Evidence and Theory
From probability to the pragmatics of
confirmation
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.
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 p be the ‘garbage can’ result, when p
fails.
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.
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.
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 .
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).
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)
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.
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?
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!
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.
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.
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.
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.
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.
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…
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 nonblack things to see if they’re ravens, it may help answer
the second.
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.
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…
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!