C. Manning & H. Sch¨ utze, ch. 2, MIT Press, 2002

Basic Statistics and Probability Theory

Based on

“Foundations of Statistical NLP”

C. Manning & H. Sch¨ utze, ch. 2, MIT Press, 2002

“Probability theory is nothing but common sense reduced to calculation.”

Pierre Simon, Marquis de Laplace (1749-1827)

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

PLAN (cont’d)

3.

Limit Theorems

• Laws of Large Numbers

• Central Limit Theorems

4.

Estimating the parameters

• 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

1. Elementary Probability Notions

• sample space: Ω (either discrete or continuous)

• event: A ⊆ Ω

– the certain event: Ω – the impossible event: ∅

– elementary event: any {ω}, where ω ∈ Ω

• 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:

∀A₁, ..., A_k disjoint events, P(∪A_i) = P

P(A_i)

Consequence: for a uniform distribution in a finite sample space:

P(A) = #favorable elementary events

#all elementary events

Conditional Probability

• P(A | B) =

P (A ∩ B) P (B )

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(A₁ ∩ A₂ ∩ . . .∩ A_n) =

P(A₁)P(A₂ | A₁)P(A₃ | A₁, A₂). . . P(A_n | A₁, A₂, . . . , A_n−1)

•

The “total probability” formula:

P (A) = P (A | B)P (B) + P (A | ¬ B )P ( ¬ B)

More generally:

if A ⊆ ∪B_i and ∀i 6= j B_i ∩B_j = ∅, then P(A) = P

i P(A | Bi)P(Bi)

•

Bayes’ Theorem:

P (B | A)

P (A | B) P (B) P (A)

or P(B | A) = P(A | B) P(B)

P(A | B)P(B) + P(A | ¬B)P(¬B) or ...

Independence of Probabilistic Events

• Independent events: P(A∩ B) = P(A)P(B)

Note: When P(B) 6= 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) 6= 0.

Note: When P(B ∩ C) 6= 0, the above definition is equivalent to P(A|B, C) = P(A|C).

2. Random Variables 2.1 Basic Definitions

Let Ω be a sample space, and

P : 2^Ω → [0,1] a probability function.

• A random variable of distribution P is a function X : Ω → Rⁿ

◦ 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})

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 P

xi∈V al(X) p(x_i) = 1.) Mean, Variance, and Standard Deviation:

• Expectation / mean of X: E(X) ^not.= E[X] = P

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 = p

Var(X).

Covariance of X and Y , two random variables of distribution P:

• Cov(X, Y) = E[(X −E[X])(Y −E[Y ])]

Exemplification:

• the Binomial distribution: b(r;n, p) = C_n^r p^r(1 −p)^n−r for r = 0, . . . , n mean: np, variance: np(1 −p)

◦ the Bernoulli distribution: b(r; 1, p)

mean: p, variance: p(1 −p), entropy: −plog₂ p −(1 − p) log₂(1 − p)

0 10 20 30 40

0.000.050.100.150.200.25

r

b(r ; n, p)

0 10 20 30 40

0.000.050.100.150.200.25

Binomial probability mass function

p = 0.5, n = 20 p = 0.7, n = 20 p = 0.5, n = 40

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r

F(r)

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Binomial cumulative distribution function

p = 0.5, n = 20 p = 0.7, n = 20 p = 0.5, n = 40

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) = Rx

−∞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 R

B p(x)dx exists, P(X⁻¹(B)) = R

B p(x)dx, where X⁻¹(B) ^not.= {ω ∈ Ω | X(ω) ∈ B}.

In particular, R+∞

−∞ p(x)dx = 1.

• Expectation / mean of X: E(X) ^not.= E[X] = R

xp(x)dx.

Exemplification:

• Normal (Gaussean) distribution: N(x;µ, σ) = √ ¹

2πσ

e

2

2σ²

mean: µ, variance: σ²

◦ 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.

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x Nµ,σ2(X=x)

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Gaussian probability density function

µ = 0, σ = 0.2 µ = 0, σ = 1.0 µ = 0, σ = 5.0 µ = −2, σ = 0.5

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x φµ,σ2(x)

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Gaussian cumulative distribution function

µ = 0, σ = 0.2 µ = 0, σ = 1.0 µ = 0, σ = 5.0 µ = −2, σ = 0.5

2.4 Basic Properties of Random Variables

Let P : 2^Ω → [0,1] be a probability function,

X : Ω → Rⁿ be a random discrete/continuous variable of distribution P.

• If g : Rⁿ → R^m is a function, then g(X) is a random variable.

If g(X) is discrete, then E(g(X)) = P

xg(x)p(x).

If g(X) is continuous, then E(g(X)) = R

g(x)p(x)dx.

◦ If g is non-linear 6⇒ E(g(X)) = g(E(X)).

• E(aX) = aE(X).

• E(X + Y) = E(X) + E(Y), therefore E[Pn

i=1a_iX_i] = Pn

i=1a_iE[X_i].

◦ Var(aX) = a²Var(X).

◦ Var(X + a) = Var(X).

• Var(X) = E(X²) − E²(X).

• Cov(X, Y) = E[XY ] −E[X]E[Y ].

2.5 Joint, Marginal and Conditional Distributions

Exemplification for the bi-variate case:

Let Ω be a sample space, P : 2^Ω → [0,1] a probability function, and V : Ω → R² 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.= pX,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) = P

y p(x, y), p_Y(y) = P

xp(x, y) for the continuous case:

pX(x) = R

y p(x, y)dy, pY(y) = R

xp(x, y)dx

• the conditional pmf/pdf of X given Y is:

pX|Y(x | y) = p_X,Y(x, y) pY(y)

2.6 Independence of Random Variables

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.

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 6⇒ 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.

3. Limit Theorems

[ Sheldon Ross, A first course in probability, 5th ed., 1998 ]

“The most important results in probability theory are limit theo- rems. 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.”

Two Probability Bounds

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 σ², then for any value a > 0,

P(| X −µ |≥ a) ≤ σ² a².

Note: As Chebyshev’s inequality is valid for all distributions of the random variable X, we cannot expect the bound of the probability to be very close to the actual probability in most cases. (See ex. 2b, pag.

397 in Ross’ book.)

The weak law of large numbers

[ Bernoulli; Khintchine ]

Let X₁, X₂, . . . , 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, P

X₁ + . . . + X_n

n −µ

≥ ǫ

→ 0 as n → ∞.

The central limit theorem for i.i.d. random variables

[ Pierre Simon, Marquis de Laplace; Liapunoff in 1901-1902 ] Let X1, X2, . . . , Xn be a sequence of independent random variables, each having finite mean µ and finite variance σ².

Then the distribution of

X₁ + . . . + X_n −nµ σ √

n

tends to be the standard normal (Gaussian) as n → ∞. That is, for −∞ < a < ∞,

P

X₁ + . . . + X_n − nµ σ √

n ≤ a

→ 1

√2π Z a

−∞

e^−x2/2dx as n → ∞

The central limit theorem

for independent random variables

Let X₁, X₂, . . . , X_n be a sequence of independent random variables having respective means µ_i and variances σ_i².

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) P∞

i=1σ_i² = ∞, then

P

Pn

i=1(X_i −µ_i) pPn

i=1σ_i² ≤ a

!

→ Φ(a) as n → ∞

where Φ is the cumulative distribution function for the standard normal (Gaussian) distribution.

The strong law of large numbers

Let X₁, X₂, . . . , 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,

X₁ + . . . + X_n

n → µ as n → ∞ That is,

P

n→∞lim (X₁ + . . . + X_n)/n = µ

= 1

Other Probability Bounds

One-sided Chebyshev inequality:

If X is a random variable with mean 0 and finite variance σ², then for any a > 0,

P(X ≥ a) ≤ σ² σ² + a² Corollary:

If E[X] = µ, Var(X) = σ², then for a > 0 P(X ≥ µ + a) ≤ σ²

σ² + a² P(X ≤ µ− a) ≤ σ²

σ² + a²

Other Probability Bounds (cont’d)

Chernoff bounds:

If X is a random variable, then M(t) ^not= E[e^tX], is called the moment generating function of X.

It can be shown that

P(X ≥ a) ≤ e^−taM(t) for all t > 0 P(X ≤ a) ≤ e^−taM(t) for all t < 0.

Chernoff bounds for the standard normal distribution:

If Z is a standard normal random variable, then M(t) ^not.= E[e^tX] ^calculus= e^t2/2.

It can be shown that

P(Z ≥ a) ≤ e^−a2/2 for all a > 0 P(Z ≤ a) ≤ e^−a2/2 for all a < 0.

Other Probability Bounds (cont’d)

Hoeffding bounds:

Let X₁, . . . , X_n be some independent random variables, each X_i being bounded by the interval [a_i, b_i].

If X¯ ^not.= _n¹ Pn

i=1 Xi, then it follows that for any t ≥ 0

P( ¯X − E[ ¯X] ≥ t) ≤ exp − 2n²t² Pn

i=1 (b_i − a_i)²

!

P(E[ ¯X] − X¯ ≥ t) ≤ exp

− 2n²t² Pn

i=1(bi − ai)²

⇒ P(|X¯ − E[ ¯X]| ≥ t) ≤ 2 exp

− 2n²t² Pn

i=1(b_i − a_i)²

.

4. Estimation/inference of the parameters of probabilistic models from data

(based on [Durbin et al, Biological Sequence Analysis, 1998], p. 311-313, 319-321)

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 parame- ters 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.

A case study: Estimation of the parameters of a categorical distribution from data

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 align- ment — 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 likeli- hood (ML) solution,

P

j n_j

not.= n_i N. Note: it is easy to show that indeed P(n | θ^ML) > P(n | θ) for any θ 6= θ^ML.

ln P(n | θ^ML)

P(n | θ) = ln Π_i(θ_i^ML)ⁿⁱ

Π_iθ_iⁿⁱ = X

i

n_iln θ_i^ML

θ_i = N X

i

θ_i^MLln θ^M_i ^L θ_i > 0

The inequality follows from the fact that the relative entropy is always positive except when the two distributions are identical.

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.

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)

θ^PME_i ^def.= Z

θP(θ | n)dθ = n_i + α_i N + P

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.

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:

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 knowl- edge, using Bayes’ theorem. (In the Bayesian framework, the parameters are themselves seen as random variables!)

B. The Maximum A posteriori Probability (MAP) Estimate

θ^{MAP d}=^ef. argmax

θ

P(θ | D, M) = argmax

θ

P(D | θ, M)P(θ | M) P(D | M)

= argmax

θ

P(D | θ, M)P(θ | 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 = Z

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

5. Elementary Information Theory Definitions:

Let X and Y be discrete random variables.

• Entropy: H(X) ^def.= P

xp(x) log₂ 1

p(x) = −P

x p(x) log₂ p(x) = E_p[−log₂ p(X)].

Convention: if p(x) = 0 then we shall consider p(x) log₂p(x) = 0.

• Specific Conditional entropy: H(Y | X = x) ^def.= −P

y∈Y p(y | x) log₂ p(y | x).

• Average conditional entropy:

H(Y | X) ^def.= P

x∈X p(x)H(Y | X = x) ^imed.= −P

x∈X

P

y∈Y p(x, y) log₂ p(y | x).

• Joint entropy:

H(X, Y)^def.= −P

x,y p(x, y) log₂ p(x, y)^dem.= H(X)+H(Y |X)^dem.= H(Y )+H(X|Y ).

• Information gain (or: Mutual information):

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).

Exemplification: Entropy of a Bernoulli Distribution

H(p) = −plog₂ p −(1 − p) log₂(1 − p)

0.0 0.2 0.4 0.6 0.8 1.0

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p

0.0 0.2 0.4 0.6 0.8 1.0

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Basic properties of

Entropy, Conditional Entropy, Joint Entropy and Information Gain / Mutual Information

• 0 ≤ H(p1, . . . , pn) ≤ H 1

n, . . . , 1 n

= log₂ 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;

IG(X;X) = H(X).

• 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;

H(X, Y|A) = H(X|A) + H(Y |A) (a conditional form).

• a chain rule: H(X₁, . . . , X_n) = H(X₁)+H(X₂|X₁)+. . .+H(X_n|X₁, . . . , X_n−1).

The Relationship between

Entropy, Conditional Entropy, Joint Entropy and Information Gain

H(X|Y) H(Y|X) H(X,Y)

H(X) H(Y)

IG(X;Y)

Other definitions

• 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∈X

p(x) log₂ q(x) = E_p

log₂ 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∈X

p(x) log₂ q(x)

p(x) = E_p

log₂ p(X) q(X)

= CH(X, q) − H(X).

The Relationship between

Entropy, Conditional Entropy, Joint Entropy, Information Gain,

Cross-Entropy and Relative Entropy (or KL divergence)

H(p )

CH(X, p )

CH(Y, p )

X

KL(p || p )

H(p )

KL(p || p )

H(X|Y) H(Y|X)

H(X,Y)

Y XY X

IG(X,Y) = KL(p || p p )

Basic properties of

cross-entropy and relative entropy

◦ CH(X, q) ≥ 0

• 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:

−Pn

i=1 pi log₂ pi ≤ −P

i pi log₂ qi.

◦ Unlike H of a discrete n-ary variable, which is bounded by log₂ 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_X,Y || p_X p_Y) = −P

x

P

y p(x, y) log₂

p(x)p(y) p(x, y)

.

Remark

◦ The quantity

V I(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 tri- angle inequality.

◦ Consider M(p, q) = 1

2(p + q).

The function JSD(p||q) = 1

2KL(p||M) + 1

2KL(q||M) is called the Jensen- Shannon divergence.

One can prove that p

JSD(p||q) defines a distance metric (the Jensen- Shannon distance).

6. Recommended Exercises

• From [Manning & Sch¨ utze, 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

Addenda:

Other Examples of Probabilistic Distributions

Multinomial distribution:

generalises the binomial distribution to the case where there are K inde- pendent outcomes with probabilities θi, i = 1, . . . , K such that PK

i=1 θi = 1.

The probability of getting n_i occurrence of outcome i is given by P(n | θ) = n!

Π^K_i=1(n_i!)Π^K_i=1θ_iⁿⁱ, where n = n₁ + . . .+ n_K, and θ = (θ₁, . . . , θ_K).

Note:

categorical distribu- tion

Example:

12!

(2!)⁶ 1

6 12

= 3.4 ×10⁻³

Poisson distribution

p(k;λ) = λ^k

k! · e^−λ, with k ∈ N and parameter λ > 0.

Mean = variance = λ.

0 5 10 15 20

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k

P(X=k)

0 5 10 15 20

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Poisson probability mass function

λ = 1 λ = 4 λ = 10

0 5 10 15 20

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k

P(X ≤ k)

0 5 10 15 20

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Poisson cumulative distribution function

λ = 1 λ = 4 λ = 10

Exponential distribution

p(x;λ) = λe^−λx for x ≥ 0 and parameter λ > 0.

Mean = λ⁻¹, variance = λ⁻².

0 1 2 3 4 5

0.00.51.01.5

x

p(x)

0 1 2 3 4 5

0.00.51.01.5

Exponential probability density function

λ = 0.5 λ = 1 λ = 1.5

0 1 2 3 4 5

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x

P(X ≤ x)

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Exponential cumulative distribution function

λ = 0.5 λ = 1 λ = 1.5

Note: The Exponential distribution is a particular case of the Gamma distribution (take k= 1 in the next slide).

p(x;k, θ) = x^k−1 e^−x/θ

Γ(k)θ^k for x ≥ 0 and parameters k > 0 (shape) and θ > 0 (scale).

Mean = kθ, variance = kθ².

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)!.)

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x

p(x)

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Gamma probability density function

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

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x

P(X ≤ x)

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Gamma cumulative distribution function

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

χ

distribution:

p(x;ν) = 1 Γ(ν/2)

1 2

ν/2

x^ν/2−1e^−12x 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.

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x

p(x)

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Chi Squared probability density function

k = 1 k = 2 k = 3 k = 4 k = 6 k = 9

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x

P(X ≤ x)

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Chi Squared cumulative distribution function

k = 1 k = 2 k = 3 k = 4 k = 6 k = 9

Laplace distribution:

p(x;µ, θ) = 1 2θe⁻

|µ −x|

θ , with θ > 0.

Mean = µ, variance = 2θ².

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x

p(x)

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Laplace probability density function

µ = 0, θ = 1 µ = 0, θ = 2 µ = 0, θ = 4 µ = −5, θ = 4

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P(X ≤ x)

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Laplace cumulative density function

µ = 0, θ = 1 µ = 0, θ = 2 µ = 0, θ = 4 µ = −5, θ = 4

Student’s distribution:

p(x;ν) = Γ

ν + 1 2

√νπ Γν 2

1 + x² ν

−ν + 1

2 for x ∈ R and ν > 0 (the “degree of freedom” param.)

Mean = 0 for ν > 1, otherwise undefined.

Variance = ν

ν −2 for ν > 2, ∞ for 1 < ν ≤ 2, otherwise undefined.

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.

Beta distribution:

p(θ;α, β) = θ^α−1(1 −θ)^β−1 B(α, β) ,

where B(α, β) is the Beta function of arguments α, β ∈ R₊

B(α, β) = Γ(α)Γ(β) Γ(α+ β),

with Γ(x) = (x −1)! for any x ∈ N^∗.

Beta distribution: p.d.f.

Dirichlet distribution:

D(θ | α) = 1

Z(α)Π^K_i=1θ_i^αi−1δ(PK

i=1 θi −1) where

α = α₁, . . . , 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(P

iθ_i − 1), and

the normalising factor can be expressed in terms of the gamma function:

Z(α) = R

Π^K_i=1θ^α_i ⁱ⁻¹δ(P

i −1)dθ = Π_iΓ(α_i) Γ(P

i α_i) Mean of θ_i: α_i

P

j α_j.

For K = 2, the Dirichlet distribution reduces to the Beta distribution.

Remark:

Concerning the multinomial and Dirichlet distributions:

The algebraic expression for the parameters θ_i is similar in the two distri- butions.

However, the multinomial is a distribution over its exponents n_i, whereas the Dirichlet is a distribution over the numbers θi that are exponentiated.

The two distributions are said to be

conjugate distributions

Similarly,

the Beta distribution is the conjugate of the Bernoulli distribution, and the Gamma distribution is the conjugate of the Poisson distribution.

Notations

Let n = the number of pages on Internet, and H and A two n ×n matrices defined by

h_ij =

1 if page j points to page i (notation: P_j ∈ B_i) 0 otherwise

aij =

1 if page i contains no outgoing links 0 otherwise

α ∈ [0; 1] (this is a parameter that was initially set to 0.85) The transition matrix of the Google Markov Chain is

G = α(H + A) + 1 − α n · 1

where 1 is the n ×n matrix whose entries are all 1

The significance of G is derived from:

• the Random Surfer model

• the definition the (relative) importance of a page: com- bining votes from the pages that point to it

I (P

) = X

P_j∈B_i

I (P

) l

where l

is the number of links pointing out from P j .

The PageRank algorithm

[Brin & Page, 1998]

G is a stochastic matrix (g_ij ∈ [0; 1], Pn

i=1 g_ij = 1),

therefore λ₁ the greatest eigenvalue of G is 1, and G has a stationary vector I (i.e., GI = I).

G is also primitive (| λ₂ |< 1, where λ₂ is the second eigenvalue of G) and irreducible (I > 0).

From the matrix calculus it follows that

I can be computed using the power method:

if I¹ = GI⁰, I² = GI¹, . . . , I^k = GI^k−1 then I^k → I. I gives the relative importance of pages.

Suggested readings

“Using Google’s PageRank algorithm to identify important attributes of genes”, G.M. Osmani, S.M. Rahman, 2006

ADDENDA

Formalisation of HMM algorithms in

“Biological Sequence Analysis” [ Durbin et al, 1998 ]

Note

A begin state was introduced. The transition probability a_0k from this begin state to state k can be thought as the probability of starting in state k.

An end state is assumed, which is the reason for a_k0 in the termination step.

If ends are not modelled, this a_k0 will disappear.

For convenience we label both begin and end states as 0. There is no conflict because you can only transit out of the begin state and only into the end state, so variables are not used more than once.

The emission probabilities are considered independent of the origin state.

(Thus te emission of (pairs of ) symbols can be seen as being done when reaching the non-end states.) The begin and end states are silent.

Forward:

1. Initialization (i = 0): f

(0) = 1; f

(0) = 0, for k > 0 2. Induction (i = 1 . . . L): f

(i) = e

(x

) P

k

f

(i − 1)a

3. Total: P (x) = P

k

f

(L)a

. Backward:

1. Initialization (i = L): b

(L) = a

, for all k 2. Induction (i = L − 1, . . . , 1: b

(i) = P

l

a

e

(x

)b

(i + 1) 3. Total: P (x) = P

l

a

e

(x

)b

(1)

Combining f and b: P (π

, x) = f

(i)b

(i)

Viterbi:

1. Initialization (i = 0): v

(0) = 1; v

(0) = 0, for k > 0 2. Induction (i = 1 . . . L):

v

(i) = e

(x

) max

(v

(i − 1)a

);

ptr

(l) = argmax

v

(i − 1)a

)

3. Termination and readout of best path:

P (x, π

) = max

(v

(L)a

);

π

= argmax

v

(L)a

, and π

= ptr

(π

), for i = L . . . 1.

Baum-Welch:

1. Initialization: Pick arbitrary model parameters 2. Induction:

For each sequence j = 1. . . n calculate f_k^j(i) and b^j_k(i) for sequence j using the forward and respectively backward algorithms.

Calculate the expected number of times each transition of emission is used, given the training sequences:

A_kl = X

j

1 P(x^j)

X

i

f_k^j(i)a_kle_l(x^j_i+1)b^j_l(i+ 1)

E_kl = X

j

1 P(x^j)

X

{i|x^j_i=b}

f_k^j(i)b^j_k(i)

Calculate the new model parameters:

a_kl = A_kl P

l^′ A_kl^′ and e_k(b) = E_k(b) P

b^′ E_k(b^′) Calculate the new log likelihood of the model.

3. Termination:

Stop is the change in log likelihood is less than some predefined threshold or the maximum number of iterations is exceeded.