Essential Probability & Statistics (Lecture for CS397-CXZ Algorithms in Bioinformatics) Jan. 23, 2004 ChengXiang Zhai Department of Computer Science University of Illinois, Urbana-Champaign

Purpose of Prob. & Statistics Deductive vs. Plausible reasoning Incomplete knowledge -> uncertainty How do we quantify inference under uncertainty? –Probability: models of random process/experiments (how data are generated) –Statistics: draw conclusions on the whole population based on samples (inference on data)

Basic Concepts in Probability Sample space: all possible outcomes, e.g., –Tossing 2 coins, S ={HH, HT, TH, TT} Event: E  S, E happens iff outcome in E, e.g., –E={HH} (all heads) –E={HH,TT} (same face) Probability of Event : 1  P(E)  0, s.t. –P(S)=1 (outcome always in S) –P(A  B)=P(A)+P(B) if (A  B)= 

Basic Concepts of Prob. (cont.) Conditional Probability :P(B|A)=P(A  B)/P(A) –P(A  B) = P(A)P(B|A) =P(B)P(B|A) –So, P(A|B)=P(B|A)P(A)/P(B) –For independent events, P(A  B) = P(A)P(B), so P(A|B)=P(A) Total probability: If A 1, …, A n form a partition of S, then –P(B)= P(B  S)=P(B  A 1 )+…+P(B  A n ) –So, P(A i |B)=P(B|A i )P(A i )/P(B) (Bayes’ Rule)

Interpretation of Bayes’ Rule Hypothesis space: H={H 1, …, H n }Evidence: E If we want to pick the most likely hypothesis H*, we can drop p(E) Posterior probability of H i Prior probability of H i Likelihood of data/evidence if H i is true

Random Variable X: S   (“measure” of outcome) Events can be defined according to X –E(X=a) = {s i |X(s i )=a} –E(X  a) = {s i |X(s i )  a} So, probabilities can be defined on X –P(X=a) = P(E(X=a)) –P(a  X) = P(E(a  X)) (f(a)=P(a>x): cumulative dist. func) Discrete vs. continuous random variable (think of “partitioning the sample space”)

An Example Think of a DNA sequence as results of tossing a 4- face die many times independently P(AATGC)=p(A)p(A)p(T)p(G)p(C) A model specifies {p(A),p(C), p(G),p(T)}, e.g., all 0.25 (random model M0) P(AATGC|M0) = 0.25*0.25*0.25*0.25*0.25 Comparing 2 models –M1: coding regions –M2: non-coding regions –Decide if AATGC is more likely a coding region

Probability Distributions Binomial: Times of successes out of N trials Gaussian: Sum of N independent R.V.’s Multinomial: Getting n i occurrences of outcome i

Parameter Estimation General setting: –Given a (hypothesized & probabilistic) model that governs the random experiment –The model gives a probability of any data p(D|  ) that depends on the parameter  –Now, given actual sample data X={x 1,…,x n }, what can we say about the value of  ? Intuitively, take your best guess of  -- “best” means “best explaining/fitting the data” Generally an optimization problem

Maximum Likelihood Estimator Data: a sequence d with counts c(w 1 ), …, c(w N ), and length |d| Model: multinomial M with parameters {p(w i )} Likelihood: p(d|M) Maximum likelihood estimator: M=argmax M p(d|M) We’ll tune p(w i ) to maximize l(d|M) Use Lagrange multiplier approach Set partial derivatives to zero ML estimate

Maximum Likelihood vs. Bayesian Maximum likelihood estimation –“Best” means “data likelihood reaches maximum” –Problem: small sample Bayesian estimation –“Best” means being consistent with our “prior” knowledge and explaining data well –Problem: how to define prior?

Bayesian Estimator ML estimator: M=argmax M p(d|M) Bayesian estimator: – First consider posterior: p(M|d) =p(d|M)p(M)/p(d) – Then, consider the mean or mode of the posterior dist. p(d|M) : Sampling distribution (of data) P(M)=p(  1,…,  N ) : our prior on the model parameters conjugate = prior can be interpreted as “extra”/“pseudo” data Dirichlet distribution is a conjugate prior for multinomial sampling distribution “extra”/“pseudo” counts e.g.,  i =  p(w i |REF)

Dirichlet Prior Smoothing (cont.) Posterior distribution of parameters: The predictive distribution is the same as the mean: Bayesian estimate (|d|  ?)

Illustration of Bayesian Estimation Prior: p(  ) Likelihood: p(D|  ) D=(c 1,…,c N ) Posterior: p(  |D)  p(D|  )p(  )    : prior mode  ml : ML estimate  : posterior mode

Basic Concepts in Information Theory Entropy: Measuring uncertainty of a random variable Kullback-Leibler divergence: comparing two distributions Mutual Information: measuring the correlation of two random variables

Entropy Entropy H(X) measures the average uncertainty of random variable X Example: Properties: H(X)>=0; Min=0; Max=log M; M is the total number of values

Interpretations of H(X) Measures the “amount of information” in X –Think of each value of X as a “message” –Think of X as a random experiment (20 questions) Minimum average number of bits to compress values of X –The more random X is, the harder to compress A fair coin has the maximum information, and is hardest to compress A biased coin has some information, and can be compressed to <1 bit on average A completely biased coin has no information, and needs only 0 bit

Cross Entropy H(p,q) What if we encode X with a code optimized for a wrong distribution q? Expected # of bits=? Intuitively, H(p,q)  H(p), and mathematically,

Kullback-Leibler Divergence D(p||q) What if we encode X with a code optimized for a wrong distribution q? How many bits would we waste? Properties: - D(p||q)  0 - D(p||q)  D(q||p) - D(p||q)=0 iff p=q KL-divergence is often used to measure the distance between two distributions Interpretation: -Fix p, D(p||q) and H(p,q) vary in the same way -If p is an empirical distribution, minimize D(p||q) or H(p,q) is equivalent to maximizing likelihood Relative entropy

Cross Entropy, KL-Div, and Likelihood Likelihood: log Likelihood: Criterion for estimating a good model

Mutual Information I(X;Y) Comparing two distributions: p(x,y) vs p(x)p(y) Conditional Entropy: H(Y|X) Properties: I(X;Y)  0; I(X;Y)=I(Y;X); I(X;Y)=0 iff X & Y are independent Interpretations: - Measures how much reduction in uncertainty of X given info. about Y - Measures correlation between X and Y

What You Should Know Probability concepts: –sample space, event, random variable, conditional prob. multinomial distribution, etc Bayes formula and its interpretation Statistics: Know how to compute maximum likelihood estimate Information theory concepts: –entropy, cross entropy, relative entropy, conditional entropy, KL-div., mutual information, and their relationship