Scientific reasoning: A philosophical toolkit
(0L350)
Tiago de Lima
2nd semester 2007/2008
Outline of this lecture
Scientific
reasoning
Bayesian
Confirmation
Theory
Today we will see a very brief overview of Bayesian
confirmation theory.
This theory tries to explain how scientists revise their
beliefs in view of observations.
This theory identifies beliefs with probabilities, and use
Bayes’ rule of conditionalization to explain scientists
behaviour.
Bayesian confirmation theory also provides some
insights on Hume’s problem of induction.
However, it is not enough to solve that problem, as we
will see in the end of the lecture.
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Probability calculus
Scientific
reasoning
Bayesian
Confirmation
Theory
In this calculus we assign probabilities to formulas.
The probabilities are supposed to model the credence
in a formula. It does not consists in its truth value.
Formulas are either true or false as before.
Formulas are meant to describe outcomes.
The calculus is defined axiomatically by the following:
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2
3
For any outcome Φ, C(Φ) ≥ 0.
For any inevitable outcome Φ, C(Φ) = 1.
For mutually exclusive outcomes Φ and Ψ,
C(Φ ∨ Ψ) = C(Φ) +C(Ψ).
Some Theorems:
1
2
3
4
5
6
C(Φ) +C(¬Φ) = 1.
C(Φ) ≤ 1.
If |= Φ ↔ Ψ then C(Φ) = C(Ψ).
C(Φ) = C(Φ ∧ Ψ) +C(Φ ∧ ¬Ψ)
If Φ |= Ψ then C(Φ) ≤ C(Φ).
If C(Φ → Ψ) = 1 then C(Φ) ≤ C(Ψ).
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Conditional probability
Scientific
reasoning
Bayesian
Confirmation
Theory
The probability of an outcome e given another outcome
d is written C(e|d). It is defined as follows. Let C(d) > 0:
C(e|d) =
C(e ∧ d)
.
C(d)
Intuitively, restrict our view to the possible worlds in
which the outcome d occurs. Imagine that these are the
only possibilities. Then the probability of e conditional
on d is the probability of e in this imaginary, restricted
universe.
Theorem of Bayes:
C(e|d) =
C(d|e)
C(e)
C(d)
Total Probability Theorem. Let d1 , d2 , . . . be mutually
exclusive and exhaustive:
C(e) = C(e|d1 )C(d1 ) +C(e|d2 )C(d2 ) + . . .
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Bayesian confirmation theory
Scientific
reasoning
Bayesian
Confirmation
Theory
Bayesian confirmation theory uses the following three
assumptions:
Scientists assign credences to different competing
hypotheses. A credence is a number between 0 and 1
reflecting the level of expectation that a hypothesis will
turn out to be true.
The credences behave mathematically like probabilities.
Scientists learn from evidence by the Bayesian
conditionalization rule.
133
Dutch book arguments
Scientific
reasoning
Bayesian
Confirmation
Theory
Is it the case that credences can be identified with
probabilities? Some people thing that the answer is yes.
They use the so-called ”Dutch book arguments” to
support it. It is roughly as follows.
First, we assume that if one’s credence for an outcome e
is p, then he/she should accept odds of up to p : (1 − p)
to bet on e and odds of up to (1 − p) : p to bet against e.
Now, consider someone whose credences violate axiom
2. It means that he/she has a credence for an inevitable
event e that is less than 1. E.g., suppose that this
person has a credence of 0.9 on e. Then, he/she is
prepared to bet against e at odds of 1 : 9. But this
person is sure to loose such bet!
Then, this person is irrational!!
Therefore, to be rational, you have to accept axiom 2.
Similar arguments are given to the other axioms.
134
From conditionalization to confirmation theory
Scientific
reasoning
Bayesian
Confirmation
Theory
The idea is to use the Bayes Theorem to calculate the
evolution of credences. Let’s see an example.
We have three competing hypotheses:
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2
3
h1 : The probability that any given raven is black is one
(roughly speaking: all ravens are black).
h2 : The probability that any given raven is black is
one-half (roughly speaking: half of all ravens are black).
h3 : The probability that any given raven is black is zero.
(roughly speaking: no raven is black).
Important: The calculus will work only if the sum of the
probabilities of all competitors is equal to 1.
Let’s give a credence of 1/3 to each one.
Now, suppose that we observe a black raven. And let’s
call it evidence e1 .
135
From conditionalization to confirmation theory
Scientific
reasoning
Bayesian
Confirmation
Theory
The new credence in our hypotheses are:
C+ (hi ) = C(hi |e1 ) =
C(e1 |hi )
C(hi )
C(e1 )
C(e1 |h1 ) = 1, because we restrict our view to the
possible worlds where the raven is black.
C(e1 |h2 ) = 1/2.
C(e1 |h3 ) = 0.
By the Total Probability Theorem we have:
C(e1 ) = C(e1 |h1 )C(h1 ) +C(e1 |h2 )C(h2 ) +C(e1 |h3 )C(h3 )
Therefore we have:
C(e1 ) = 1/3 + 1/6 + 0 = 1/2
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From conditionalization to confirmation theory
Scientific
reasoning
Bayesian
Confirmation
Theory
The new credences are:
1
2
1/3 =
1/2
3
1
1/2
C+ (h2 ) =
1/3 =
1/2
3
0
C+ (h3 ) =
1/2 = 0
1/2
C+ (h1 ) =
If we continue observing black ravens then we have:
n P(h1 ) C(h2 ) C(h3 ) C(en )
0 1/3
1/3
1/3
1/2
1 2/3
1/3
0
5/6
2 4/5
1/5
0
9/10
3 8/9
1/9
0
?
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Some interesting properties
Scientific
reasoning
Bayesian
Confirmation
Theory
If h entails e then the observation of e confirms h (it
increases the credence in h).
The higher the probability that h assigns to e, the more
strongly h is confirmed to e.
Hypotheses that assign equal probabilities to an
evidence are equally strongly confirmed by that
evidence.
No matter what evidence is observed, the probabilities
of a set of mutually exclusive, exhaustive hypotheses
will always sum to one.
The order in which the evidence is observed does not
alter the cumulative effect on the probability of a
hypothesis.
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Hume’s problem of induction revisited
Scientific
reasoning
Bayesian
Confirmation
Theory
The Hume’s problem of induction is the problem of
finding objective grounds for preferring some
hypotheses to others on the basis of observations.
Bayesian confirmation theory made us make some
progress. For instance, this theory limits inductive
reasoning.
However, it does not solve this problem completely.
First, to use this theory we have at least to accept (1)
The axioms of probability and (2) Bayes’
conditionalization rule.
Second, the credence on the hypotheses depends on
the probabilities assigned to the hypotheses before any
evidence (prior credences).
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More criticism
Scientific
reasoning
Bayesian
Confirmation
Theory
There are many critics to Bayesian confirmation theory.
I discuss here only two of them.
The connection between the betting behaviour and
credences may not be strong enough. E.g., what about
the possibility that an aversion to gambling distort this
relation?
No one can be blamed for failing to arrange their
credences in accordance with the axioms. For instance,
in order to follow axiom 2, you should know which
outcomes are inevitable. That is, you should know all
the theorems of the underlying logic!
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