Everett and Evidence

1. Everett and Evidence Hilary Greaves and Wayne Myrvold

2. 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

3. 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.

4. 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

5. 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.

6. 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.

7. The challenge • Can we tell a similar story if the coin flip is regarded as a branching event?

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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

14. 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.

15. 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 inT2

16. 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.

17. 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.