitative belief revision Conditional probability Qualitative belief revision helps see conceptual options in probabilistic context, arising in critical zone Also maps in reverse direction Still much to explore, for both belief revision and probability 33