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

Visual Break Continuous Random Variable Image rendered using PBRT Realistic Image Synthesis SS2018

Visual Break 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 � 48 Realistic Image Synthesis SS2018

Probability Density Functions � 49 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. � 50 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 � 51 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 � 52 Realistic Image Synthesis SS2018

Probability density function Uniform distribution Non-uniform distribution constant pdf � 53 Realistic Image Synthesis SS2018

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

Probability density function Some properties of PDFs: p ( x ) > 0 Z ∞ p ( x ) dx = 1 −∞ � 55 Realistic Image Synthesis SS2018

Probability density function Z b x ∈ [ a, b ) p ( x ) dx = 1 a constant pdf a b � 56 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 1 C = b − a � 57 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 1 p ( x ) = C dx = 1 b − a a a b C ( b − a ) = 1 1 C = b − a � 58 Realistic Image Synthesis SS2018

Cumulative distribution function • The PDF is the derivative of the random variable's CDF: p ( x ) � 59 Realistic Image Synthesis SS2018

Cumulative distribution function • The PDF is the derivative of the random variable's CDF: p ( x ) p ( x ) = dP ( x ) dx : cumulative distribution function (CDF) , P ( x ) also called cumulative density function � 60 Realistic Image Synthesis SS2018

Cumulative distribution function • The PDF is the derivative of the random variable's CDF: p ( x ) Z x p ( x ) = dP ( x ) P ( x ) = p ( x ) dx dx −∞ : cumulative distribution function (CDF) , P ( x ) also called cumulative density function � 61 Realistic Image Synthesis SS2018

Cumulative distribution function ( Z x 1 x ∈ [0 , 1) p ( x ) = P ( x ) = p ( x ) dx 0 otherwise −∞ constant pdf 1 � 62 Realistic Image Synthesis SS2018

Cumulative distribution function Z x p ( x ) P ( x ) = p ( x ) dx −∞ Non-constant pdf 1 � 63 Realistic Image Synthesis SS2018

Visual Break � 64 Image rendered using PBRT Realistic Image Synthesis SS2018

Probability: Integral of PDF • Given the arbitrary interval in the domain, integrating the PDF gives [ a, b ] the probability that a RV lies inside that interval: Z b P ( x ∈ [ a, b ]) = p ( x ) dx a p ( x ) a b � 65 Realistic Image Synthesis SS2018

Examples: Sampling PDFs � 66 Realistic Image Synthesis SS2018

Constant Sampling PDFs Random 2D Jittered 2D 1 1 0 1 0 1 � 67 Realistic Image Synthesis SS2018

Constant Sampling PDFs Random 1D 0 1 ξ ∈ [0 , 1) Sampling a unit domain with uniform random samples � 68 Realistic Image Synthesis SS2018

Constant Sampling PDFs Random 1D Random 1D 0 1 ξ ∈ [0 , 1) Sampling a unit domain with uniform random samples � 69 Realistic Image Synthesis SS2018

Constant Sampling PDFs Random 1D ( C x ∈ [0 , 1) p ( x ) = 0 otherwise 0 1 ξ ∈ [0 , 1) Sampling a unit domain with uniform random samples � 70 Realistic Image Synthesis SS2018

Constant Sampling PDFs Jittered 1D 0 1 Sampling each stratum with uniform random samples � 71 Realistic Image Synthesis SS2018

Constant Sampling PDFs Jittered 1D ∆ 0 1 ∆ = 1 N Sampling each stratum with uniform random samples � 72 Realistic Image Synthesis SS2018

Constant Sampling PDFs Jittered 1D Probability density of generating a sample in an -th stratum is given by: i ∆ i p ( x i ) = ??? 0 1 ∆ = 1 N Sampling each stratum with uniform random samples � 73 Realistic Image Synthesis SS2018

Constant Sampling PDFs Jittered 1D Probability density of generating a sample in an -th stratum is given by: i ∆ i ( x ∈ [ i N , i +1 N ) N p ( x i ) = 0 1 ∆ = 1 0 otherwise N Sampling each stratum with uniform random samples � 74 Realistic Image Synthesis SS2018

Joint PDFs Jittered 1D ∆ i First, we divide the domain into equal strata. 0 1 Second, we sample the domain. ∆ = 1 N This implies that two samples are correlated to each other. � 75 Realistic Image Synthesis SS2018

Joint PDFs Jittered 1D ∆ i First, we divide the domain into equal strata. 0 1 Second, we sample the domain. ∆ = 1 N This implies that two samples are correlated to each other. For two di ff erent strata and what is the joint PDF for jittered sampling ? i j , p ( x i , x j ) = ??? � 76 Realistic Image Synthesis SS2018

Conditional and Marginal PDFs � 77 Realistic Image Synthesis SS2018

Joint PDF For two random variables and , the joint PDF is given by: p ( x 1 , x 2 ) X 1 X 2 � 78 Realistic Image Synthesis SS2018

Joint PDF For two random variables and , the joint PDF is given by: p ( x 1 , x 2 ) X 1 X 2 p ( x 1 , x 2 ) = p ( x 2 | x 1 ) p ( x 1 ) � 79 Realistic Image Synthesis SS2018

Joint PDF For two random variables and , the joint PDF is given by: p ( x 1 , x 2 ) X 1 X 2 p ( x 1 , x 2 ) = p ( x 2 | x 1 ) p ( x 1 ) where, : conditional density function p ( x 2 | x 1 ) X 1 = x 1 : marginal density function p ( x 1 ) X 2 = x 2 � 80 Realistic Image Synthesis SS2018

Joint PDF For two random variables and , the joint PDF is given by: p ( x 1 , x 2 ) X 1 X 2 p ( x 1 , x 2 ) = p ( x 2 | x 1 ) p ( x 1 ) where, : conditional density function p ( x 2 | x 1 ) X 1 = x 1 : marginal density function p ( x 1 ) X 2 = x 2 � 81 Realistic Image Synthesis SS2018

Joint PDF For two random variables and , the joint PDF is given by: p ( x 1 , x 2 ) X 1 X 2 p ( x 1 , x 2 ) = p ( x 1 | x 2 ) p ( x 2 ) where, : conditional density function p ( x 1 | x 2 ) X 1 = x 1 : marginal density function p ( x 2 ) X 2 = x 2 � 82 Realistic Image Synthesis SS2018

Marginal PDF Z p ( x 1 ) = p ( x 1 , x 2 ) dx 2 R Z p ( x 2 ) = p ( x 1 , x 2 ) dx 1 R We integrate out one of the variable. � 83 Realistic Image Synthesis SS2018

Conditional PDF p ( x 1 | x 2 ) = p ( x 1 , x 2 ) p ( x 2 ) p ( x 2 | x 1 ) = p ( x 1 , x 2 ) p ( x 1 ) The conditional density function is the density function for given that x i some particular has been chosen. x j � 84 Realistic Image Synthesis SS2018

Conditional PDF If both and are independent then: x 1 x 2 p ( x 1 | x 2 ) = p ( x 1 ) p ( x 2 | x 1 ) = p ( x 2 ) � 85 Realistic Image Synthesis SS2018

Conditional PDF If both and are independent then: x 1 x 2 p ( x 1 | x 2 ) = p ( x 1 ) p ( x 2 | x 1 ) = p ( x 2 ) That gives: p ( x 1 , x 2 ) = p ( x 1 ) p ( x 2 ) � 86 Realistic Image Synthesis SS2018

Joint PDF of Jittered 1D Sampling i j 0 1 For two di ff erent strata and what is the joint PDF for jittered sampling ? i j , p ( x i , x j ) = ??? � 87 Realistic Image Synthesis SS2018

Joint PDF of Jittered 1D Sampling i j 0 1 p ( x 1 , x 2 ) = p ( x 1 | x 2 ) p ( x 2 ) � 88 Realistic Image Synthesis SS2018

Joint PDF of Jittered 1D Sampling i j 0 1 p ( x 1 , x 2 ) = p ( x 1 | x 2 ) p ( x 2 ) p ( x 1 , x 2 ) = p ( x 1 ) p ( x 2 ) � 89 Realistic Image Synthesis SS2018

Joint PDF of Jittered 1D Sampling i j 0 1 ( p ( x i ) p ( x j ) i 6 = j p ( x i , x j ) = 0 otherwise � 90 Realistic Image Synthesis SS2018

Joint PDF of Jittered 1D Sampling i j 0 1 ( p ( x i ) p ( x j ) i 6 = j p ( x i , x j ) = 0 otherwise ( N 2 i 6 = j p ( x i , x j ) = Since , p ( x i ) = N 0 otherwise � 91 Realistic Image Synthesis SS2018

Visual Break � 92 Image rendered using PBRT Realistic Image Synthesis SS2018

Expected Value � 93 Realistic Image Synthesis SS2018

Expected value • Expected value: average value of the variable N X E [ X ] = x i p i i =1 • example: rolling a die E [ X ] = 1 · 1 6 + 2 · 1 6 + 3 · 1 6 + 4 · 1 6 + 5 · 1 6 + 6 · 1 6 = 3 . 5 � 94 Realistic Image Synthesis SS2018

Expected value • Expected value: average value of the variable N X E [ X ] = x i p i i =1 • example: rolling a die E [ X ] = 1 · 1 6 + 2 · 1 6 + 3 · 1 6 + 4 · 1 6 + 5 · 1 6 + 6 · 1 6 = 3 . 5 � 95 Realistic Image Synthesis SS2018

Expected value • Properties: E [ X + Y ] = E [ X ] + E [ Y ] E [ X + c ] = E [ X ] + c E [ cX ] = cE [ X ] � 96 Realistic Image Synthesis SS2018

Expected value • Properties: E [ X + Y ] = E [ X ] + E [ Y ] E [ X + c ] = E [ X ] + c E [ cX ] = cE [ X ] � 97 Realistic Image Synthesis SS2018

Expected value • Properties: E [ X + Y ] = E [ X ] + E [ Y ] E [ X + c ] = E [ X ] + c E [ cX ] = cE [ X ] � 98 Realistic Image Synthesis SS2018

Estimating expected values • To estimate the expected value of a variable • choose a set of random values based on the probability • average their results N E [ X ] ≈ 1 X x i N i =1 • example: rolling a die roll 3 times: {3, 1, 6} → E [x] ≈ (3 + 1 + 6)/3 = 3.33 • roll 9 times: {3, 1, 6, 2, 5, 3, 4, 6, 2} → E [x] ≈ 3.51 • � 99 Realistic Image Synthesis SS2018

Estimating expected values • To estimate the expected value of a variable • choose a set of random values based on the probability • average their results N E [ X ] ≈ 1 X x i N i =1 • example: rolling a die roll 3 times: {3, 1, 6} → E [x] ≈ (3 + 1 + 6)/3 = 3.33 • roll 9 times: {3, 1, 6, 2, 5, 3, 4, 6, 2} → E [x] ≈ 3.51 • � 100 Realistic Image Synthesis SS2018