Basics Of Probability And Probability Distributions - IIT Kanpur

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Basics of Probability and Probability DistributionsPiyush Rai(IITK)Basics of Probability and Probability Distributions1

Some Basic Concepts You Should Know AboutRandom variables (discrete and continuous)Probability distributions over discrete/continuous r.v.’sNotions of joint, marginal, and conditional probability distributionsProperties of random variables (and of functions of random variables)Expectation and variance/covariance of random variablesExamples of probability distributions and their propertiesMultivariate Gaussian distribution and its properties (very important)Note: These slides provide only a (very!) quick review of these things. Please refer to a text such asPRML (Bishop) Chapter 2 Appendix B, or MLAPP (Murphy) Chapter 2 for more detailsNote: Some other pre-requisites (e.g., concepts from information theory, linear algebra, optimization,etc.) will be introduced as and when they are required(IITK)Basics of Probability and Probability Distributions2

Random VariablesInformally, a random variable (r.v.) X denotes possible outcomes of an eventCan be discrete (i.e., finite many possible outcomes) or continuousSome examples of discrete r.v.A random variable X {0, 1} denoting outcomes of a coin-tossA random variable X {1, 2, . . . , 6} denoteing outcome of a dice rollSome examples of continuous r.v.A random variable X (0, 1) denoting the bias of a coinA random variable X denoting heights of students in this classA random variable X denoting time to get to your hall from the department(IITK)Basics of Probability and Probability Distributions3

Discrete Random VariablesFor a discrete r.v. X , p(x) denotes the probability that p(X x)p(x) is called the probability mass function (PMF)Xp(x) 0p(x) 1p(x)1 x(IITK)Basics of Probability and Probability Distributions4

Continuous Random VariablesFor a continuous r.v. X , a probability p(X x) is meaninglessInstead we use p(X x) or p(x) to denote the probability density at X xFor a continuous r.v. X , we can only talk about probability within an interval X (x, x δx)p(x)δx is the probability that X (x, x δx) as δx 0The probability density p(x) satisfies the followingZp(x) 0 andp(x)dx 1(note: for continuous r.v., p(x) can be 1)x(IITK)Basics of Probability and Probability Distributions5

A word about notation.p(.) can mean different things depending on the contextp(X ) denotes the distribution (PMF/PDF) of an r.v. Xp(X x) or p(x) denotes the probability or probability density at point xActual meaning should be clear from the context (but be careful)Exercise the same care when p(.) is a specific distribution (Bernoulli, Beta, Gaussian, etc.)The following means drawing a random sample from the distribution p(X )x p(X )(IITK)Basics of Probability and Probability Distributions6

Joint Probability DistributionJoint probability distribution p(X , Y ) models probability of co-occurrence of two r.v. X , YFor discrete r.v., the joint PMF p(X , Y ) is like a table (that sums to 1)XXp(X x, Y y ) 1xyFor continuous r.v., we have joint PDF p(X , Y )Z Zp(X x, Y y )dxdy 1x(IITK)yBasics of Probability and Probability Distributions7

Marginal Probability DistributionIntuitively, the probability distribution of one r.v. regardless of the value the other r.v. takesPPFor discrete r.v.’s: p(X ) y p(X , Y y ), p(Y ) x p(X x, Y )For discrete r.v. it is the sum of the PMF table along the rows/columnsFor continuous r.v.: p(X ) Ryp(X , Y y )dy ,p(Y ) Rxp(X x, Y )dxNote: Marginalization is also called “integrating out”(IITK)Basics of Probability and Probability Distributions8

Conditional Probability Distribution- Probability distribution of one r.v. given the value of the other r.v.- Conditional probability p(X Y y ) or p(Y X x): like taking a slice of p(X , Y )- For a discrete distribution:- For a continuous distribution1 :1 Picture courtesy: Computer vision: models, learning and inference (Simon Price)(IITK)Basics of Probability and Probability Distributions9

Some Basic RulesSum rule: Gives the marginal probability distribution from joint probability distributionFor discrete r.v.: p(X ) PFor continuous r.v.: p(X ) p(X , Y )YRYp(X , Y )dYProduct rule: p(X , Y ) p(Y X )p(X ) p(X Y )p(Y )Bayes rule: Gives conditional probabilityp(Y X ) For discrete r.v.: p(Y X ) p(X Y )p(Y )p(X )Pp(X Y )p(Y )Y p(X Y )p(Y )For continuous r.v.: p(Y X ) R p(X Y )p(Y )Y p(X Y )p(Y )dYAlso remember the chain rulep(X1 , X2 , . . . , XN ) p(X1 )p(X2 X1 ) . . . p(XN X1 , . . . , XN 1 )(IITK)Basics of Probability and Probability Distributions10

Independence Y ) when knowing one tells nothing about the otherX and Y are independent (X p(X Y y ) p(X )p(Y X x) p(Y )p(X , Y ) p(X )p(Y ) Y is also called marginal independenceX Conditional independence (X Y Z ): independence given the value of another r.v. Zp(X , Y Z z) p(X Z z)p(Y Z z)(IITK)Basics of Probability and Probability Distributions11

ExpectationExpectation or mean µ of an r.v. with PMF/PDF p(X )XE[X ] xp(x)(for discrete distributions)xZE[X ] xp(x)dx(for continuous distributions)xNote: The definition applies to functions of r.v. too (e.g., E[f (X )])Linearity of expectationE[αf (X ) βg (Y )] αE[f (X )] βE[g (Y )](a very useful property, true even if X and Y are not independent)Note: Expectations are always w.r.t. the underlying probability distribution of the random variableinvolved, so sometimes we’ll write this explicitly as Ep() [.], unless it is clear from the context(IITK)Basics of Probability and Probability Distributions12

Variance and CovarianceVariance σ 2 (or “spread” around mean µ) of an r.v. with PMF/PDF p(X )var[X ] E[(X µ)2 ] E[X 2 ] µ2pStandard deviation: std[X ] var[X ] σFor two scalar r.v.’s x and y , the covariance is defined bycov[x, y ] E [{x E[x]}{y E[y ]}] E[xy ] E[x]E[y ]For vector r.v. x and y , the covariance matrix is defined as cov[x, y ] E {x E[x]}{y T E[y T ]} E[xy T ] E[x]E[y ]Cov. of components of a vector r.v. x: cov[x] cov[x, x]Note: The definitions apply to functions of r.v. too (e.g., var[f (X )])Note: Variance of sum of independent r.v.’s: var[X Y ] var[X ] var[Y ](IITK)Basics of Probability and Probability Distributions13

Transformation of Random VariablesSuppose y f (x) Ax b be a linear function of an r.v. xSuppose E[x] µ and cov[x] ΣExpectation of yE[y ] E[Ax b] Aµ bCovariance of ycov[y ] cov[Ax b] AΣATLikewise if y f (x) a T x b is a scalar-valued linear function of an r.v. x:E[y ] E[a T x b] a T µ bvar[y ] var[a T x b] a T ΣaAnother very useful property worth remembering(IITK)Basics of Probability and Probability Distributions14

Common Probability DistributionsImportant: We will use these extensively to model data as well as parametersSome discrete distributions and what they can model:Bernoulli: Binary numbers, e.g., outcome (head/tail, 0/1) of a coin tossBinomial: Bounded non-negative integers, e.g., # of heads in n coin tossesMultinomial: One of K ( 2) possibilities, e.g., outcome of a dice rollPoisson: Non-negative integers, e.g., # of words in a document. and many othersSome continuous distributions and what they can model:Uniform: numbers defined over a fixed rangeBeta: numbers between 0 and 1, e.g., probability of head for a biased coinGamma: Positive unbounded real numbersDirichlet: vectors that sum of 1 (fraction of data points in different clusters)Gaussian: real-valued numbers or real-valued vectors. and many others(IITK)Basics of Probability and Probability Distributions15

Discrete Distributions(IITK)Basics of Probability and Probability Distributions16

Bernoulli DistributionDistribution over a binary r.v. x {0, 1}, like a coin-toss outcomeDefined by a probability parameter p (0, 1)P(x 1) pDistribution defined as: Bernoulli(x; p) p x (1 p)1 xMean: E[x] pVariance: var[x] p(1 p)(IITK)Basics of Probability and Probability Distributions17

Binomial DistributionDistribution over number of successes m (an r.v.) in a number of trialsDefined by two parameters: total number of trials (N) and probability of each success p (0, 1)Can think of Binomial as multiple independent Bernoulli trials N mDistribution defined asBinomial(m; N, p) p (1 p)N mmMean: E[m] NpVariance: var[m] Np(1 p)(IITK)Basics of Probability and Probability Distributions18

Multinoulli DistributionAlso known as the categorical distribution (models categorical variables)ThinkPof a random assignment of an item to one of K bins - a K dim. binary r.v. x with single 1K(i.e., k 1 xk 1): Modeled by a multinoulli[0 00.0{z10length K0]}Let vector p [p1 , p2 , . . . , pK ] define the probability of going to each binpk (0, 1) is the probability that xk 1 (assigned to bin k)PKk 1 pk 1The multinoulli is defined as: Multinoulli(x; p) QKk 1pkxkMean: E[xk ] pkVariance: var[xk ] pk (1 pk )(IITK)Basics of Probability and Probability Distributions19

Multinomial DistributionThink of repeating the Multinoulli N timesLike distributing N items to K bins. Suppose xk is count in bin kKX0 xk N k 1, . . . , K ,xk Nk 1Assume probability of going to each bin: p [p1 , p2 , . . . , pK ]Multonomial models the bin allocations via a discrete vector x of size K[x1x2. . . xk 1xkxk 1 . . .xK ]Nx1 x2 . . . xK YKDistribution defined as Multinomial(x; N, p) Mean: E[xk ] Npkpkxkk 1Variance: var[xk ] Npk (1 pk )Note: For N 1, multinomial is the same as multinoulli(IITK)Basics of Probability and Probability Distributions20

Poisson DistributionUsed to model a non-negative integer (count) r.v. kExamples: number of words in a document, number of events in a fixed interval of time, etc.Defined by a positive rate parameter λDistribution defined asPoisson(k; λ) λk e λk!k 0, 1, 2, . . .Mean: E[k] λVariance: var[k] λ(IITK)Basics of Probability and Probability Distributions21

Continuous Distributions(IITK)Basics of Probability and Probability Distributions22

Uniform DistributionModels a continuous r.v. x distributed uniformly over a finite interval [a, b]Uniform(x; a, b) 1b a(b a)22var[x] (b a)12Mean: E[x] Variance:(IITK)Basics of Probability and Probability Distributions23

Beta DistributionUsed to model an r.v. p between 0 and 1 (e.g., a probability)Defined by two shape parameters α and βBeta(p; α, β) Mean: E[p] Γ(α β) α 1p(1 p)β 1Γ(α)Γ(β)αα βVariance: var[p] αβ(α β)2 (α β 1)Often used to model the probability parameter of a Bernoulli or Binomial (also conjugate to thesedistributions)(IITK)Basics of Probability and Probability Distributions24

Gamma DistributionUsed to model positive real-valued r.v. xDefined by a shape parameters k and a scale parameter θxGamma(x; k, θ) x k 1 e θθk Γ(k)Mean: E[x] kθVariance: var[x] kθ2Often used to model the rate parameter of Poisson or exponential distribution (conjugate to both),or to model the inverse variance (precision) of a Gaussian (conjuate to Gaussian if mean known)Note: There is another equivalent parameterization of gamma in terms of shape and rate parameters (rate 1/scale). Another related distribution: Inverse gamma.(IITK)Basics of Probability and Probability Distributions25

Dirichlet DistributionUsed to model non-negative r.v. vectors p [p1 , . . . , pK ] that sum to 1KX0 pk 1, k 1, . . . , K andpk 1k 1Equivalent to a distribution over the K 1 dimensional simplexDefined by a K size vector α [α1 , . . . , αK ] of positive realsPKKDistribution defined asΓ( k 1 αk ) Y αk 1pkDirichlet(p; α) QKk 1 Γ(αk ) k 1Often used to model the probability vector parameters of Multinoulli/Multinomial distributionDirichlet is conjugate to Multinoulli/MultinomialNote: Dirichlet can be seen as a generalization of the Beta distribution. Normalizing a bunch ofGamma r.v.’s gives an r.v. that is Dirichlet distributed.(IITK)Basics of Probability and Probability Distributions26

Dirichlet Distribution- For p [p1 , p2 , . . . , pK ] drawn from Dirichlet(α1 , α2 , . . . , αK )Mean: E[pk ] PKαkk 1Variance: var[pk ] αkαk (α0 αkα20 (α0 1)where α0 PKk 1αk- Note: p is a point on (K 1)-simplexPK- Note: α0 k 1 αk controls how peaked the distribution is- Note: αk ’s control where the peak(s) occurPlot of a 3 dim. Dirichlet (2 dim. simplex) for various values of α:Picture courtesy: (IITK)Computer vision: models, learning and inference (SimonPrice)Basicsof Probability and Probability Distributions27

Now comes theGaussian (Normal) distribution.(IITK)Basics of Probability and Probability Distributions28

Univariate Gaussian DistributionDistribution over real-valued scalar r.v. xDefined by a scalar mean µ and a scalar variance σ 2Distribution defined asN (x; µ, σ 2 ) 12πσ 2e (x µ)22σ 2Mean: E[x] µVariance: var[x] σ 2Precision (inverse variance) β 1/σ 2(IITK)Basics of Probability and Probability Distributions29

Multivariate Gaussian DistributionDistribution over a multivariate r.v. vector x RD of real numbersDefined by a mean vector µ RD and a D D covariance matrix ΣN (x; µ, Σ) p1(2π)D Σ 1e 2 (x µ) Σ 1 (x µ)The covariance matrix Σ must be symmetric and positive definiteAll eigenvalues are positivez Σz 0 for any real vector zOften we parameterize a multivariate Gaussian using the inverse of the covariance matrix, i.e., theprecision matrix Λ Σ 1(IITK)Basics of Probability and Probability Distributions30

Multivariate Gaussian: The Covariance MatrixThe covariance matrix can be spherical, diagonal, or full(IITK)Basics of Probability and Probability Distributions31

Some nice properties of theGaussian distribution.(IITK)Basics of Probability and Probability Distributions32

Multivariate Gaussian: Marginals and ConditionalsGiven x having multivariate Gaussian distribution N (x µ, Σ) with Λ Σ 1 . SupposeThe marginal distribution is simplyp(x a ) N (x a µa , Σaa )The conditional distribution is given byThus marginals and conditionalsof Gaussians are Gaussians(IITK)Basics of Probability and Probability Distributions33

Multivariate Gaussian: Marginals and ConditionalsGiven the conditional of an r.v. y and marginal of r.v. x, y is conditioned onMarginal of y and “reverse” conditional are given bywhere Σ (Λ A LA) 1Note that the “reverse conditional” p(x y ) is basically the posterior of x is the prior is p(x)Also note that the marginal p(y ) is the predictive distribution of y after integrating out xVery useful property for probabilistic models with Gaussian likelihoods and/or priors. Also veryhandly for computing marginal likelihoods.(IITK)Basics of Probability and Probability Distributions34

Gaussians: Product of GaussiansPointwise multiplication of two Gaussians is another (unnormalized) Gaussian(IITK)Basics of Probability and Probability Distributions35

Multivariate Gaussian: Linear TransformationsGiven a x Rd with a multivariate Gaussian distributionN (x; µ, Σ)Consider a linear transform of x into y RDy Ax bwhere A is D d and b RDy RD will have a multivariate Gaussian distributionN (y ; Aµ b, AΣA )(IITK)Basics of Probability and Probability Distributions36

Some Other Important DistributionsWishart Distribution and Inverse Wishart (IW) Distribution: Used to model D D p.s.d. matricesWishart often used as a conjugate prior for modeling precision matrices, IW for covariance matricesFor D 1, Wishart is the same as gamma dist., IW is the same as inverse gamma (IG) dist.Normal-Wishart Distribution: Used to model mean and precision matrix of a multivar. GaussianNormal-Inverse Wishart (NIW): : Used to model mean and cov. matrix of a multivar. GaussianFor D 1, the corresponding distr. are Normal-Gamma and Normal-Inverse Gamma (NIG)Student-t Distribution (a more robust version of Normal distribution)Can be thought of as a mixture of infinite many Gaussians with different precisions (or a singleGaussian with its precision/precision matrix given a gamma/Wishart prior and integrated out)Please refer to PRML (Bishop) Chapter 2 Appendix B, or MLAPP (Murphy) Chapter 2 for moredetails(IITK)Basics of Probability and Probability Distributions37

Random variables (discrete and continuous) . concepts from information theory, linear algebra, optimization, etc.) will be introduced as and when they are required (IITK) Basics of Probability and Probability Distributions 2. Random Variables . Uniform: numbers de ned over a xed range Beta: numbers between 0 and 1, e.g., probability of head .

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