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Chapter7.1_Bayesianism_Aprimer-createdfor3rdedition.ppt

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Chapter 7.1:
Bayesianism: A primer

Three key ideas of Bayesianism

Belief comes in degrees.

Degrees of belief can be modeled as probabilities, and so have to confirm to the basic principles of the probability calculus.

Learning takes place by updating probabilities according to Bayes’s Rule.

Cognitive Science  José Luis Bermúdez / Cambridge University Press 2020

Degrees of belief

Belief is often viewed as a two-valued states: true/false

For Bayesians, belief comes in degrees:

Cognitive Science  José Luis Bermúdez / Cambridge University Press 2020

total lack of confidence

completely confident

fairly confident

slightly confident

0.5

Cognitive Science  José Luis Bermúdez / Cambridge University Press 2020

Degrees of beliefs as probabilities

More precise than the language of “not”, “slightly”, ”fairly”, ”completely”.

Obey basic rules of probability calculus

Cognitive Science  José Luis Bermúdez / Cambridge University Press 2020

Rules of probability calculus

Basic rule 1: Probabilities are numbers between 0 and 1

Basic rule 2: All impossible sentences have probability 0

Basic rule 3: All necessary truths (such as “2 + 2 = 4”) have probability 1

Basic rule 4: If sentences P and Q are logically equivalent, then p(P) = p(Q)

Cognitive Science  José Luis Bermúdez / Cambridge University Press 2020

Rules of probability calculus (cont.)

The negation rule
If sentence S has probability p, then its negation not-S (S) has probability 1 – p

The disjunction rule (restricted)
If sentences R and S are mutually exclusive, then the probability of R or S is p(R) + p(S)

The conjunction rule (restricted)
If sentences R and S are independent of each other (i.e. the presence of one does not make the other more likely), then the probability of R and S is p(R)  p(S)

Cognitive Science  José Luis Bermúdez / Cambridge University Press 2020

Conditional probability

p(A/B)

The probability of A, conditional upon B, is the probability that A holds, relative to the assumption that B holds.

E.g. ‘A’ stands for ‘There is thunder’ and ‘B’ for ‘It is raining’, then p(A/B) is the probability that there is thunder, if we assume that it is raining.

Cognitive Science  José Luis Bermúdez / Cambridge University Press 2020

Formal definition

Cognitive Science  José Luis Bermúdez / Cambridge University Press 2020

Bayes’s rule

You have some evidence for a hypothesis.

Question: How strong is the evidence?

-> Given that the evidence holds, how likely is it that the hypothesis is true?

-> p(Hypothesis/evidence) or p(H/E)

Cognitive Science  José Luis Bermúdez / Cambridge University Press 2020

Illustration: a medical case

You test positive for a nasty disease. The hypothesis (H) is that you actually have the disease. The evidence (E) is the positive test.

Suppose you know the frequency of the disease in the population (the prior probability of the hypothesis). p(H)=0.0001

Also suppose you know how reliable the test is, i.e., how likely it is that you will test positive if you actually do have the disease (the likelihood of the evidence). p(E/H)=0.99

Question: How likely is it that you have the disease, given that you have tested positive (the posterior probability of the hypothesis) ? p(H/E)=?

Cognitive Science  José Luis Bermúdez / Cambridge University Press 2020

Formulation of Bayes’s rule (short version)

Posterior probability of the hypothesis =

Likelihood of the evidence  Prior probability of the hypothesis

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Probability of the evidence

For a Bayesian, you update your belief by switching from the prior to the posterior probability of the hypothesis.

Cognitive Science  José Luis Bermúdez / Cambridge University Press 2020

Formulation of Bayes’s rule (long version)

Cognitive Science  José Luis Bermúdez / Cambridge University Press 2020