Probability in the Everett interpretation: How to live without uncertainty

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Probability in the Everett interpretation How
to live without uncertainty

• or, How to avoid doing semantics

Hilary Greaves New Directions in the Foundations
of Physics April 29, 2006
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Aims of the talk

• Raise and solve the epistemic problem for
  many-worlds quantum mechanics (MWQM).
• Solve it without relying on contentious language
  of uncertainty.
• Conclude that worries about probability do not
  provide a reason to reject the many-worlds
  interpretation.

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Outline of the talk

• The many-worlds interpretation probability
• First problem of probability (practical)
• Solution to the practical problem
  (Deutsch/Wallace)
• Interlude On the semantics of branching
• Second problem of probability (epistemic)
• Solution to the epistemic problem
• Concluding remarks

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1.1 Many-worlds interpretations (MWI) introduced
Cat goes into mixed state
Pointer goes into mixed state
Measurement occurs
M
M
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1.1 Many-worlds interpretations (MWI) introduced
Branch 1
Branch 2

• A first pass When a quantum measurement is
  performed, the world splits into multiple
  branches, and each possible outcome is realized
  in some branch

Splitting occurs
M
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1.2 MWI via consistent histories
?

• What there is ? (????), undergoing unitary
  evolution
• How the macroworld supervenes on ? via a
  decomposition into histories
• Preferred basis problem which history set?
• Use dynamical decoherence (Zurek, Zeh, Gell-Mann
  and Hartle, Saunders, Wallace)
• Emergent branching structure

t
P2?(t2)
P2?(t2)
P1(t1)
t
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1.3 The problem of probability
a12
an2

• If one postulates that all of the histories are
  realised then no role has been assigned to the
  probabilities, and there seems no obvious way of
  introducing further assumptions which would allow
  probabilistic statements to be deduced.
• (Dowker Kent (1994))

P2?(t2)
P2?(t2)
P1(t1)

• Quantum weight of ith branch,
• ai2 C?i ??2
• The quantum weights
• satisfy the axioms of probability
• but mean...??

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1.4 Everett on probability in MWI

• Everett (1957), DeWitt (1973) in the limit, the
  quantum measure of deviant branches goes to
  zero
• This is true, but its not enough (as in
  classical case)

Normal statistics
N
4
3
2
1
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1.5 My claim (1)
Worries about probability do not provide reasons
to reject the many-worlds interpretation.
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Outline of the talk

• The many-worlds interpretation probability
• First problem of probability (practical)
• Solution to the practical problem
  (Deutsch/Wallace)
• Interlude On the semantics of branching
• Second problem of probability (epistemic)
• Solution to the epistemic problem
• Concluding remarks

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2.1 The practical problem How to use QM as a
guide to life

• Nuclear power plant design A
• p(disaster) 0.0000..07.
• Nuclear power plant design B
• p(disaster) 0.9999..94.
• What to do?

a12
a22
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Outline of the talk

• The many-worlds interpretation probability
• First problem of probability (practical)
• Solution to the practical problem
  (Deutsch/Wallace)
• Interlude On the semantics of branching
• Second problem of probability (epistemic)
• Solution to the epistemic problem
• Concluding remarks

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3.1 Deutschs/Wallaces program

• Quantum games (??, X, P)
• Utility function, U
• Probability function, p decision-theoretic
  branch weights
• Structural claim Maximization of expected
  utility (MEU)
• Quantitative claim decision-theoretic branch
  weight quantum branch weight
• So The rational agent acts as if the Born rule
  were true. (Deutsch (1999))

a2
b2
P(½)100
P(-½) 1000
100
1000
X (½) ???? (-½) ????
?? a?? b??
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Outline of the talk

• The many-worlds interpretation probability
• First problem of probability (practical)
• Solution to the practical problem
  (Deutsch/Wallace)
• Interlude On the semantics of branching
• Second problem of probability (epistemic)
• Solution to the epistemic problem
• Concluding remarks

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4.1 Semantics A Subjective uncertainty

Thing 1 might happen is true iff Thing 1
happens on some branch to the future.
Thing 1 happens on this branch
Thing 2 happens on this branch
Thing 1 might happen Thing 2 might happen. I am
uncertain about which will happen.
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4.2 Semantics B The fission program (or, How
to live without uncertainty)
Thing 1 will happen is true iff Thing 1
happens on some branch to the future.

Thing 1 happens on this branch
Thing 2 happens on this branch
Thing 1 will (certainly) happen Thing 2 will
(certainly) happen. There is nothing for me to be
uncertain about.
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4.3 Two ways to understand maximization of
expected utility (MEU)

• Subjective uncertainty (SU) despite knowing
  that the world will undergo branching, the agent
  is uncertain about which outcome will occur.
• The probability measure quantifies the rational
  agents degree of belief in each branch.
• The fission program (FP) Theres nothing for
  the agent to be uncertain about she knows that
  all branches will be real.
• The probability measure (caring measure)
  quantifies the rational agents degree of concern
  for each branch.

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4.4 Mere semantics?

• Two questions

what is the right semantics for talking about
branching?
is MWQM an acceptable physical theory?

• What hangs on the SU/FP debate?

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4.4 Mere semantics?

• What hangs on the SU/FP debate?
• -The applicability of decision theory? (No.)
• -The epistemological acceptability of a
  many-worlds interpretation?? (I will argue no.)

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Outline of the talk

• The many-worlds interpretation probability
• First problem of probability (practical)
• Solution to the practical problem
  (Deutsch/Wallace)
• Interlude Subjective uncertainty and the
  fission program
• Second problem of probability (epistemic)
• Solution to the epistemic problem
• Concluding remarks

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5.1 The epistemic problem Why believe QM in the
first place?

• e.g. 2-slit experiment
• The problem Knowing how rationally to bet on the
  assumption that MWQM is true does not amount to
  knowing whether or not MWQM is true.
• We need two things from quantum probability MEU
  is only one of them

This confirms quantum mechanics
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5.2 The confirmation challenge for MWQM

• Compare and contrast
• Quantum mechanics predicted that the relative
  frequency would approximately equal R with very
  high probability. We observed relative frequency
  R. This gives us a reason to regard QM as
  empirically confirmed.
• Seems fine
• MWQM predicted that the relative frequency would
  approximately equal R on the majority of branches
  according to the caring measure. We observed
  relative frequency R. This gives us a reason to
  regard MWQM as empirically confirmed.
• ???
• Empirical incoherence coming to believe the
  theory undermines ones reason for believing
  anything like it
• Is MWQM empirically incoherent?

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5.3 My claims (2 3)
(2) The epistemic problem (not only the practical
problem) can be solved in a way favorable to
MWQM and (3) This is the case regardless of which
is the right semantics for branching.
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Outline of the talk

• The many-worlds interpretation probability
• First problem of probability (practical)
• Solution to the practical problem
  (Deutsch/Wallace)
• Interlude Subjective uncertainty and the
  fission program
• Second problem of probability (epistemic)
• Solution to the epistemic problem
• Concluding remarks

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6.1 Strategy for solving the epistemic problem

• Ask how exactly do we deal with the epistemic
  issue in the non-MW case?
• Dynamics of rational belief A Bayesian model of
  common-or-garden empirical confirmation
• Illustrate how 2-slit experiments (etc) confirm
  QM
• Argue that the same solution (mutatis mutandis)
  works for MWQM
• Work out the dynamics of rational belief for an
  agent who has non-zero credence in MWQM
• Deduce that 2-slit experiments (etc) confirm MWQM

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6.2 Bayesian confirmation theory (non-branching
case)

• Suppose I have two theories, QM and T
• Suppose I perform an experiment with two possible
  outcomes, R and ?R
• Four possible worlds
• WT?R, T??R, QM?R, QM??R
• Credence function Cr0 at time t0, prior to
  experiment
• Cr0 obeys the Principal Principle, i.e.
• Cr0(?T) ChT(?)
• Cr0(?QM) ChQM(?)

T?R
T??R
T?R
T??R
T?R
T??R
QM?R
QM??R
T?R
T??R
QM?R
QM??R
t2
M
M
M
M
t0
T?R
T??R
QM?R
QM??R
T
QM
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6.2 Bayesian confirmation theory (non-branching
case)

Centered world in which the agent adopts credence
function Cr2R over W
T?R
T??R
T?R
T??R
T?R
T??R
QM?R
QM??R
T?R
T??R
QM?R
QM??R
t2
Centered world in which the agent adopts credence
function Cr2?R over W
M
M
M
M
t0
T?R
T??R
QM?R
QM??R
T
QM
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6.3 How to update beliefs choosing Cr2R and Cr2?R

• Conditionalization on observed outcome use
  posterior credence functions Cr2RCr0(?R), Cr2?R
  Cr0(??R)
• IF
• Cr0 obeys the Principal Principle, and
• the agent updates by conditionalization
• THEN observing R increases credence in QM at the
  expense of credence in T
• This is why observing R counts as confirmatory of
  QM

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6.4 Generalized Bayesian confirmation theory
(branching case)

• Candidate theories MWQM, T
• Possible worlds
• W T?R, T??R, MWQM
• Centered possible worlds at time t2
• WC T?R, T??R, MWQM?R, MWQM??R

T?R
T??R
MW?R
MW??R
T?R
T??R
MW?R
MW??R
t2
M
M
M
t0
T?R
T??R
MWQM
T
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6.4 Generalized Bayesian confirmation theory
(branching case)
Centered world in which the agent adopts credence
function Cr2R over W
T?R
T??R
T?R
T??R
MW?R
MW??R
t2
Centered world in which the agent adopts credence
function Cr2?R over W
M
M
M
t0
T?R
T??R
MWQM
T
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6.5 Choosing Cr2R and Cr2?R in the branching case

• Two prima facie plausible updating policies
• Naïve conditionalization
• Extended conditionalization
• Both of these are generalizations of ordinary
  conditionalization

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6.6 Naïve conditionalization

• Some very natural, but pernicious intuitions
• Caring measure has nothing to do with credence
• The agents credence that R occurs is given by
  Cr0(R) Cr0(T?R) Cr0(MWQM)
• How to conditionalize
• Cr2R(?) Cr0(?R)
• Cr0(??R)/Cr0(R)

T?R
T??R
MWQM?R
MWQM??R
T?R
T??R
MW?R
MW??R
T?R
T??R
MWQM
R definitely happens in this possible world (and
so does ?R)
R does not happen in this possible world
R happens in this possible world
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6.7 Naïve conditionalization is bizarre

• Observation Naïve conditionalization has the
  consequence that credence in MWQM increases at
  the expense of credence in T, regardless of
  whether R or ?R occurs
• i.e. Cr2R(MWQM) gt Cr0(MWQM)
• and Cr2?R(MWQM) gt Cr0(MWQM)
• This is not surprising
• Auxiliary premise No rational updating policy
  can allow any theory to enjoy this sort of free
  ticket to confirmation
• Conclusion Naïve conditionalization is not the
  rational updating policy for an agent who has
  nonzero credence in a branching-universe theory

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6.8 Defining Extended Conditionalization

• We have the resources to define an updating
  policy according to which evidence supports
  belief in MWQM in the same way that it supports
  belief in QM
• Construct an effective credence function, Cr'0
  (defined on WC), from Cr0 and Car0
• Cr0(T?R) Cr0(T?R)
• Cr0(T??R) Cr0(T??R)
• Cr0(MWQM?R) Cr0(MWQM)?Car0(R)
• Cr0(MWQM??R) Cr0(MWQM)?Car0(?R)
• Updating policy obtained by conditionalizing the
  effective credence function on R and on ?R
• This policy would have the effect that credence
  in MWQM responds to evidence in just the same way
  that credence in QM responds to evidence

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6.9 Defending Extended Conditionalization

• Is Extended Conditionalization the rational
  updating policy for an agent who thinks the
  universe might be branching?
• Yes
• All the arguments we have in favour of
  conditionalization in the ordinary case apply
  just as well in the branching case, and favour
  Extended Conditionalization over Naïve
  Conditionalization

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6.10 Defending (ordinary) conditionalization The
(diachronic) Dutch Book argument

• Assume that degrees of belief give betting
  quotients
• This holds because the agent is an expected
  utility maximizer
• A fair bet is a bet with zero net expected
  utility
• If the agent updates other than by
  conditionalization, a Dutch Book can be made
  against her

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6.11 Defending Extended Conditionalization
diachronic) Dutch Book argument

• If the agent is an expected-utility maximizer in
  Deutschs/Wallaces sense (), her betting
  quotients are given by her effective credence
  function, Cr'0
• If the agent updates other than by Extended
  Conditionalization, a Dutch Book can be made
  against her
• (Other arguments for conditionalization can be
  generalized in the same sort of way)

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6.12 On black magic

• How these arguments manage to connect a caring
  measure to credences
• Cast the confirmation question in terms of
  rational belief-updating
• Choosing an updating policy is an epistemic
  action
• Epistemic action is a species of action
• The caring measure is relevant to all choices of
  actions, including epistemic ones

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Outline of the talk

• The many-worlds interpretation probability
• First problem of probability (practical)
• Solution to the practical problem
  (Deutsch/Wallace)
• Interlude Subjective uncertainty and the
  fission program
• Second problem of probability (epistemic)
• Solution to the epistemic problem
• Concluding remarks

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7 Concluding remarks

• There exists a natural measure over Everett
  branches, given by the Born rule (we knew this
  already)
• The measure governs
• rational action (Deutsch/Wallace have argued) so
  we know how to use the theory as a guide to life
• rational belief (I have argued) so we are
  justified in believing the theory on the basis of
  our empirical data, just as in the non-MW case
• The subjective uncertainty semantics is not
  required for any of the above.
• This time, we can do philosophy of physics
  without doing semantics.
• Worries about probability are not a reason to
  reject the many-worlds interpretation.