# Bayesian Epistemology - PowerPoint by CQkc23E

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```									Bayesian Epistemology

Kareem Khalifa
Department of Philosophy
Middlebury College
The twin pillars of Bayesianism
• Inductive reasoning can be represented in
terms of probability theory:
– Probability places coherence constraints on
rational degrees of belief
– Inductive reasoning proceeds through a
probabilistic rule of inference called
conditionalization
• A rule of reasoning is justified if it passes
the “pragmatic self-defeat” test (Dutch
Book)
What is probability?
• Several interpretations
• Logical interpretation
– Probability: approximation of deductive entailment/
truth
• Frequency interpretation
– Probability: how often frequently something happens
• Propensity interpretation
– Probability: a physical system’s disposition to produce
a certain outcome
• Subjective interpretation = Bayesian
– Probability: Degree of belief
Coherence
• Property of a synchronic set of beliefs
– Synchronic set of beliefs = beliefs held at one time
• Ex. A theory
• Criterion of rationality
– Coherent set of beliefs = rational
– Incoherent set of beliefs = irrational
wagering policies (‘pragmatic self-defeat’) in the
long run
Conditionalization
• Bayes Theorem: P(h/e) = probability of h given e
– Ex. P (There is fire / There is smoke) = 0.9 means it’s
generally true that where there’s smoke, there’s fire.
• Simple Rule of Conditionalization: Pf(h) = Pi(h/e)
– Pi = prior probability = the degree of belief that one
has in a claim prior to collecting evidence e
– Pf = posterior probability = the degree of belief that
one has in a claim after collecting evidence e
Conditionalization: “3 Stages”
• Throw yourself into a situation; assign
prior probabilities
• Gather evidence; assign a posterior
probability of 1 to your evidence
• Conditionalize to find posterior of
hypothesis
Conditionalization: “Stage 1”
• Stage 1: You are thrown into a world with
degrees of belief in various claims (priors),
including:
– Pi(h/e) = probability a hypothesis is true given
a certain piece of evidence
– Ex. For an early Australian explorer
• Pi (Platypuses exist / Platypuses are observed) =
0.9
Conditionalization: “Stage 2”
• Stage 2: Gather the evidence. For all
evidence gathered, assign a posterior
probability Pf(e) = 1 (i.e., you are now
certain that e is true)
– Ex. You observe a platypus. Pf(Platypuses
are observed) = 1
Conditionalization: “Stage 3”
• Ascertain how well this new evidence
confirms your hypothesis using the rule of
conditionalization.
• Pf(h) = Pi(h/e)
• Ex. Pf (Platypuses exist) = Pi (Platypuses
exist / Platypuses are observed) = 0.9
Conditionalization and Modus
Ponens
• Modus Ponens: Deductive Rule of Inference:
– If e, then h
–e
– Therefore h
• Conditionalization: Inductive Rule of Inference
– If e, then probably h, i.e., Pi(e/h) = p.
– e, i.e., Pf(e) = 1.
– Therefore probably h, i.e., Pf(h) = Pi(e/h) = p.
Bayesian Confirmation
• Pf(h) = Pi(h/e) = Pi(e/h) x Pi(h) / Pi(e)
– Corollary of Simple Conditionalization
• Admits of degrees of confirmation
• Three stages still obtain:
– Throw yourself into a situation; assign prior
probabilities
– Gather evidence; assign a posterior
probability of 1 to your evidence
– Conditionalize to find posterior of hypothesis
Bayesian confirmation: Stage 1
• More priors to be assigned:
• Pi(e/h) = probability of evidence e given
the truth of the hypothesis h
– Called the likelihood of h on e
– The converse of the prior examined in Simple
Conditionalization
• Pi(e) = probability that evidence obtains
• Pi(h) = probability that hypothesis is true
Bayesian confirmation:
Stages 2 & 3
• Stage 2 = Gather evidence, assign Pf(e) =
1
– Just as with Simple Conditionalization
• Stage 3: Calculate Pf(h) = Pi(e/h)P(h)/Pi(e)
– Like Simple Conditionalization, but more
interesting
– Less like Modus Ponens
The Platypus Example
• Stage 1: Assign priors
– Pi(Platypus observed / Platypus exists) = .6
– Pi(Platypus observed) = .4
– Pi(Platypus exists) = .5
• Stage 2: Gather evidence
– Pf(Platypus observed) = 1
• Stage 3: Calculate posterior for hypothesis
– Pf(Platypus exists) = (.6)(.5)/(.4) = .75
Some Bayesian truisms
• Evidence e confirms hypothesis h if and only if
Pi(h/e) > Pi(h)
• Evidence e disconfirms hypothesis h if and only
if Pi(h/e) < Pi(h)
• If h deductively entails e, then e confirms h and
~e disconfirms h (by reducing Pf(h) = 0)
– Echoes HD and PF
• Comparing hypotheses:
– Pf(ha)/Pf(hb) = [Pi(e/ha) x Pi(ha)] / [Pi(e/hb) x Pi(hb)
Intuitive consequences of
Bayesianism
• The higher the likelihood, the better
confirmed the hypothesis
• The higher the prior of the hypothesis, the
better confirmed the hypothesis
– Gives higher confirmation to hypotheses that
• The lower the prior of the evidence, the
better confirmed the hypothesis
– Privileges bold predictions
Bayesian virtues
• Demarcation
• Induction
– Underdetermination
– Grue
• Induction and Decision
• Vagueness
Bayesian Demarcation
• Both laypeople and scientists can be
treated as Bayesians
• Differences between laypeople and
scientists:
– Evidence
– Testing
– How priors are assigned (possibly)
Bayesians vs. Underdetermination
• Underdetermination of theory by data:
Many hypotheses can entail the same set
of data (such hypotheses are empirically
equivalent)
• However, not all of them are equally
probable given that data
• Bayesian solution: difference in likelihoods
• Pi(non-black, non-raven / all ravens are
black) would be low
• Pi(black raven / all ravens are black) would
be high
• So non-black, non-ravens don’t have
equal confirmatory power for the
hypothesis that all ravens are black.
Bayesian Holism
• Scientific testing consists of a complex
conjunction of theoretical (“core”) and auxiliary
hypotheses about the workings of instruments,
measurements, definitions of concepts, etc.
• Theoretical and auxiliary hypotheses (even
empirical statements) can be:
– Probabilistically dependent upon each other
– Of differing probabilities
• Thus, an experiment may more strongly
disconfirm one statement than another
Bayesians vs. Grue
• Let h1= All emeralds are green.
• Let h2= All emeralds are grue, i.e., green
now, but will turn blue on May 10, 3776
(my 2000th birthday!)
• Why choose h1 over h2?
• The higher the prior of the hypothesis, the
better confirmed the hypothesis.
– Our prior for h1 is higher than our gruesome
h2 hypothesis.
Bayesian Precision
• Popper, Kuhn, and Thagard all suffered
from vagueness in their core concepts