Classical AI and ML research ignored this phenomena Another example

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1 Wat is tis? Classical AI and ML researc ignored tis enomena Anoter eamle you want to catc a fligt at 0:00am from Pitt to SF, can I make it if I leave at 8am and take a Marta at Gatec? artial observability road state, oter drivers' lans, etc. noisy sensors radio traffic reorts uncertainty in action outcomes flat tire, etc. immense comleity of modeling and redicting traffic

2 Basic Probability Concets A samle sace S is te set of all ossible outcomes of a concetual or ysical, reeatable eeriment. S can be finite or infinite. E.g., S may be te set of all ossible outcomes of a dice roll: S E.g., S may be te set of all ossible nucleotides of a DNA site: S A, T, C, G E.g., S may be te set of all ossible time-sace ositions o of a aircraft on a radar screen: S { 0, R } { 0, 360 } { 0, } An event A is any subset of S :,,3,4,5,6 Seeing "" or "6" in a dice roll; observing a "G" at a site; UA007 in sace-time interval An event sace E is te ossible worlds te outcomes can aen All dice-rolls, reading a genome, monitoring te radar signal ma

3 Probability A robability PA is a function tat mas an event A onto te interval [0, ]. PA is also called te robability measure or robability mass of A. Samle sace of all ossible worlds. Its area is Worlds in wic A is true Worlds in wic A is false Pa is te area of te oval

4 Kolmogorov Aioms All robabilities are between 0 and 0 PA PE = PΦ=0 Te robability of a disunction is given by PA B = PA + PB PA B A B B A B? A B A

5 Wy use robability? Tere ave been attemts to develo different metodologies for uncertainty: Fuzzy logic Qualitative reasoning Qualitative ysics Probability teory is noting but common sense reduced to calculation Pierre Lalace, 8. In 93, de Finetti roved tat it is irrational to ave beliefs tat violate tese aioms, in te following sense: If you bet in accordance wit your beliefs, but your beliefs violate te aioms, ten you can be guaranteed to lose money to an oonent wose beliefs more accurately reflect te true state of te world. Here, betting and money are roies for decision making and utilities. Wat if you refuse to bet? Tis is like refusing to allow time to ass: every action including inaction is a bet

6 Random Variable A random variable is a function tat associates a unique numerical value a token wit every outcome of an eeriment. Te value of te r.v. will vary from trial to trial as te eeriment is reeated w S Discrete r.v.: Te outcome of a dice-roll Te outcome of reading a nt at site i: Binary event and indicator variable: Seeing an "A" at a site =, o/w =0. Tis describes te true or false outcome a random event. Can we describe ricer outcomes in te same way? i.e., =,, 3, 4, for being A, C, G, T --- tink about wat would aen if we take eectation of. Continuous r.v.: Te outcome of recording te true location of an aircraft: Te outcome of observing te measured location of an aircraft i true obs w

7 Discrete Prob. Distribution A robability distribution P defined on a discrete samle sace S is an assignment of a non-negative real number Ps to eac samle ss suc tat S ss Ps=. 0Ps intuitively, Ps corresonds to te frequency or te likeliood of getting a articular samle s in te eeriments, if reeated multile times. call q s = Ps te arameters in a discrete robability distribution A discrete robability distribution is sometimes called a robability model, in articular if several different distributions are under consideration write models as M, M, robabilities as P M, P M e.g., M may be te aroriate rob. dist. if is from "fair dice", M is for te "loaded dice". M is usually a two-tule of {dist. family, dist. arameters}

8 Bernoulli distribution: Ber Multinomial distribution: Mult,q Multinomial indicator variable:., w.. and ],, [ were, [,...,6] [,...,6] q q {, were inde te dice-face} k k k P q q Discrete Distributions 0 P for for P

9 Multinomial distribution: Multn,q Count variable: Discrete Distributions K n were, K K K n n K q q q q!!!!!!!!

10 Continuous Prob. Distribution A continuous random variable is defined on a continuous samle sace: an interval on te real line, a region in a ig dimensional sace, etc. usually corresonds to a real-valued measurements of some roerty, e.g., lengt, osition, It is meaningless to talk about te robability of te random variable assuming a articular value --- P = 0 Instead, we talk about te robability of te random variable assuming a value witin a given interval, or alf interval, or arbitrary Boolean combination of basic roositions. P, P P, P,, 3 4

11 Probability Density If te rob. of falling into [, +d] is given by d for d, ten is called te robability density over. If te robability P is differentiable, ten te robability density over is te derivative of P. Te robability of te random variable assuming a value witin some given interval from to is equivalent to te area under te gra of te robability density function between and. Probability mass: P d,, note tat d. Cumulative distribution function CDF: P P ' d ' Probability density function PDF: d P d d ; 0, Car flow on Liberty Bridge cooked u!

12 Te intuitive meaning of If = a and = b, ten wen a value is samled from te distribution wit density, you are a/b times as likely to find tat is very close to tan tat is very close to. Tat is: b a d d P P 0 lim

13 Continuous Distributions Uniform Density Function / b a for a b 0 elsewere Normal Gaussian Density Function f m / s e s Te distribution is symmetric, and is often illustrated as a bell-saed curve. m Two arameters, m mean and s standard deviation, determine te location and sae of te distribution. Te igest oint on te normal curve is at te mean, wic is also te median and mode. Eonential Distribution PDF:, m / m o e CDF: P e / m 0 f.4.3 P < = area = Time Between Successive Arrivals mins.

14 Gaussian Normal density in D If Nµ, σ, te robability density function df of is defined as m / s e s We will often use te recision λ = /σ instead of te variance σ. Here is ow we lot te df in matlab s=-3:0.0:3; lots,normdfs,mu,sigma; Note tat a density evaluated at a oint can be larger tan.

15 Gaussian CDF If Z N0,, te cumulative density function is defined as Φ z dz z / Tis as no closed form eression, but is built in to most software ackages eg. normcdf in matlab stats toolbo. e dz

16 More on Gaussian Distribution If Nµ, σ, ten Z = µ/σ N0,. How muc mass is contained inside te [-σ,σ] interval? Since Z = normcdf = 0.05 we ave P σ < µ < σ 0.05 = 0.95 s m s m s m s m a b b a Φ Φ Z P b a P

17 Eectation: te centre of mass, mean value, first moment: Samle mean: Variance: te sreadness: Samle variance continuous discrete d E S i i i continuous ] [ discrete ] [ d E E Var S i i Statistical Caracterizations

18 Central limit teorem If,, n are i.i.d. continuous random variables Define As n infinity, f,,..., Gaussian wit mean E[ i ] and variance Var[ i ] n n i n i Somewat of a ustification for assuming Gaussian noise is common