Evidence and Theory

1. Evidence and Theory From probability to the pragmatics of confirmation

2. Bayesianism • This has been the biggest, most influential program in thinking about evidence and how it supports scientific theories over the last 30 years or so. • It takes its starting point from a simple theorem of probability theory. • Probability theory itself grew out of mathematical thinking about games of chance, beginning in the 17th century.

3. Probability • We can think of probability as a measure of a rational inclination to make bets. • A collection of exhaustive and exclusive results for a bet must be such that we know one and only one will obtain in the end. • Let p be a result that we can evaluate at some point. • Let pbe the ‘garbage can’ result, when p fails.

4. From 0 to 1 inclusive • A bet on ‘p or p’ pays if we get the result p and if the result p fails to hold– that is, it pays regardless of what happens. • Let’s assign any such result (i.e. a result that is known in advance to hold) the probability 1. • At the other extreme, let’s assign any result that is known in advance to fail the probability 0.

5. Betting • What should you pay for a bet returning $X net if the result you bet on comes through? • Suppose the result has probability 1, that is, you know in advance that the result will come through. • Then you should be willing to pay any price at all for the certain gain of $X. • Of course, if the result has probability 0, there will be no price low enough to entice you.

6. Betting odds • Between the extreme probabilities we get more interesting results. • What would you pay for a bet returning $X on a fair coin flip? • $X would be a fair price (better than you’d get in Vegas). • The probability can be defined as the (highest/break-even) price you’d pay for the bet, divided by the sum of the price and the payout if your result comes through: in this case, ½ .

7. Basic Principles • Pr(A) = 1 – Pr(A) • Pr(A v B) = Pr(A) + Pr(B) – Pr(A&B) • Pr(A&B) = Pr(A)*Pr(B/A) = Pr(B)*Pr(A/B) • For exclusive (mutually incompatible) results A and B, P(A v B) = Pr(A) +Pr(B) • For independent results (when Pr(A) = Pr(A/B)), Pr(A & B)= Pr(A)* Pr(B) • Pr(A/B) = Pr(A&B)/Pr(B) (if Pr(B) 0).

8. Baye’s Rule • This is a straightforward outcome of the rule giving Pr(A&B): • Pr(A&B)=Pr(A)*Pr(B/A) = Pr(B)*Pr(A/B). • But If Pr(A)*Pr(B/A) = Pr(B)*Pr(A/B), then Pr(B/A) = [Pr(B)*Pr(A/B)]/Pr(A) • Or, using h for our hypothesis and e for the evidence, • Pr(h/e) = Pr(h) * Pr(e/h)/Pr(A)

9. Games of Chance • This works perfectly in games of chance– e.g., what is the probability (given a standard deck of cards & a fair shuffle) that I will draw a King? • Pr(K) = 4/52 = 1/13. • Suppose I know that I’ve drawn a face card (here’s our new evidence). • Then Pr(K/e) = Pr(K) * Pr(e/K)/ Pr(e) • But Pr(e) = 12/52, and Pr(e/K) = 1. • So Pr(K/e) = 4/52 / 12/52 = 1/3.

10. Monte Hall • It also works nicely on the Monte Hall puzzle. • Monte asks us to choose between 3 doors. (Behind one of them is a wonderful prize.) • Once we’ve chosen one, Monte announces he will pick one of the other two doors & show us that the prize isn’t behind that other door; he then shows us the empty space behind that door. • Then Monte offers a choice: Switch from our door to the unopened other door, or stand pat. • What should we do?

11. Apply Bayes’ Theorem! • Let A be the result that our door hides the prize. Let B1 and B2 be the results that the other doors do. • Initially, Pr(A) = Pr(B1) = Pr(B2) = 1/3 • But then we learn e: B1. Now calculate: • P(A/e) = P(A)*P(e/A) / P(e) = 1/3*1/2/1/2=1/3 • P(B2/e)= P(B2)*P(e/B2)/P(e) = 1/3*1/1/2=2/3 • We’d better switch doors!

12. In general • Probability theory has applications in many areas: quantum measurements, statistical mechanics, randomized sampling and causal studies, and games of chance. • In these areas, Bayes’theorem is very useful, because we have (what we take to be) good grounds for assigning particular values to the probabilities we need to apply it.

13. But when we’re choosing hypotheses • Here we don’t have the repetition of similar but distinct cases that can ground frequency-based evaluations of probabilities. • Neither do we have basic symmetries of fundamental equations like those of quantum mechanics that could underwrite probability assignments. • So applying Bayes’ theorem gets hard. • Subjectivism is one response to these problems.

14. Subjectivism • There are no grounds beyond the basic principles of probability theory that constrain rational degrees of belief. • Dutch book arguments underwrite the idea that rational degrees of belief (treated as minimally rational betting ratios) must respect the probability calculus. • For the true believers, Bayes’ theorem and its generalization (to cases where our evidence doesn’t receive probability 1) are the only rational ways to change our degrees of belief.

15. Convergence • Given time and shared opinions on likelihoods (the probability of evidence given a hypothesis), as evidence accumulates, degrees of belief will converge. • But if we start enough far apart, convergence can be put off indefinitely. • And it’s not clear why we can expect subjectivists to agree on likelihoods, either.

16. Grue • One Bayesian response to the grue puzzle is to assume a low prior for the hypothesis that emeralds are grue. This makes emeralds observed to be green support for the grue hypothesis, but only weak support– the green hypothesis wins out. • But this seems a bit ad hoc: nothing in subjectivism demands or explains this low prior probability.

17. A different (realist) view of evidence • GS sees testing as an effort to find grounds for choosing between hypotheses about the ‘hidden structure’ of the world. • ‘Eliminative induction’ seems to work this way– we eliminate hypotheses via falsification, but we don’t assume that there is an unlimited range of hypotheses out there, so the elimination also manages to confirm the surviving hypothesis(es). • And scientists actually do this kind of reasoning! • Even its failures reflect real errors that occur sometimes in science, when important possibilities are neglected…

18. Procedure: A practical turn • Testing claims about Ravens requires a procedure, not just particular observations! Consider: • What % of Ravens are black? • Are all Ravens black? • If we find a black raven by testing black things to see if they’re ravens, the observation is useless. • But if we come upon a black raven in the course of observing ravens to see how many are black, it can help us to answer both questions. • And if we come upon a white shoe while testing non-black things to see if they’re ravens, it may help answer the second.

19. Beyond Mere Empiricism: The pragmatics of evidence • The force of our evidence depends on how we came by it, not just what it says about the world. • This connects to the grue puzzle, says GS. • When a hypothesis links what it says about the world to our observations, it can make a difference to the support our observations give it. • The ‘emeralds are grue’ hypothesis does this: whether a green emerald confirms it depends on when we observe that emerald. • Observing green emeralds is different from observing grue ones– observing grue emeralds assumes that, had we observed them later, they would have had a different colour.

20. Doubts • I’m not optimistic about this resolution of the grue puzzle. • Grue and green are symmetrical; to someone who thinks in terms of grue and bleen, it’s us who believe that the ‘colour’ of an emerald will differ, depending on when we observe it. • I think a better solution to the problem lies in pursuing this symmetry to the point where either it breaks down or the difference actually disappears…

21. Sympathies • But I do think the practical business of observation matters. • Observations don’t arise out of nowhere– they are the product of deliberate efforts, and arise in a context where there is a lot of background & structure already in place that affects what we conclude from our observations. • Consider Hume’s Porter & how we really test for causal regularities!