Transcript Inductive Bayesian Logic

ILLC 2005
Interfacing Probabilistic and Epistemic Update
Dutch books and epistemic events
Jan-Willem Romeijn
Psychological Methods
University of Amsterdam
Outline
 Updating by conditioning
 Violations of conditioning
 External shocks to the probability
 Meaning shifts in epistemic updates
 A Bayesian model of epistemic updates
 No-representation theorem
 Concluding remarks
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 Updating by conditioning
Updating by conditioning is a consistency constraint for
incorporating new facts in a probability assignment.
probability assignment p
events A, B, C, ...
conditioning on events
probabilistic conclusions p(
 | ABC...)
If probability theory is seen as a logic, updating functions
like a deductive inference rule.
-3-

Muddy Venn diagrams
Conditioning on the fact that A is like zooming in on the
probability assignment p within the set of possible worlds A.
A
p
p(  | A)
A
Probability is represented by the size of rectangulars. Apart
from normalising the probability of A, no changes are
induced by the update operation.
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 Violating conditioning
Bayesian conditioning is violated if, in the course of the
update, we also change the probabilities within A.
p(  | A )
A
B
B
B
B
pA(  )
B
B
The updated probability is pA(B) < p(B|A). This difference
makes vulnerable for a Dutch book.
-5-

Rational violations?
In particular cases, violations of conditioning may seem
rational.
 Violations of the likelihood principle in classical
statistics, model selection problems.
 Epistemic updates: incorporating facts about
knowledge states.
Can we make sense of such violations from within a
Bayesian perspective?
-6-

Possible resolution
Violations are understandable if they result from changes
in meaning. On learning A we may reinterpret B as B'.
p( B | A )
B
B
p ( B' | A )
 B'
B'
p( B | A' )
?
B
B
Can we represent such a meaning shift as Bayesian
update, saying that we actually learned A' ?
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 Probability shocks
Violations of conditioning can be understood as an
external shock to the probability assignment p.
A
B
p= 1/4
p= 1/4
B
p'= 3/8
p'= 3/8
B
p= 1/4
p= 1/4
A
B
p'= 1/8
p'= 1/8
The events are associated with the same possible
worlds, denoted •, but these worlds are assigned
probabilities p', according to a new constraint .
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
Restricting the shock
External shocks to the probability assignment may be
governed by further formal criteria, such as minimal
distance  between p and p'.


p

p'
Such criteria may be conservative, but they are not
consistent.
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
Choosing premises
From a logical point of view, the update procedure comes
down to choosing new premises.
premise p
premise p'
events A, B, C, ...
events A, B, C, ...
conclusion p(
 | ABC...)
conclusion p'(
 | ABC...)
This is the extra-logical domain of objective Bayesianism:
formally constrained prior probabilities.
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 Meaning shifts
The update operation can also be seen as a change to the
semantics: p (B' | A) < p (B | A).
A
B
p= 1/4
p= 1/4
 B'
p= 1/4
p= 1/4
B
p= 1/4
p= 1/4
B'
p= 1/4
p= 1/4
A
The probabilities of possible worlds remain the same, but
the update induces an implicit change of the facts involved.
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
Epistemic updates
Consider two research groups, 1 and 2, that try to discover
which of A, B, or C holds:
 D1
D1
A
p= 1/3
B
p= 1/3
 D2
C
p= 1/3
D2
The groups use different methods, delivering doubt or
certainty in differing sets of possible worlds.
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
Conditional probability
According to the standard definition of conditional probability,
we have p( D2 | D1) = 1/2:
 D1
D1
A
p= 1/3
 D2
B
p= 1/3
C
p= 1/3
D2
D1
A
p= 1/2
 D2
But is this also the appropriate updated probability?
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B
p= 1/2
D2

Updated probability
It seems that after an update with D1, the second research
group has very little to doubt about:
D1
A
p= 1/2
 D2
D1
A
p= 1/2
B
p= 1/2
B
p= 1/2
 D'2
D2
Updating induces a meaning shift D2  D'2 , and the correct
updated probability is p ( D'2 | D1) = 0.
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 Epistemic events
The meaning shift D2  D'2 can be understood by
including epistemic states into the semantics.
C
D1
A
B
C

B
B
2
A
D2
A
B
1
C
The diagram shows the accessible epistemic states in
the world-state B.
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
External states
After learning that D1, we may exclude world-state C
from the state space.
C
C
B
B
C
2
2
B
A
A
A
B
1
C
C
B
A
A
W
A
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B
1
C
W

Epistemic update
But a full update also comprises conditioning on the
accessible epistemic states of both research groups.
C
C
B
B
2
2
B
A
A
A
B
1
B
A
A
C
A
B
1
C
This latter step brings about the event change D2  D'2.
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
Bayesian conditioning
There is no violation of conditioning in the example.
D1
D'1
C
C
?
B
B
C
2
2
B
A
A
A
B
1
C
C
B
A
A
W
A
B
1
C
W
It is simply unclear which event we are supposed to update
with upon learning that group 1 is in doubt: D1 or D'1.
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 Choosing semantics
Many puzzles on the applicability of Bayesian updating
can be dealt with by making explicit the exact events
we update upon.
A
B
p= 1/4
p= 1/4
B
p= 1/4
p= 1/4
?
B
p= 1/4
p= 1/4
B
p= 1/4
p= 1/4
We must choose the semantics so as to include all
these events. Is that always possible?
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A'

Judy Benjamin updates
In updating a probability p to p by distance minimisation
under a partition of constraints , we may have
p  

p ( B)  p( B)
p  

p ( B)  p( B)
for some B and all . Now suppose that we can
associate the constraints with a partition of events G:
p (  )  p(  | G ),
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 p(G ) d  1.

No-representation theorem
In Bayesian conditioning on events A from a partition, the
prior is always a convex combination of the posteriors:
p( B)   p( A ) p( B | A ) d .
But because p(B|G) > p(B) for all but one , we have
p( B)   p(G ) p( B | G ) d .
It thus seems that there is no set of events G that can
mimic distance minimisation on the constraints .
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 In closing
Some considerations for further research:
• There is a large gap between the epistemic puzzles
and cases like model selection.
• It is unclear what kind of event is behind violations
of the likelihood principle, as in the stopping rule.
• Probabilistic consistency may not be the only virtue
if we object to a principled distinction between
epistemology and logic.
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