Probability: Theory and practice Philipp Slusallek Karol - PowerPoint PPT Presentation

-Algebra σ • For continuous set : Ω A. The associated sigma algebras are the Borel sets over , i.e., the collection Ω of all open sets over omega that can be generated via countable unions, countable intersections, and complement of open sets I = [ p, q ) , p, q ∈ R Fixed half-interval T = [ α , β ) ⊆ [ p, q ) Collection of all half-intervals Here, is not a -algebra because, generally speaking, neither the union T σ nor the di ff erence of two half-intervals is a half-interval. � 30 Realistic Image Synthesis SS2018

-Algebra σ It is the mathematical construct that allows defining a measure � 31 Realistic Image Synthesis SS2018

Measure • In probability theory, it plays the role of a probability distribution � 32 Realistic Image Synthesis SS2018

Measure • In probability theory, it plays the role of a probability distribution • A real-valued set function defined on a sigma-algebra that assigns each subset of a sigma-algebra a non-negative real number. � 32 Realistic Image Synthesis SS2018

Measure • In probability theory, it plays the role of a probability distribution • A real-valued set function defined on a sigma-algebra that assigns each subset of a sigma-algebra a non-negative real number. • A sigma-additive set function: i.e., the measure of the union of disjoint sets is equal to the sum of the measures of the individual sets � 32 Realistic Image Synthesis SS2018

Lebesgue Measure • Standard way of assigning measure to subsets of n-dimensional Euclidean space. � 33 Realistic Image Synthesis SS2018

Lebesgue Measure • Standard way of assigning measure to subsets of n-dimensional Euclidean space. • For n = 1,2 or 3, it coincides with the standard measure of length, area or volume, respectively. Length Area Volume � 33 Realistic Image Synthesis SS2018

Random Variable • Central concept in probability theory σ � 34 Realistic Image Synthesis SS2018

Random Variable • Central concept in probability theory • Enables to construct a simpler probability space from a rather complex one σ � 34 Realistic Image Synthesis SS2018

Random Variable • Central concept in probability theory • Enables to construct a simpler probability space from a rather complex one • Correspond to a measurable function defined on a -algebra that σ assigns each element to a real number � 34 Realistic Image Synthesis SS2018

Random Variable • A random variable is a value chosen by some random process X f X Y = f ( X ) � 35 Realistic Image Synthesis SS2018

Random Variable • A random variable is a value chosen by some random process X • Random variables are always drawn from a domain: discrete (e.g., a fixed set of probabilities) or continuous (e.g., real numbers) f X Y = f ( X ) � 35 Realistic Image Synthesis SS2018

Random Variable • A random variable is a value chosen by some random process X • Random variables are always drawn from a domain: discrete (e.g., a fixed set of probabilities) or continuous (e.g., real numbers) • Applying a function to a random variable results in a new random f X variable Y = f ( X ) � 35 Realistic Image Synthesis SS2018

Discrete Probability Space � 36 Realistic Image Synthesis SS2018

Discrete Random Variable • Random variable (RV): Ω = { x 1 , x 2 , . . . , x n } X : Ω → E • Probabilities: { p 1 , p 2 , . . . , p n } N X p i = 1 i =1 � 37 Realistic Image Synthesis SS2018

Discrete Random Variable • Example: Rolling a Die 
 x 1 = 1 , x 2 = 2 , x 3 = 3 , x 4 = 4 , x 5 = 5 , x 6 = 6 • Probability of each event: 
 p i = 1/6 for i = 1, …, 6 � 38 Realistic Image Synthesis SS2018

Discrete Random Variable • Example: Rolling a Die 
 x 1 = 1 , x 2 = 2 , x 3 = 3 , x 4 = 4 , x 5 = 5 , x 6 = 6 • Probability of each event: 
 P ( X = i ) = 1 p i = 1/6 for i = 1, …, 6 6 � 38 Realistic Image Synthesis SS2018

Discrete Random Variable 4 X P (2 ≤ X ≤ 4) = P ( X = i ) i =2 � 39 Realistic Image Synthesis SS2018

Discrete Random Variable 4 X P (2 ≤ X ≤ 4) = P ( X = i ) i =2 4 1 6 = 1 X = 2 i =2 � 39 Realistic Image Synthesis SS2018

Probability mass function • PMF is a function that gives the probability that a discrete RV is exactly equal to some value. � 40 Realistic Image Synthesis SS2018

Probability mass function • PMF is a function that gives the probability that a discrete RV is exactly equal to some value. • PMF is di ff erent from PDF (probability density function) which is for continuous RVs. � 40 Realistic Image Synthesis SS2018

Probability mass function Constant PMF Non-uniform PMF 0.4 0.3 1 1 1 1 1 1 6 6 6 6 6 6 0.15 0.1 0.05 � 41 Realistic Image Synthesis SS2018

Continuous Probability Space � 42 Realistic Image Synthesis SS2018

Continuous Random Variable • In rendering, discrete random variables are less common than continuous random variables ξ � 43 Realistic Image Synthesis SS2018

Continuous Random Variable • In rendering, discrete random variables are less common than continuous random variables • Continuous random variables take on values that ranges of continuous domains (e.g. real numbers or directions on the unit sphere) ξ � 43 Realistic Image Synthesis SS2018

Continuous Random Variable • In rendering, discrete random variables are less common than continuous random variables • Continuous random variables take on values that ranges of continuous domains (e.g. real numbers or directions on the unit sphere) • A particularly important random variable is the canonical uniform random variable, which we write as ξ � 43 Realistic Image Synthesis SS2018

Continuous Random Variable 0 1 ξ ∈ [0 , 1) � 44 Realistic Image Synthesis SS2018

Continuous Random Variable 0 1 ξ ∈ [0 , 1) � 44 Realistic Image Synthesis SS2018

Continuous Random Variable 0 1 • We can take a continuous, uniformly distributed random variable ξ ∈ [0 , 1) and map to a discrete random variable, choosing if: X i � 45 Realistic Image Synthesis SS2018

Continuous Random Variable 0 1 • We can take a continuous, uniformly distributed random variable ξ ∈ [0 , 1) and map to a discrete random variable, choosing if: X i i − 1 i X X p j < ξ ≤ p j j =1 j =1 X i = { 1 , 2 , 3 , 4 , 5 , 6 } � 46 Realistic Image Synthesis SS2018

Continuous Random Variable Questions ? Image rendered using PBRT Realistic Image Synthesis SS2018

Continuous Random Variable Questions ? Image rendered using PBRT Realistic Image Synthesis SS2018

Continuous Random Variable Image rendered using PBRT Realistic Image Synthesis SS2018

Continuous Random Variable • For lighting application, we might want to define probability of sampling illumination from each light source in the scene based on its power Φ i Φ i p i = P j Φ j Here, the probability is relative to the total power � 49 Realistic Image Synthesis SS2018

Probability Density Functions � 50 Realistic Image Synthesis SS2018

Probability density function 0 2 x • Consider a continuous RV that ranges over real numbers: , where [0 , 2) the probability of taking on any particular value is proportional to the x value 2 − x � 51 Realistic Image Synthesis SS2018

Probability density function 0 2 x • Consider a continuous RV that ranges over real numbers: , where [0 , 2) the probability of taking on any particular value is proportional to the x value 2 − x • It is twice as likely for this random variable to take on a value around 0 as it is to take around 1, and so forth. � 51 Realistic Image Synthesis SS2018

Probability density function • The probability density function (PDF) formalizes this idea: it describes the relative probability of a RV taking on a particular value. � 52 Realistic Image Synthesis SS2018

Probability density function • The probability density function (PDF) formalizes this idea: it describes the relative probability of a RV taking on a particular value. • Unlike PMF , the values of the PDFs are not the probabilities as such: a PDF must be integrated over an interval to yield a probability � 52 Realistic Image Synthesis SS2018

Probability density function For uniform random variables: For non-uniform random variables: ( 1 x ∈ [0 , 1) p ( x ) could be any function p ( x ) = 0 otherwise � 53 Realistic Image Synthesis SS2018

Probability density function Uniform distribution Non-uniform distribution constant pdf � 54 Realistic Image Synthesis SS2018

Probability density function Uniform distribution Non-uniform distribution constant pdf � 54 Realistic Image Synthesis SS2018

Probability density function Uniform distribution Non-uniform distribution constant pdf � 55 Realistic Image Synthesis SS2018

Probability density function Uniform distribution Non-uniform distribution constant pdf � 55 Realistic Image Synthesis SS2018

Probability density function Some properties of PDFs: p ( x ) > 0 Z ∞ p ( x ) dx = 1 −∞ � 56 Realistic Image Synthesis SS2018

Probability density function Z b x ∈ [ a, b ) p ( x ) dx = 1 a constant pdf a b � 57 Realistic Image Synthesis SS2018

Probability density function Z b x ∈ [ a, b ) p ( x ) dx = 1 a constant pdf C p ( x ) = C a b � 58 Realistic Image Synthesis SS2018

Probability density function Z b x ∈ [ a, b ) p ( x ) dx = 1 a constant pdf Z b C C dx = 1 p ( x ) = C a a b � 58 Realistic Image Synthesis SS2018

Probability density function Z b x ∈ [ a, b ) p ( x ) dx = 1 a constant pdf Z b C C dx = 1 p ( x ) = C a Z b C dx = 1 a a b � 58 Realistic Image Synthesis SS2018

Probability density function Z b x ∈ [ a, b ) p ( x ) dx = 1 a constant pdf Z b C C dx = 1 p ( x ) = C a Z b C dx = 1 a a b C ( b − a ) = 1 � 58 Realistic Image Synthesis SS2018