# Introduction to Bayesian statistics - Bolstad M.

ISBN 0-471-27020-2

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• The conditional probability of event B given event A is given by

P (B|A) = ^

• The event B is unobservable. The event A is observable. We could nominally write the conditional probability formula for P(A | B), but the relationship is not

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LOGIC, PROBABILITY, AND UNCERTAINTY

used in that form. We do not treat the events symmetrically. The multiplication rule is the definition of conditional probability cleared of the fraction.

P(A n B) = P(B) x P(A|B).

It is used to assign probabilities to compound events.

• The law of total probability says that given events Bb...,Bn that partition the sample space (universe), and another event A, then

n

P(A) = £ P(Bj n A)

j=i

because probability is additive over the disjoint events, (A n Bi)... (A n Bn). When we find the probability of each of the intersections A n Bj by the multiplication rule, we get

P(A) = £ P(Bj) x P(A|Bj). j

Bayes’ theorem is the key to Bayesian statistics:

P(Bj) x P(A|Bj)

P (Bj|A) =

P(Bj) x P(A|Bj) •

This comes from the definition of conditional probability. The marginal probability of the event A is found by the law of total probability, and each of the joint probabilities is found from the multiplication rule. P(B*) is called the prior probability of event B*, and P(B*|A) is called the posterior probability of event B*.

In the Bayesian universe, the unobservable events B1,...,Bn which partition the universe are horizontal slices, and the observable event A is a vertical slice. The probability P(A) is found by summing the P(A n B*) down the column. Each of the P(A n B*) is found by multiplying the prior P(B*) times the likelihood P(A|B*). So Bayes’ theorem can be summarized by saying the posterior probability is the prior times likelihood divided by the sum of the prior times likelihood.

The Bayesian universe has two dimensions. The sample space forms the observable (horizontal) dimension of the Bayesian universe. The parameter space is the unobservable (vertical) dimension. In Bayesian statistics, the probabilities are defined on both dimensions of the Bayesian universe.

The odds ratio of an event A is the ratio of the probability of the event to the probability of its complement:

odds(A) =

P (A)

EXERCISES 73

If it is found before analyzing the data, it is the prior odds ratio. If it is found after analyzing the data, it is the posterior odds ratio.

• The Bayes factor is the amount of evidence in the data that changes the prior odds to the posterior odds:

prior odds = B x posterior odds .

Exercises

4.1 There are two events A and B. P(A) = .4 and P(B) = .5. The events A and B are independent.

(a) Find P(A).

(b) Find P(A n B).

(c) Find P(A U B).

4.2 There are two events A and B. P(A) = .5 and P(B) = .3. The events A and B are independent.

(a) Find P(A).

(b) Find P(A n B).

(c) Find P(A U B).

4.3 There are two events A and B. P(A) = .4 and P(B) = .4. P(A n B) = .24.

(a) Are A and B independent events? Explain why or why not.

(b) Find P(A U B).

4.4 There are two events A and B. P(A) = .7 and P(B) = .8. P(A n B) = .1.

(a) Are A and B independent events? Explain why or why not.

(b) Find P(A U B).

4.5 A single fair die is rolled. Let the event A be "the face showing is even." Let the event B be "the face showing is divisible by 3."

(a) List out the sample space of the experiment.

(b) List the outcomes in A, and find P(A).

(c) List the outcomes in B, and find P(B).

(d) List the outcomes in A n B, and find P(A n B).

(e) Are the events A and B independent? Explain why or why not.

4.6 Two fair dice, one red and one green, are rolled. Let the event A be "the sum of the faces showing is equal to seven." Let the event B be "the faces showing on the two dice are equal."

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LOGIC, PROBABILITY, AND UNCERTAINTY

(a) List out the sample space of the experiment.

(b) List the outcomes in A, and find P(A).

(c) List the outcomes in B, and find P(B).

(d) List the outcomes in A П B, and find P (A П B).

(e) Are the events A and B independent? Explain why or why not.

(f) How would you describe the relationship between event A and event B?

4.7 Two fair dice, one red and one green, are rolled. Let the event A be "the sum of the faces showing is an even number." Let the event B be "the sum of the faces showing is divisible by 3."

(a) List the outcomes in A, and find P(A).

(b) List the outcomes in B, and find P(B).

(c) List the outcomes in A П B, and find P (A П B).

(d) Are the events A and B independent? Explain why or why not.

4.8 Two dice are rolled. The red die has been loaded. Its probabilities are P(1) = P(2) = P(3) = P(4) = 5 and P(5) = P(6) = ^. The green die is fair. Let the event A be "the sum of the faces showing is an even number." Let the event B be "the sum of the faces showing is divisible by 3."

(a) List the outcomes in A, and find P(A).

(b) List the outcomes in B, and find P(B).

(c) List the outcomes in A П B, and find P (A П B).

(d) Are the events A and B independent? Explain why or why not.

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