Department of Philosophy

Carnegie Mellon University

- An
- p&q
- p or q
- not p
- not q
- P(h) >= 0;
- P(T) = 1;
- a is inconsistent with b ===> P(a or b) = P(a) + P(b).

T = the vacuous proposition.

A **probability function **P on an algebra is an assignment of numbers
to propositions such that:

- P(h|e) = P(h &
e)/P(e), if P(e) > 0.

- P(h|e) = [P(h)P(e|h)]/P(e).

Proof:

P(e|h) = P(e &
h)/P(h) [definition
of conditional probability].

P(e & h) = P(e|h)P(h)
[multiply both sides by P(h)].

P(h|e) = P(h
& e)/P(e) [definition
of conditional probability].

P(h|e) = P(e|h)P(h)/P(e)
[substitution].

- P(e) = SUM

Proof:

e = OR_{i} (e & h_{i})
[logic]

P(OR_{i} (e & h_{i})) =
SUM_{i} P(e & h_{i}) [second axiom of probability]

P(e|h_{i}) = P(e & h_{i})/P(h_{i}) [definition
of conditional probability]

P(e & h_{i}) = P(e|h_{i}i)P(h_{i})
[multiply both sides by P(h)].

P(e) = P(OR_{i} (e & h_{i}))
= SUM_{i} P(e & h_{i})
= SUM_{i} P(e|h_{i}i)P(h_{i})
[preceding lines]

e is

The preceding definition seems rather technical. The following recharacterization makes intuitive sense. Probabilistic independence is irrelevance: learning the one event wouldn't change your belief in the other.

**Theorem: probabilistic independence is informational irrelevance:**

e is independent of h if and only if P(h|e) = P(h).

Proof: Suppose P(h|e) = P(h). Then P(h & e) = P(h|e)P(e)
= P(h)P(e).

Suppose P(h & e) = P(h)P(e). Then P(h|e) = P(h & e)/P(e)
= P(h)P(e)/P(e) = P(h).

By Bayes' theorem, the new degree of belief in h after seeing e is

- P(h|e) = [P(h)P(e|h)]/P(e).

**Prior probability **of h = P(h).
This may be quite subjective, reflecting a theory's initial "plausibility"
prior to scientific investigation. This plausibility depends on such factors
as intelligibility, simplicity, and whether the mechanism posited by the
theory has been observed to operate elsewhere in nature (e.g., uniformitarian
vs. catastrophist geology). In the 19th c. it was proposed that only causes
observed to operate in nature could be invoked in new theories. This reflects
prior probability.

**Prior probability **of e = P(e).
This is subjective and very hard to specify. Using total probability,

- P(e) = SUM

P(h|e)/P(h'|e) = [P(h)/P(h')][P(e|h)P(e|h')].The ratio [P(h)/P(h')] is the

**Refutation is fatal:** If consistent e is inconsistent with h,
then P(h|e) = 0.

- Proof:

Note e & h = not T, since the two are inconsistent.

Also, P(not T) + P(T) = 1 by axiom (3).

P(T) = 1 by axiom (2).

Hence, P(not T) = 0. Now we have:

P(h|e) = P(h & e)/P(e) = P(not T)/P(e) = 0/P(e) = 0.

**Diminishing returns of repeated testing:** Once P(e) is expected,
by the preceding argument confirmation is reduced.

**Strong explanations are good, initial plausibilities being similar:**
The ratio P(h|e)/P(h'|e) changes through time entirely as a function of
the ratio of relative strength of explanation P(e|h)/P(e|h'), for

P(h_{1}|e)/P(h_{2}|e) = [P(h_{1})/P(h_{2})][P(e|h_{1})/P(e|h_{2})] = k[P(e|h_{1})/P(e|h_{2})].

P(e & e') = P(e)P(e').Now suppose that e and e' remain independent given h

h & e' --> e. Thus

P(e & e'|hSo_{1}) = P(e'|h_{1})P(e|h_{1}) and

P(e & e'|h_{2}) = P(e|h_{2}).

P(hNow there is no reason to suppose that P(e|h_{1}|e & e')/P(h_{2}|e & e') =

[P(h_{1})/P(h_{2})][P(e & e'|h_{1})/P(e & e'|h_{2})] =

k[P(e & e'|h_{1})/P(e & e'|h_{2})] =

k[P(e|h_{1})P(e'|h_{1})/P(e|h_{2})].

**Saying more lowers probability: **h entails h' ==> P(h) __<__
P(h').

**Conflict turns explanatory strength into an asset:** Didn't we
just say that strong explanations are good??? That is true if the initial
plausibilities are similar. But if one theory entails the other, they won't
be. Thus, *unification-style arguments only work if the competing theories
are mutually contradictory!*

**Scientific method should not consider subjective, prior plausibilities.
**That's
just the kind of blind, pre-paradigm science Kuhn ridicules as being sterile.
Without prior plausibilities to guide inquiry, no useful experiments would
ever be performed.

**Priors should be flat.** What is flat? If we are uncertain about
the size of a cube, should we be indifferent about

- the possible volumes,
- the areas of the sides, or
- the lengths of the sides?

**It isn't clear that numbers like P(e) even exist.** One can respond
with a protocol for eliciting such numbers, but in practice it doesn't
always work. One can say that the subjects are "irrational", but the audience
can always blame Bayesianism instead of the subjects.

**The old evidence problem.** If e is already known, then P(h|e)
= P(h) P(e|h)/P(e) = P(h). So old evidence never "confirms" a hypothesis.

**Responses:**