Understanding Confirmation with Probability

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Understanding Confirmation with Probability

• December 8, 2008

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Kolmogorovs Axioms
1. P(A) 0 for all A e S. 2. If T is a logical
truth, then P(T) 1 3. P(A ? B) P(A) P(B)
for all A, B such that (A ? B) is contradictory.
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Conditional Probability
P(HE) the probability of H, given E P(HE)
P(H ? E) P(E) Degree of TOTAL
CONFIRMATION P(HE) Degree of INCREMENTAL
CONFIRMATION P(HE) P(H)
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Example
P(H ? E) .01 P(H ? E) .04 P(H ? E)
.02 P(H ? E) .93
The degree of TOTAL confirmation that H has is
the probability that H is true, assuming that E
is true, where E is all your evidence. If E
makes H more likely, than H is on its own, then E
INCREMENTALLY confirms H.
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Bayes Theorem
P(HE) P(EH) P(H)
P(EH) P(H) P(EH) P(H)
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Bayes Theorem
P(HE) P(EH) P(H)
P(EH) P(H) P(EH) P(H)
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Bayes Theorem
P(HE) P(EH) P(H)
P(EH) P(H) P(EH) P(H)
This all seems to work well. But where do we get
the probabilities from?
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Interpretations of Probability

• Logical
• Frequency
• Objective Chance
• Subjective Degrees of Belief

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Logical Probability
Probabilities are assigned to different
possibilities using some logical method.
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Logical Probability
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Logical Probability
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Logical Probability
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Logical Probability
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H Everything is F. E Fb P(HE) P(H ?
E)/P(E) (1/3)/(1/2) 2/3 gt 1/3 P(H). Thus,
our evidence confirms our hypothesis.
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Problems for Logical Interpretation
1. Why this distribution of probability? There
are other distributions, so how is this
logical?   2. Why this language to describe
things? If we describe things in a different
language, we can get different results. This is
the lesson from Goodman.   3. If degree of
confirmation depends on distribution of
probability and language chosen, then how can
this guide science?
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Frequency and Objective Chance
We dont have time, but these arent going to be
satisfactory. They wont assign probabilities to
things that we want probabilities assigned to.
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Subjective Degree of Belief
Probabilities are identified with degrees of
confidence, or degrees of beliefs of suitable
agents. Suitable agents are those that are
rational.
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What is a degree of belief?
Betting Interpretation Your degree of belief in
some proposition, Q, is equal to X/(X1) where
X1 are the odds you deem fair for a bet on Q.
Example Your fair odds for a bet on Seabiscuit
winning are 31. Let S Seabiscuit wins.
Pyou(S) 3/(31) ¾ 0.75
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Understanding Confirmation with Probability

• December 10, 2008

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Basic Notions
TOTAL CONFIRMATION P(HE) INCREMENTAL
CONFIRMATION P(HE) gt P(H)
The degree of TOTAL confirmation that H has is
the probability that H is true, assuming that E
is true, where E is all your evidence. If E
makes H more likely, than H is on its own, then E
INCREMENTALLY confirms H.
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What are these Probabilities?
A Degrees of belief/degrees of confidence of
rational agents.
What is a degree of belief?
A Degree of belief in P is understood in terms
of the odds that you deem fair for a bet on P
being true.
Pyou(P) X/(X1) when X1 are the odds you deem
fair for a bet on P.
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Example
Lets say that you believe that theres a 50-50
chance of rain (R). Further, you believe that
Brian Lapis is a pretty good weatherman. You
think that the probability that L Lapis says it
will rain, given R that it will rain, is fairly
high. Finally, since youre such an avid viewer
of 22News, you know that Lapis tends to predict
rain (L) about 45 of the time. We have P(R)
.5P(L/R) .8P(L) .45
P(R/L) P(L/R)P(R) 0.8 0.5 ? .89
P(L) 0.45
P(R/L) gt P(R)
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What are rational degrees of belief?
Rational degrees of belief are those that satisfy
the Kolmogorov axioms. For example, rational
degrees of belief are such that P(H) P(H)
1 Why?
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Dutch Book Argument

• Assume Pjeff(S) Pjeff(S) ¾.
• Thus, Pjeff(S) Pjeff(S) gt 1. (So, I violate
  Kolmogorovs Axioms).
• Given my degrees of belief, and the betting
  interpretation of degrees of belief, I will
  accept 31 odds for a bet on S and 31 odds for a
  bet on S.
• Thus, I will accept the following two
  bets (Bet 1) Pay 3 for a chance to win 1 if S
  is true. (Bet 2) Pay 3 for a chance to win 1
  if S is true.
• If S is true, Ill win 1 from Bet 1, but lose
  3 on Bet 2, for net loss of 2.
• If S is true, Ill win 1 from Bet 2, but lose
  3 on Bet 2, for net loss of 2.

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Dutch Book Argument
Basically 1. If you dont meet the Axioms, then
you can be Dutch-Booked. 2. If you can be
Dutch-Booked, then you arent rational. ----------
--------------------------------------------------
----------------- 3. Thus, if you dont meet the
Axioms, then you arent rational. Which is
equivalent to 3. If you are rational, then you
meet the Axioms.
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Advantages

• Assigns probabilities to events we care
  about. (So, better than frequency and objective
  chance.)
• No problem with how to assign probabilities. (So,
  better than logical approach.)

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Potential Problems
i. Problem of the priors ii. Problem of
likelihoods iii. Idealizations in modeling degree
of belief iv. Language Incommensurability not
addressed
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Problem of the Priors
If everyone gives different hypotheses different
probabilities, then how is this an advantage over
logical probability? Answer Convergence
Theorems Given that we agree on the likelihoods
P(EH)s but disagree about how probable
different hypotheses are, there is some amount of
evidence that will lead us to have arbitrarily
close agreement about the probability of
hypotheses.
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Convergence
0 ravens 1 raven 3 ravens 6 ravens 12 ravens 60
ravens
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Problem of the Likelihoods
Why think that there will be agreement about
likelihoods P(E/H) if there is not agreement
about anything else? Partial Response Some
likelihoods appear to be given by the theory
itself.
all should agree (at least approximately) on
the values of the likelihoods of evidence claims
given by specific hypotheses. For, the
likelihoods represent the empirical content of a
hypothesis - what the hypothesis says about
evidence claims. James Hawthorne, A Better
Bayesian Convergence Theorem
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Convergence Arguments
Rational agents will converge to the same degrees
of belief.
with positive probability in the same
propositions
in propositions the evidence distinguishes
between
in the long run, believe with probability 1 that
they will
with probability 1
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The Dialectic
Problem of the Priors. Response Convergence
Theorems. Problem of the Likelihoods. Respons
e Just what the theory says. Q But why do
priors matter?
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Goodmans Grue and Priors
E All emeralds observed have been green. G
All emeralds are green. g All emeralds are
grue.
P(EG) gt P(E). why? But P(Eg) gt P(E), too.
why? Further, P(EG) P(Eg). why? From this
it follows that P(EG)/P(E) P(Eg)/P(E)
why? Now, we want to know whether to believe G
or g. So we need to look at P(GE) and P(gE).
P(GE) P(EG)/P(E) P(G) P(gE)
P(Eg)/P(E) P(g)
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Further Problems
iii. Idealizations in the model. iv. Language
Incommensurability. If scientists really
describe the evidence differently, then much of
the Bayesian machinery will not work.