Probability and Inference

Transcription

1 Deniz Yuret ECOE 554 Lecture 3

2 Outline 1 Probabilities and ensembles 2 3

3 Ensemble An ensemble X is a triple (x, A X, P X ), where the outcome x is the value of a random variable, which takes on one of a set of possible values, A X = {a 1, a 2,..., a i,..., a I }, having probabilities P X = {p 1, p 2,..., p I }, with P(x = a i ) = p i, p i 0 and a i A X P(x = a i ) = 1.

4 Probability of a subset Probability of a subset. If T is a subset of A X then: P(T ) = P(x T ) = a i T P(x = a i ).

5 Joint ensemble A joint ensemble XY is an ensemble in which each outcome is an ordered pair x, y with x A X = {a 1,..., a I } and y A Y = {b 1,..., b J }. We call P(x, y) the joint probability of x and y.

6 Marginal probability Marginal probability. We can obtain the marginal probability P(x) from the joint probability P(x, y) by summation: P(x = a i ) y A Y P(x = a i, y).

7 Conditional probability P(x = a i y = b j ) P(x = a i, y = b j ) P(y = b j ) if P(y = b j ) 0.

8 Product rule Product rule obtained from the definition of conditional probability: P(x, y H) = P(x y, H)P(y H) = P(y x, H)P(x H). This rule is also known as the chain rule.

9 Sum rule Sum rule a rewriting of the marginal probability definition: P(x H) = y = y P(x, y H) P(x y, H)P(y H).

10 Bayes theorem Bayes theorem obtained from the product rule: P(y x, H) = = P(x y, H)P(y H) P(x H) P(x y, H)P(y H) y P(x y, H)P(y H).

11 Independence Independence Two random variables X and Y are independent (sometimes written X Y ) if and only if P(x, y) = P(x)P(y).

12 The Cox axioms Notation Let the degree of belief in proposition x be denoted by B(x). The negation of x (not-x) is written x. The degree of belief in a conditional proposition, x, assuming proposition y to be true, is represented by B(x y).

13 The Cox axioms Axiom 1 Degrees of belief can be ordered; if B(x) is greater than B(y), and B(y) is greater than B(z), then B(x) is greater than B(z). [Consequence: beliefs can be mapped onto real numbers.] Axiom 2 The degree of belief in a proposition x and its negation x are related. There is a function f such that B(x) = f [B(x)]. Axiom 3 The degree of belief in a conjunction of propositions x, y (x and y) is related to the degree of belief in the conditional proposition x y and the degree of belief in the proposition y. There is a function g such that B(x, y) = g [B(x y), B(y)].

14 The Cox axioms If a set of beliefs satisfy these axioms then they can be mapped onto probabilities satisfying P(false) = 0, P(true) = 1, 0 P(x) 1, and the rules of probability: and P(x) = 1 P(x), P(x, y) = P(x y)p(y).

15 Forward probabilities Example: An urn contains K balls, of which B are black and W = K B are white. Fred draws a ball at random from the urn and replaces it, N times. 1 What is the probability distribution of the number of times a black ball is drawn, n B? 2 What is the expectation of n B? What is the variance of n B? What is the standard deviation of n B? Give numerical answers for the cases N = 5 and N = 400, when B = 2 and K = 10.

16 Inverse probabilities - example Example: There are eleven urns labelled by u {0, 1, 2,..., 10}, each containing ten balls. Urn u contains u black balls and 10 u white balls. Fred selects an urn u at random and draws N times with replacement from that urn, obtaining n B blacks and N n B whites. Fred s friend, Bill, looks on. If after N = 10 draws n B = 3 blacks have been drawn, what is the probability that the urn Fred is using is urn u, from Bill s point of view? (Bill doesn t know the value of u.)

17 Inverse probabilities - example Joint probability of u and n B after N = 10 draws.

18 Inverse probabilities - example Conditional probability of u given n B = 3 after N = 10 draws.

19 Inverse probabilities - terminology P(u n B, N) = P(n B u, N)P(u) P(n B N) P(θ D, H) = posterior = P(D θ, H)P(θ H) P(D H) likelihood x prior evidence

20 Inference as inverse probability Example: Bill tosses a bent coin N times, obtaining a sequence of heads and tails. We assume that the coin has a probability f H of coming up heads; we do not know f H. If n H heads have occurred in N tosses, what is the probability distribution of f H? (For example, N might be 10, and n H might be 3; or, after a lot more tossing, we might have N = 300 and n H = 29.) What is the probability that the N +1th outcome will be a head, given n H heads in N tosses?

21 The likelihood principle The likelihood principle: given a generative model for data d given parameters θ, P(d θ), and having observed a particular outcome d 1, all inferences and predictions should depend only on the function P(d 1 θ).

22 The likelihood principle - example 1 Urn A contains three balls: one black, and two white; urn B contains three balls: two black, and one white. One of the urns is selected at random and one ball is drawn. The ball is black. What is the probability that the selected urn is urn A? 2 Urn A contains five balls: one black, two white, one green and one pink; urn B contains five hundred balls: two hundred black, one hundred white, 50 yellow, 40 cyan, 30 sienna, 25 green, 25 silver, 20 gold, and 10 purple. [One fifth of A s balls are black; two-fifths of B s are black.] One of the urns is selected at random and one ball is drawn. The ball is black. What is the probability that the urn is urn A?