Dutch books and epistemic events

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Dutch books and epistemic events
ILLC 2005 Interfacing Probabilistic and Epistemic
Update

• Jan-Willem Romeijn
• Psychological Methods
• University of Amsterdam

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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.
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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
pA( ? )
? B
B
? B
B
? B
B
The updated probability is pA(B) lt p(BA). This
difference makes vulnerable for a Dutch book.
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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?

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Possible resolution
?
Violations are understandable if they result from
changes in meaning. On learning A we may
reinterpret B as B'.
p ( B' A )
p( B A' )
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
A
? B
? B
B
B
p 1/4
p 1/4
p' 3/8
p' 1/8
p 1/4
p 1/4
p' 3/8
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) lt p (B A).
A
A
? B'
? B
B'
B
p 1/4
p 1/4
p 1/4
p 1/4
p 1/4
p 1/4
p 1/4
p 1/4
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
B
C
p 1/3
p 1/3
p 1/3
? D2
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
? D1
A
B
A
B
C
p 1/2
p 1/2
p 1/3
p 1/3
p 1/3
? D2
? D2
D2
D2
But is this also the appropriate updated
probability?
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Updated probability
?
It seems that after an update with D1, the second
research group has very little to doubt about
D1
D1
A
B
A
B
p 1/2
p 1/2
p 1/2
p 1/2
? D2
D2
? D'2
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
B
D1
A
B
C
B
?
2 ?
A
D2
A
B
C
1 ?
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
C
2 ?
B
B
2 ?
?
?
A
A
A
W
A
W
A
B
C
A
B
C
1 ?
1 ?
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Epistemic update
?
But a full update also comprises conditioning on
the accessible epistemic states of both research
groups.
C
C
B
B
B
B
2 ?
2 ?
A
A
A
A
A
B
C
A
B
C
1 ?
1 ?
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
C
B
B
2 ?
2 ?
?
?
A
A
A
A
W
W
C
A
B
C
A
B
1 ?
1 ?
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
A'
? B
B
? B
B
?
p 1/4
p 1/4
p 1/4
p 1/4
p 1/4
p 1/4
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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Judy Benjamin updates
?
In updating a probability p to p? by distance
minimisation under a partition of constraints ??,
we may have
for some B and all ?. Now suppose that we can
associate the constraints with a partition of
events G?
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No-representation theorem
?
In Bayesian conditioning on events A? from a
partition, the prior is always a convex
combination of the posteriors
But because p(BG?) gt p(B) for all but one ?, we
have
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.