Everett and Evidence
Hilary Greaves and Wayne Myrvold
Evidence for QM
• On standard interpretation, QM yields
probabilities for experimental outcomes.
• Much of the evidence for QM is statistical
in nature:
 Relative frequencies of outcomes in repeated
experiments are compared with calculated
probabilities
 Close agreement is evidence that QM is getting
the probabilities right
Everett Interpretation
• All possible experimental outcomes occur
on some branches
• No obvious sense in which we can speak
about the probability that the result will be,
say, spin-up.
Probabilities: who needs
them?
• Threat of undermining much of the reason
we have for taking QM seriously in the first
place
• Required: either a way of making sense of
probability in an Everettian context, or of
finding a substitute that plays a parallel role
in confirmation
Evidence about chances
• Suppose I want to determine whether a
coin-flip is biased
• Start with some degrees of belief about the
chance of landing heads
• Perform repeated flips
• Update degrees of belief by conditionalizing
on the results.
Updating belief about
chances
• Suppose my prior degrees of belief about the
chance of heads are represented by a density
function f(x).
• I observe N flips, of which m are heads and n = N
- m are tails,
• Density f gets multiplied by likelihood function
l(x) = xm (1 - x)n
• This is peaked at the observed relative frequency
m/N; more sharply peaked, the larger N is.
The challenge
• Can we tell a similar story if the coin flip is
regarded as a branching event?
The Framework
• A wager is an association of payoffs with
subsets of the outcome space of an
experiment.
• Savage axioms (P1-P6) entail that the
agent’s preferences between wagers are as if
she is maximizing expected utility.
Learning from experience
• Learning the result of an experiment may
alter your preferences between wagers on
outcomes of future experiments
• Preference ordering on wagers induces a
preference order on updating strategy.
• P7: In learning experiences, our agent
adopts the updating strategy that ranks
highest on her current preferences.
Repeated Experiments
• No two experiments are exactly alike.
• A sequence of experiments will be called
exchangeable if preferences between
wagers on sets of experiments in the
sequence are unchanged on permutation of
payoffs associated with experiments in the
sequence.
De Finetti representation
theorem
• If your preferences between wagers make a
series of experiments an exchangeable one,
then they are as if you believe that
associated with each outcome a chance,
which is the same for all elements of the
sequence.
• Your degree of belief in an outcome is an
epistemically weighted mean of the possible
chances of the outcome.
Non-dogmatism
• Everything so far is consistent with
continuing to bet at even odds, even after
observing long series of heads.
• P8: Don’t exclude a priori any open set of
chance space.
• That is, be prepared to have your degrees of
belief converge, in the long run, to the
observed relative frequency.
The Upshot
• As long as the observed relative frequency
of a given outcome in a finite string of
experiments is not in an interval to which
you have assigned zero prior degree of
belief, you will boost your degrees of belief
in chances that are near the observed
relative frequency
We claim
• The postulates P1-P8 can be taken as constraints
on reasonable preferences whether or not one
thinks of the experiments in the usual way, or as
branching events.
• Preferences will be as if the agent thinks of
subsets of outcome space having associated with
them “branch weights,” about which she can learn
by performing repeated experiments.
• Relative frequency data becomes evidence about
branch weights, just as it is for chances.
An analogy: classical fission
• A faulty transporter creates three copies of
you, identical in all respects (visible or
invisible), except for letter on your T-shirts.
• T1: 2 As, 1 B
• T2: 1 A, 2 Bs
• You go through this process, look down,
and see an A.
• This raises your degree of belief in T1,
lowers degree of belief in T2
A disanalogy
• We don’t think that there are determinate
numbers of branches.
• “Number of branches” must be replaced by
a measure on sets of branches.
Conclusion
• Take Everettian QM as a theory that posits
branching structure, and Born rule weights
as branch weights.
• This will be empirically confirmed in a
manner exactly parallel to the way ordinary
QM, which posits Born rule chances, is.