Basic Statistics and Probability Theory

Transcription

1 0. Basic Statistics and Probability Theory Based on Foundations of Statistical NLP C. Manning & H. Schütze, ch. 2, MIT Press, 2002 Probability theory is nothing but common sense reduced to calculation. Pierre Simon, Marquis de Laplace ( )

2 1. PLAN 1. Elementary Probability Notions: Sample Space, Event Space, and Probability Function Conditional Probability Bayes Theorem Independence of Probabilistic Events 2. Random Variables: Discrete Variables and Continuous Variables Mean, Variance and Standard Deviation Standard Distributions Joint, Marginal and and Conditional Distributions Independence of Random Variables

3 PLAN (cont d) Limit Theorems Laws of Large Numbers Central Limit Theorems 4. Estimating the parameters of probabilistic models from data Maximum Likelihood Estimation (MLE) Maximum A Posteriori (MAP) Estimation 5. Elementary Information Theory Entropy; Conditional Entropy; Joint Entropy Information Gain / Mutual Information Cross-Entropy Relative Entropy / Kullback-Leibler (KL) Divergence Properties: bounds, chain rules, (non-)symmetries, properties pertaining to independence

4 1. Elementary Probability Notions 3. sample space: Ω (either discrete or continuous) event: A Ω the certain event: Ω the impossible event: event space: F = 2 Ω (or a subspace of 2 Ω that contains and is closed under complement and countable union) probability function/distribution: P : F [0, 1] such that: P(Ω) = 1 the countable additivity property: A 1,...,A k disjoint events, P( A i ) = P(A i ) Consequence: for a uniform distribution in a finite sample space: P(A) = #favorable events #all events

5 4. P(A B) = P(A B) P(B) Conditional Probability Note: P(A B) is called the a posteriory probability of A, given B. The multiplication rule: P(A B) = P(A B)P(B) = P(B A)P(A) The chain rule: P(A 1 A 2... A n ) = P(A 1 )P(A 2 A 1 )P(A 3 A 1,A 2 )...P(A n A 1,A 2,...,A n 1 )

6 5. The total probability formula: P(A) = P(A B)P(B)+P(A B)P( B) More generally: if A B i and i j B i B j =, then P(A) = i P(A B i)p(b i ) Bayes Theorem: P(B A) = or P(B A) = or... P(A B) P(B) P(A) P(A B) P(B) P(A B)P(B)+P(A B)P( B)

7 6. Independence of Probabilistic Events Independent events: P(A B) = P(A)P(B) Note: When P(B) 0, the above definition is equivalent to P(A B) = P(A). Conditionally independent events: P(A B C) = P(A C)P(B C), assuming, of course, that P(C) 0. Note: When P(B C) 0, the above definition is equivalent to P(A B,C) = P(A C).

8 7. Let Ω be a sample space, and P : 2 Ω [0,1] a probability function. 2. Random Variables 2.1 Basic Definitions A random variable of distribution P is a function X : Ω R n For now, let us consider n = 1. The cumulative distribution function of X is F : R [0, ) defined by F(x) = P(X x) = P({ω Ω X(ω) x})

9 2.2 Discrete Random Variables 8. Definition: Let P : 2 Ω [0,1] be a probability function, and X be a random variable of distribution P. If Val(X) is either finite or unfinite countable, then X is called a discrete random variable. For such a variable we define the probability mass function (pmf) p : R [0,1] as p(x) not. = p(x = x) def. = P({ω Ω X(ω) = x}). (Obviously, it follows that x i Val(X) p(x i) = 1.) Mean, Variance, and Standard Deviation: Expectation / mean of X: E(X) not. = E[X] = xxp(x) if X is a discrete random variable. Variance of X: Var(X) not. = Var[X] = E((X E(X)) 2 ). Standard deviation: σ = Var(X). Covariance of X and Y, two random variables of distribution P: Cov(X,Y) = E[(X E[X])(Y E[Y])]

10 Exemplification: the Binomial distribution: b(r;n,p) = C r n pr (1 p) n r for r = 1,...,n mean: np, variance: np(1 p) the Bernoulli distribution: b(r; 1, p) mean: p, variance: p(1 p), entropy: plog 2 p (1 p)log 2 (1 p) 9. Binomial probability mass function Binomial cumulative distribution function b(r ; n, p) p = 0.5, n = 20 p = 0.7, n = 20 p = 0.5, n = 40 F(r) p = 0.5, n = 20 p = 0.7, n = 20 p = 0.5, n = r r

11 2.3 Continuous Random Variables Definitions: Let P : 2 Ω [0,1] be a probability function, and X : Ω R be a random variable of distribution P. If Val(X) is unfinite non-countable set, and F, the cumulative distribution function of X is continuous, then X is called a continuous random variable. (It follows, naturally, that P(X = x) = 0, for all x R.) If there exists p : R [0, ) such that F(x) = x p(t)dt, then X is called absolutely continuous. In such a case, p is called the probability density function (pdf) of X. For B R for which p(x)dx exists, B P(X 1 (B)) = p(x)dx, B where X 1 (B) not. = {ω Ω X(ω) B}. In particular, + p(x)dx = 1. Expectation / mean of X: E(X) not. = E[X] = xp(x)dx. 10.

12 11. Exemplification: Normal (Gaussean) distribution: N(x;µ,σ) = 1 2πσ e (x µ)2 2σ 2 mean: µ, variance: σ 2 Standard Normal distribution: N(x; 0, 1) Remark: For n,p such that np(1 p) > 5, the Binomial distributions can be approximated by Normal distributions.

13 12. Gaussian probability density function Gaussian cumulative distribution function N µ,σ 2(X=x) µ = 0, σ = 0.2 µ = 0, σ = 1.0 µ = 0, σ = 5.0 µ = 2, σ = 0.5 φ µ,σ 2(x) µ = 0, σ = 0.2 µ = 0, σ = 1.0 µ = 0, σ = 5.0 µ = 2, σ = x x

14 2.4 Basic Properties of Random Variables 13. Let P : 2 Ω [0,1] be a probability function, X : Ω R n be a random discrete/continuous variable of distribution P. If g : R n R m is a function, then g(x) is a random variable. If g(x) is discrete, then E(g(X)) = x g(x)p(x). If g(x) is continuous, then E(g(X)) = g(x)p(x)dx. If g is non-linear E(g(X)) = g(e(x)). E(aX +b) = ae(x)+b. E(X +Y) = E(X)+E(Y), therefore E[ n i=1 a ix i ] = n i=1 a ie[x i ]. Var(aX) = a 2 Var(X). Var(X +a) = Var(X). Var(X) = E(X 2 ) E 2 (X). Cov(X,Y) = E[XY] E[X]E[Y].

15 2.5 Joint, Marginal and Conditional Distributions Exemplification for the bi-variate case: 14. Let Ω be a sample space, P : 2 Ω [0,1] a probability function, and V : Ω R 2 be a random variable of distribution P. One could naturally see V as a pair of two random variables X : Ω R and Y : Ω R. (More precisely, V(ω) = (x,y) = (X(ω),Y(ω)).) the joint pmf/pdf of X and Y is defined by p(x,y) not. = p X,Y (x,y) = P(X = x,y = y) = P(ω Ω X(ω) = x,y(ω) = y). the marginal pmf/pdf functions of X and Y are: for the discrete case: p X (x) = y p(x,y), p Y(y) = x p(x,y) for the continuous case: p X (x) = y p(x,y)dy, p Y(y) = x p(x,y)dx the conditional pmf/pdf of X given Y is: p X Y (x y) = p X,Y(x,y) p Y (y)

16 2.6 Independence of Random Variables 15. Definitions: Let X,Y be random variables of the same type (i.e. either discrete or continuous), and p X,Y their joint pmf/pdf. X and Y are said to be independent if p X,Y (x,y) = p X (x) p Y (y) for all possible values x and y of X and Y respectively. Similarly, let X,Y and Z be random variables of the same type, and p their joint pmf/pdf. X and Y are conditionally independent given Z if p X,Y Z (x,y z) = p X Z (x z) p Y Z (y z) for all possible values x,y and z of X,Y and Z respectively.

17 16. Properties of random variables pertaining to independence If X,Y are independent, then Var(X +Y) = Var(X) + Var(Y). If X,Y are independent, then E(XY) = E(X)E(Y), i.e. Cov(X,Y) = 0. Cov(X,Y) = 0 X,Y are independent. The covariance matrix corresponding to a vector of random variables is symmetric and positive semi-definite. If the covariance matrix of a multi-variate Gaussian distribution is diagonal, then the marginal distributions are independent.

18 Limit Theorems [ Sheldon Ross, A first course in probability, 5th ed., 1998 ] The most important results in probability theory are limit theorems. Of these, the most important are... laws of large numbers, concerned with stating conditions under which the average of a sequence of random variables converges (in some sense) to the expected average; central limit theorems, concerned with determining the conditions under which the sum of a large number of random variables has a probability distribution that is approximately normal.

19 Two basic inequalities and the weak law of large numbers 18. Markov s inequality: If X is a random variable that takes only non-negative values, then for any value a > 0, P(X a) E[X] a Chebyshev s inequality: If X is a random variable with finite mean µ and variance σ 2, then for any value k > 0, P( X µ k) σ2 k 2 The weak law of large numbers (Bernoulli; Khintchine): Let X 1,X 2,...,X n be a sequence of independent and identically distributed random variables, each having a finite mean E[X i ] = µ. Then, for any value ǫ > 0, ( X X n P n ) µ ǫ 0 as n

20 19. The central limit theorem for i.i.d. random variables [ Pierre Simon, Marquis de Laplace; Liapunoff in ] Let X 1,X 2,...,X n be a sequence of independent random variables, each having mean µ and variance σ 2. Then the distribution of X X n nµ σ n tends to be the standard normal (Gaussian) as n. That is, for < a <, ( X X n nµ P σ n ) a 1 2π a e x2 /2 dx as n

21 The central limit theorem for independent random variables 20. Let X 1,X 2,...,X n be a sequence of independent and identically distributed random variables having respective means µ i and variances σ 2 i. If (a) the variables X i are uniformly bounded, i.e. for some M R + P( X i < M) = 1 for all i, and (b) i=1 σ2 i =, then P ( n i=1 (X i µ i ) ) n i=1 σ2 i a Φ(a) as n where Φ is the cumulative distribution function for the standard normal (Gaussian) distribution.

22 21. The strong law of large numbers Let X 1,X 2,...,X n be a sequence of independent and identically distributed random variables, each having a finite mean E[X i ] = µ. Then, with probability 1, X X n n µ as n That is, P ( ) lim (X X n )/n = µ n = 1

23 Other inequalities One-sided Chebyshev inequality: If X is a random variable with mean 0 and finite variance σ 2, then for any a > 0, P(X a) σ2 σ 2 +a Corollary: If E[X] = µ, Var(X) = σ 2, then for a > 0 P(X µ+a) σ2 σ 2 +a 2 Chernoff bounds: Let M(t) not = E[e tx ]. Then P(X µ a) σ2 σ 2 +a 2 P(X a) e ta M(t) for all t > 0 P(X a) e ta M(t) for all t < 0

24 Estimation/inference of the parameters of probabilistic models from data (based on [Durbin et al, Biological Sequence Analysis, 1998], p , ) A probabilistic model can be anything from a simple distribution to a complex stochastic grammar with many implicit probability distributions. Once the type of the model is chosen, the parameters have to be inferred from data. We will first consider the case of the categorical distribution, and then we will present the different strategies that can be used in general.

25 A case study: Estimation of the parameters of a categorical distribution from data 24. Assume that the observations for example, when rolling a die about which we don t know whether it is fair or not, or when counting the number of times the amino acid i occurs in a column of a multiple sequence alignment can be expressed as counts n i for each outcome i (i = 1,l...,K), and we want to estimate the probabilities θ i of the underlying distribution. Case 1: When we have plenty of data, it is natural to use the maximum likelihood (ML) solution, i.e. the observed frequency θi ML = n i not. = n i j n j N. Note: it is easy to show that indeed P(n θ ML ) > P(n θ) for any θ θ ML. log P(n θml ) P(n θ) = log Π i(θi ML ) n i Π i θ n i i = i n i log θml i = N θ i i θ ML i log θml i > 0 θ i The inequality follows from the fact that the relative entropy is always positive except when the two distributions are identical.

26 Case 2: When the data is scarce, it is not clear what is the best estimate. In general, we should use prior knowledge, via Bayesian statistics. For instance, one can use the Dirichlet distribution with parameters α. 25. P(θ n) = P(n θ)d(θ α) P(n) It can be shown (see calculus on R. Durbin et. al. BSA book, pag. 320) that the posterior mean estimation (PME) of the parameters is θ PME i def. = θp(θ n)dθ = n i +α i N + j α j The α s are like pseudocounts added to the real counts. (If we think of the α s as extra observations added to the real ones, this is precisely the ML estimate!) This makes the Dirichlet regulariser very intuitive. How to use the pseudocounts: If it is fairly obvious that a certain residue, let s say i, is very common, than we should give it a very high pseudocount α i ; if the residue j is generaly rare, we should give it a low pseudocount.

27 Strategies to be used in the general case A. The Maximum Likelihood (ML) Estimate When we wish to infer the parameters θ = (θ i ) for a model M from a set of data D, the most obvious strategy is to maximise P(D θ,m) over all possible values of θ. Formally: 26. θ ML = argmax θ P(D θ,m) Note: Generally speaking, when we treat P(x y) as a function of x (and y is fixed), we refer to it as a probability. When we treat P(x y) as a function of y (and x is fixed), we call it a likelihood. Note that a likelihood is not a probability distribution or density; it is simply a function of the variable y. A serious drawback of maximum likelihood is that it gives poor results when data is scarce. The solution then is to introduce more prior knowledge, using Bayes theorem. (In the Bayesian framework, the parameters are themselves seen as random variables!)

28 B. The Maximum A posteriori Probability (MAP) Estimate 27. MAP def. θ = argmax θ = argmax θ P(θ D,M) = argmax θ P(D θ,m)p(θ M) P(D θ,m)p(θ M) P(D M) The prior probability P(θ M) has to be chosen in some reasonable manner, and this is the art of Bayesian estimation (although this freedom to choose a prior has made Bayesian statistics controversial at times...). C. The Posterior Mean Estimator (PME) θ PME = θp(θ D,M)dθ where the integral is over all probability vectors, i.e. all those that sum to one. D. Yet another solution is to use the posterior probability P(θ D,M) to sample from it (see [Durbin et al, 1998], section 11.4) and thereby locate regions of high probability for the model parameters.

29 5. Elementary Information Theory Definitions: Let X and Y be discrete random variables. Entropy: H(X) def. = 1 xp(x) log p(x) = x p(x)logp(x) = E p[ logp(x)]. Convention: if p(x) = 0 then we shall consider p(x)logp(x) = 0. Specific Conditional entropy: H(Y X = x) def. = y Y p(y x)logp(y x). Average conditional entropy: H(Y X) def. = imed. x X p(x)h(y X = x) = x X y Y p(x,y)logp(y x). Joint entropy: H(X,Y) def. = x,yp(x,y) logp(x,y)dem. = H(X)+H(Y X) dem. = H(Y)+H(X Y). Information gain (or: Mutual information): 28. IG(X;Y) def. = H(X) H(X Y) imed. = H(Y) H(Y X) imed. = H(X,Y) H(X Y) H(Y X) = IG(Y;X).

30 Exemplification: Entropy of a Bernoulli Distribution 29. H(p) = plog 2 p (1 p)log 2 (1 p) p

31 Basic properties of Entropy, Conditional Entropy, Joint Entropy and Information Gain / Mutual Information 30. ( 1 0 H(p 1,...,p n ) H n,..., 1 ) = logn; n H(X) = 0 iff X is a constant random variable. IG(X;Y) 0; IG(X;Y) = 0 iff X and Y are independent. H(X Y) H(X) H(X Y) = H(X) iff X and Y are independent; H(X,Y) H(X)+H(Y); H(X,Y) = H(X)+H(Y) iff X and Y are independent. a chain rule: H(X 1,...,X n ) = H(X 1 )+H(X 2 X 1 )+...+H(X n X 1,...,X n 1 ).

32 The Relationship between Entropy, Conditional Entropy, Joint Entropy and 31. Mutual Information H(X,Y) H(X Y) IG(X,Y) H(Y X) H(X) H(Y)

33 Other definitions 32. Let X be a discrete random variable, p its pmf and q another pmf (usually a model of p). Cross-entropy: CH(X,q) = x X p(x)logq(x) = E p [log ] 1 q(x) Let X and Y be discrete random variables, and p and q their respective pmf s. Relative entropy (or, Kullback-Leibler divergence): KL(p q) = x X p(x)log q(x) p(x) = E p = CH(X,q) H(X). [ log p(x) q(x) ]

34 33. The Relationship between Entropy, Conditional Entropy, Joint Entropy, Information Gain, Cross-Entropy and Relative Entropy (or KL divergence) CH(X, p ) H(X,Y) CH(Y, p ) Y X KL(p p ) X Y H(X Y) H(Y X) KL(p p ) Y X H(p ) X IG(X,Y) = KL(p p p ) XY X Y H(p ) Y

35 CH(X,q) 0 Basic properties of cross-entropy and relative entropy KL(p q) 0 for all p and q; KL(p q) = 0 iff p and q are identical. [Consequence:] If X is a discrete random variable, p its pmf, and q another pmf, then CH(X,q) H(X) 0. The first of these two inequations is also known as Gibbs inequation: n i=1 p ilog 2 p i i p ilog 2 q i. Unlike H of a discrete n-ary variable, which is bounded by log 2 n, there is no (general) upper bound for CH. (However, KL is upper-bounded.) Unlike H(X,Y), which is symmetric in its arguments, CH and KL are not! Therefore KL is NOT a distance metric! (See the next slide.) IG(X;Y) = KL(p XY p X p Y ) = ( ) p(x)p(y) x y p(x,y)log. p(x, y) 34.

36 Remark 35. The quantity VI(X,Y) def = H(X,Y) IG(X;Y) = H(X)+H(Y) 2IG(X;Y) = H(X Y)+H(Y X) known as variation of information, is a distance metric, i.e. it is nonengative, symmetric, implies indiscernability, and satisfies the triangle inequality. Consider M(p,q) = 1 2 (p+q). The function JSD(p q) = 1 2 KL(p M)+ 1 KL(q M) is called the Jensen- 2 Shannon divergence. One can prove that JSD(p q) defines a distance metric (the Jensen- Shannon distance).

37 Recommended Exercises From [Manning & Schütze, 2002, ch. 2:] Examples 1, 2, 4, 5, 7, 8, 9 Exercises 2.1, 2.3, 2.4, 2.5 From [Sheldon Ross, 1998, ch. 8:] Examples 2a, 2b, 3a, 3b, 3c, 5a, 5b

38 Addenda: Other Examples of Probabilistic Distributions 37.

39 Multinomial distribution: generalises the binomial distribution to the case where there are K independent outcomes with probabilities θ i, i = 1,...,K such that K i=1 θ i = 1. The probability of getting n i occurrence of outcome i is given by P(n θ) = n! Π K i=1 (n i!) ΠK i=1θ n i i, 38. where n = n n K, and θ = (θ 1,...,θ K ). Note: The particular case n = 1 represents the categorical distribution. This is a generalisation of the Bernoulli distribution. Example: The outcome of rolling a die n times is described by a categorical distribution. The probabilities of each of the 6 outcomes are θ 1,...,θ 6. For a fair die, θ 1 =... = θ 6, and the probability of rolling it 12 times and getting each outcome twice is: 12! (2!) 6 ( ) 12 1 =

40 Poisson distribution (or, Poisson law of small numbers): 39. p(k;λ) = λk k! e λ, with k N and parameter λ > 0. Mean = variance = λ. Poisson probability mass function Poisson cumulative distribution function P(X=k) λ = 1 λ = 4 λ = 10 P(X k) λ = 1 λ = 4 λ = k k

41 40. Exponential distribution (a.k.a. negative exponential distribution): p(x;λ) = λe λx for x 0 and parameter λ > 0. Mean = λ 1, variance = λ 2. Exponential probability density function Exponential cumulative distribution function p(x) λ = 0.5 λ = 1 λ = 1.5 P(X x) λ = 0.5 λ = 1 λ = x x

42 Gamma distribution: p(x;k,θ) = x k 1 e x/θ Mean = kθ, variance = kθ 2. for x 0 and parameters k > 0 (shape) and θ > 0 (scale). Γ(k)θk The gamma function is a generalisation of the factorial function to real values. For any positive real number x, Γ(x+1) = xγ(x). (Thus, for integers Γ(n) = (n 1)!.) 41. Gamma probability density function Gamma cumulative distribution function p(x) k = 1.0, θ = 2.0 k = 2.0, θ = 2.0 k = 3.0, θ = 2.0 k = 5.0, θ = 1.0 k = 9.0, θ = 0.5 k = 7.5, θ = 1.0 k = 0.5, θ = 1.0 P(X x) k = 1.0, θ = 2.0 k = 2.0, θ = 2.0 k = 3.0, θ = 2.0 k = 5.0, θ = 1.0 k = 9.0, θ = 0.5 k = 7.5, θ = 1.0 k = 0.5, θ = x x

43 χ 2 distribution: p(x;ν) = 1 Γ(ν/2) ( 1 2 ) ν/2 x ν/2 1 e 1 2 x for x 0 and ν a positive integer. It is obtained from Gamma distribution by taking k = ν/2 and θ = 2. Mean = ν, variance = 2ν. 42. Chi Squared probability density function Chi Squared cumulative distribution function p(x) k = 1 k = 2 k = 3 k = 4 k = 6 k = 9 P(X x) k = 1 k = 2 k = 3 k = 4 k = 6 k = x x

44 Laplace distribution: p(x;µ,θ) = 1 2θ e µ x θ. Mean = µ, variance = 2θ Laplace probability density function Laplace cumulative density function p(x) µ = 0, θ = 1 µ = 0, θ = 2 µ = 0, θ = 4 µ = 5, θ = 4 P(X x) µ = 0, θ = 1 µ = 0, θ = 2 µ = 0, θ = 4 µ = 5, θ = x x

45 Student s distribution: ( ) ν +1 Γ 2 p(x;ν) = ( ν (1+ νπ Γ 2) x2 ν ) ν +1 2 for x R and ν > 0 (the degree of freedom param.) 44. Mean = 0 for ν > 1, otherwise undefined. Variance = ν for ν > 2, for 1 < ν 2, otherwise undefined. ν 2 The probability density function and the cumulative distribution function: Note [from Wiki]: The t-distribution is symmetric and bell-shaped, like the normal distribution, but it has havier tails, meaning that it is more prone to producing values that fall far from its mean.

46 45. Dirichlet distribution: D(θ α) = 1 Z(α) ΠK i=1 θα i 1 i δ( K i=1 θ i 1) where α = α 1,...,α K with α i > 0 are the parameters, θ i satisfy 0 θ i 1 and sum to 1, this being indicated by the delta function term δ( i θ i 1), and the normalising factor can be expressed in terms of the gamma function: Z(α) = Π K i=1 θα i 1 i δ( i 1)dθ = Π iγ(α i ) Γ( i α i) α i Mean of θ i : j α. j For K = 2, the Dirichlet distribution reduces to the more widely known beta distribution, and the normalising constant is the beta function.

47 46. Remark: Concerning the multinomial and Dirichlet distributions: The algebraic expression for the parameters θ i is similar in the two distributions. However, the multinomial is a distribution over its exponents n i, whereas the Dirichlet is a distribution over the numbers θ i that are exponentiated. The two distributions are said to be conjugate distributions and their close formal relationship leads to a harmonious interplay in many estimation problems. Similarly, the beta distribution is the conjugate of the Bernoulli distribution, and the gamma distribution is the conjugate of the Poisson distribution.