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Probability: Theory and practice Philipp Slusallek Karol Myszkowski Gurprit Singh 1 Realistic Image Synthesis SS2018 A la Carte algebra and measure - Random Variables Probability distribution functions (PDFs and PMFs)

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Realistic Image Synthesis SS2018

Probability: Theory and practice

Philipp Slusallek Karol Myszkowski Gurprit Singh

  • 1
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A la Carte

2

  • algebra and measure
  • Random Variables
  • Probability distribution functions (PDFs and PMFs)
  • Conditional and Marginal PDFs
  • Expected value and Variance of a random variable

σ-

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Motivation: Ray Tracing

3

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Scene designed by David Coeurjolly

Ray Tracing

Image Plane

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Ray Tracing

Image Plane Scene designed by David Coeurjolly

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Direct Illumination

4 spp

Image rendered using PBRT

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Direct Illumination

256 spp

Image rendered using PBRT

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Direct and Indirect Illumination

4096 spp

Image rendered using PBRT

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Path Tracing

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Path Tracing

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Path Tracing

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Path Tracing

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4 spp

Direct and Indirect Illumination

Image rendered using PBRT

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How can we analyze the noise present in the images ?

14

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Probability Theory and/or Number Theory

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Probability Theory

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  • Discrete Probability Space
  • Continuous Probability Space

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Rolling a fair dice

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Ω = {1, 2, 3, 4, 5, 6}

  • Finite outcomes: discrete random experiment
  • Can ask the outcome is a number: 1 or 6
  • Can ask the outcome is a subset, e.g. all prime numbers:

{2, 3, 5}

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Rolling a fair dice

19

Ω = {1, 2, 3, 4, 5, 6}

  • R1: Apart from elementary values, the focus lies on subsets of
  • R2:

A probability assigns each element or each subset of a positive real value 


Ω Ω

The first requirement leads to the concept of -algebra

σ

The second to the mathematical construct of a measure

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Random number in [0,1]

20

1

  • Uncountably infinite outcomes: continuous random experiment
  • Does not make sense to ask for one number as output, e.g. 0.245
  • We need to ask for the probability of a region, e.g. [0.2,0.4] or [0.36,0.89]

Ω

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Random number in [0,1]

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  • R1: As in discrete case, focus lies on subsets of , also called events
  • R2: A probability assigns each subset of a positive real value.

Ω Ω

The first requirement leads to the concept of Borel -algebra

σ

The second to the mathematical construct of a Lebesgue measure

Ω

1

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  • Algebra

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  • Mathematical construct used in probability and measure theory
  • 1. Take on the role of system of events in probability theory
  • Simply spoken: Collection of subsets of a given set
  • A. A non-empty collection of subsets of that is closed under the set

theoretical operations of: countable unions, countable intersections, and complement

σ

Ω

Ω

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  • Algebra

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  • For discrete set :
  • 1. The sigma-algebra corresponds to the power set of omega (set of all

subsets)

σ

Ω

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  • Algebra

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  • For discrete set :
  • 1. The sigma-algebra corresponds to the power set of omega (set of all

subsets)

σ

Ω

Σ = {{φ}, {0}, {1}, {0, 1}} Ω = {0, 1}

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  • Algebra

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  • For discrete set :
  • 1. The sigma-algebra corresponds to the power set of omega (set of all

subsets)

σ

Ω

Ω = {a, b, c, d} Σ = {{φ}, {0}, {1}, {0, 1}} Ω = {0, 1} Σ = {{φ}, {a, b}, {c, d}, {a, b, c, d}}

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  • Algebra

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σ

Ω

  • For continuous set :
  • A. The associated sigma algebras are the Borel sets over , i.e., the collection
  • f all open sets over omega that can be generated via countable unions,

countable intersections, and complement of open sets

Ω

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  • Algebra

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σ

Ω

  • For continuous set :
  • A. The associated sigma algebras are the Borel sets over , i.e., the collection
  • f 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

Ω

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  • Algebra

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σ

Ω

  • For continuous set :
  • A. The associated sigma algebras are the Borel sets over , i.e., the collection
  • f 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 T = [α, β) ⊆ [p, q)

Fixed half-interval Collection of all half-intervals

Ω

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  • Algebra

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σ

Ω

  • For continuous set :
  • A. The associated sigma algebras are the Borel sets over , i.e., the collection
  • f 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 T = [α, β) ⊆ [p, q)

Fixed half-interval Collection of all half-intervals

Here, is not a -algebra because, generally speaking, neither the union nor the difference of two half-intervals is a half-interval.

T σ

Ω

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  • Algebra

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σ

It is the mathematical construct that allows defining a measure

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Measure

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  • 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

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Lebesgue Measure

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  • 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

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Random Variable

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  • Central concept in probability theory
  • Enables to construct a simpler probability space from a rather complex
  • ne
  • Correspond to a measurable function defined on a -algebra that

assigns each element to a real number

σ

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Random Variable

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  • A random variable is a value chosen by some random process
  • 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

variable

X f X Y = f(X)

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Discrete Probability Space

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Discrete Random Variable

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  • Random variable (RV):
  • Probabilities:

{p1, p2, . . . , pn}

N

X

i=1

pi = 1

X : Ω → E Ω = {x1, x2, . . . , xn}

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Discrete Random Variable

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  • Example: Rolling a Die

  • Probability of each event:


pi = 1/6 for i = 1, …, 6

x1 = 1, x2 = 2, x3 = 3, x4 = 4, x5 = 5, x6 = 6

P(X = i) = 1 6

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Discrete Random Variable

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P(2 ≤ X ≤ 4) =

4

X

i=2

P(X = i) =

4

X

i=2

1 6 = 1 2

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Probability mass function

  • PMF is a function that gives the probability that a discrete

RV is exactly equal to some value.

  • PMF is different from PDF (probability density function)

which is for continuous RVs.

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Probability mass function

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1 6 1 6 1 6 1 6 1 6 1 6

0.4 0.15 0.3 0.05 0.1

Constant PMF Non-uniform PMF

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Continuous Probability Space

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Continuous Random Variable

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  • 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

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Continuous Random Variable

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1

ξ ∈ [0, 1)

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  • We can take a continuous, uniformly distributed random variable

and map to a discrete random variable, choosing if:

ξ ∈ [0, 1) Xi

Continuous Random Variable

1

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  • We can take a continuous, uniformly distributed random variable

and map to a discrete random variable, choosing if:

ξ ∈ [0, 1) Xi

i−1

X

j=1

pj < ξ ≤

i

X

j=1

pj

Xi = {1, 2, 3, 4, 5, 6}

Continuous Random Variable

1

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Continuous Random Variable

Image rendered using PBRT

Visual Break

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Continuous Random Variable

Image rendered using PBRT

Visual Break

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Continuous Random Variable

pi = Φi P

j Φj

Φi

  • For lighting application, we might want to define probability of sampling

illumination from each light source in the scene based on its power Here, the probability is relative to the total power

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Probability Density Functions

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Probability density function

  • Consider a continuous RV that ranges over real numbers: , where

the probability of taking on any particular value is proportional to the value

2

[0, 2) x 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

x

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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

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p(x) = ( 1 x ∈ [0, 1)

  • therwise

For uniform random variables: For non-uniform random variables:

p(x) could be any function

Probability density function

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constant pdf

Uniform distribution Non-uniform distribution

Probability density function

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constant pdf

Uniform distribution Non-uniform distribution

Probability density function

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p(x) > 0

Z ∞

−∞

p(x)dx = 1

Some properties of PDFs:

Probability density function

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Z b

a

p(x)dx = 1 x ∈ [a, b)

constant pdf

b a

Probability density function

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Z b

a

p(x)dx = 1 x ∈ [a, b)

constant pdf

b a

Z b

a

C dx = 1

C Z b

a

dx = 1 C(b − a) = 1 p(x) = C C C = 1 b − a

Probability density function

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x ∈ [a, b)

constant pdf

b a

Z b

a

p(x)dx = 1

Z b

a

C dx = 1

C Z b

a

dx = 1 C(b − a) = 1 p(x) = 1 b − a p(x) = C C

Probability density function

C = 1 b − a

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Cumulative distribution function

  • The PDF is the derivative of the random variable's CDF:

p(x)

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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) , also called cumulative density function

P(x)

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Cumulative distribution function

  • The PDF is the derivative of the random variable's CDF:

p(x) p(x) = dP(x) dx P(x) P(x) = Z x

−∞

p(x)dx

: cumulative distribution function (CDF) , also called cumulative density function

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1

Cumulative distribution function

P(x) = Z x

−∞

p(x)dx p(x) = ( 1 x ∈ [0, 1)

  • therwise

constant pdf

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Non-constant pdf

1

Cumulative distribution function

p(x) P(x) = Z x

−∞

p(x)dx

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Image rendered using PBRT

Visual Break

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Probability: Integral of PDF

P(x ∈ [a, b]) = Z b

a

p(x)dx

  • Given the arbitrary interval in the domain, integrating the PDF gives

the probability that a RV lies inside that interval:

[a, b] p(x)

a b

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Examples: Sampling PDFs

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Constant Sampling PDFs

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Random 2D

1 1

Jittered 2D

1 1

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Sampling a unit domain with uniform random samples

1

ξ ∈ [0, 1)

Random 1D

Constant Sampling PDFs

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Sampling a unit domain with uniform random samples

Random 1D

Constant Sampling PDFs

1

ξ ∈ [0, 1)

Random 1D

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Sampling a unit domain with uniform random samples

p(x) = ( C x ∈ [0, 1)

  • therwise

1

ξ ∈ [0, 1)

Random 1D

Constant Sampling PDFs

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Sampling each stratum with uniform random samples

1

Jittered 1D

Constant Sampling PDFs

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Sampling each stratum with uniform random samples

1

Jittered 1D

∆ = 1 N

∆

Constant Sampling PDFs

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Sampling each stratum with uniform random samples

1

Jittered 1D

∆ = 1 N

∆

i

Constant Sampling PDFs

p(xi) = ???

Probability density of generating a sample in an -th stratum is given by:

i

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Constant Sampling PDFs

1

Jittered 1D

∆ = 1 N

∆

Probability density of generating a sample in an -th stratum is given by:

i i

Sampling each stratum with uniform random samples

p(xi) = ( N x ∈ [ i

N , i+1 N )

  • therwise
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1

Jittered 1D

∆ = 1 N

∆

i

Joint PDFs

First, we divide the domain into equal strata. Second, we sample the domain. This implies that two samples are correlated to each other.

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1

Jittered 1D

∆ = 1 N

∆

i

Joint PDFs

p(xi, xj) = ???

what is the joint PDF for jittered sampling ?

i j

For two different strata and

,

First, we divide the domain into equal strata. Second, we sample the domain. This implies that two samples are correlated to each other.

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Conditional and Marginal PDFs

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Joint PDF

For two random variables and , the joint PDF is given by:

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X1 X2 p(x1, x2)

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Joint PDF

For two random variables and , the joint PDF is given by:

79

X1 X2 p(x1, x2) p(x1, x2) = p(x2|x1)p(x1)

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Joint PDF

For two random variables and , the joint PDF is given by:

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X1 X2 X1 = x1 X2 = x2 p(x1, x2) p(x1, x2) = p(x2|x1)p(x1)

where,

p(x2|x1) p(x1)

: conditional density function : marginal density function

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Joint PDF

For two random variables and , the joint PDF is given by:

81

X1 X2 X1 = x1 X2 = x2 p(x1, x2) p(x1, x2) = p(x2|x1)p(x1)

where,

p(x2|x1) p(x1)

: conditional density function : marginal density function

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Joint PDF

For two random variables and , the joint PDF is given by:

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X1 X2 X1 = x1 X2 = x2 p(x1, x2) p(x1, x2) = p(x1|x2)p(x2)

where,

p(x1|x2) p(x2)

: conditional density function : marginal density function

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Marginal PDF

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p(x1) = Z

R

p(x1, x2)dx2 p(x2) = Z

R

p(x1, x2)dx1

We integrate out one of the variable.

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Conditional PDF

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p(x1|x2) = p(x1, x2) p(x2) p(x2|x1) = p(x1, x2) p(x1)

The conditional density function is the density function for given that some particular has been chosen.

xi xj

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Conditional PDF

85

If both and are independent then:

p(x1|x2) = p(x1) p(x2|x1) = p(x2) x1 x2

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Conditional PDF

86

If both and are independent then:

p(x1|x2) = p(x1) p(x2|x1) = p(x2) x1 x2 p(x1, x2) = p(x1)p(x2)

That gives:

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1

Joint PDF of Jittered 1D Sampling

p(xi, xj) = ???

what is the joint PDF for jittered sampling ?

i j

For two different strata and

,

i

j

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1

Joint PDF of Jittered 1D Sampling

p(x1, x2) = p(x1|x2)p(x2) i

j

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1

Joint PDF of Jittered 1D Sampling

p(x1, x2) = p(x1|x2)p(x2) p(x1, x2) = p(x1)p(x2) i

j

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1

Joint PDF of Jittered 1D Sampling

p(xi, xj) = ( p(xi)p(xj) i 6= j

  • therwise

i

j

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1

Joint PDF of Jittered 1D Sampling

p(xi, xj) = ( p(xi)p(xj) i 6= j

  • therwise

p(xi, xj) = ( N 2 i 6= j

  • therwise

p(xi) = N

Since,

i

j

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Image rendered using PBRT

Visual Break

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Expected Value

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Expected value

94

  • Expected value: average value of the variable
  • 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

E[X] =

N

X

i=1

xipi

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Expected value

95

  • Expected value: average value of the variable
  • 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

E[X] =

N

X

i=1

xipi

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Expected value

96

  • Properties:

E[X + Y ] = E[X] + E[Y ] E[X + c] = E[X] + c E[cX] = cE[X]

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Expected value

97

  • Properties:

E[X + Y ] = E[X] + E[Y ] E[X + c] = E[X] + c E[cX] = cE[X]

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Expected value

98

  • Properties:

E[X + Y ] = E[X] + E[Y ] E[X + c] = E[X] + c E[cX] = cE[X]

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Estimating expected values

  • To estimate the expected value of a variable
  • choose a set of random values based on the probability
  • average their results
  • 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

E[X] ≈ 1 N

N

X

i=1

xi

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Estimating expected values

  • To estimate the expected value of a variable
  • choose a set of random values based on the probability
  • average their results
  • 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

E[X] ≈ 1 N

N

X

i=1

xi

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Estimating expected values

  • To estimate the expected value of a variable
  • choose a set of random values based on the probability
  • average their results
  • 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

101

E[X] ≈ 1 N

N

X

i=1

xi

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Estimating expected values

  • To estimate the expected value of a variable
  • choose a set of random values based on the probability
  • average their results
  • 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

102

E[X] ≈ 1 N

N

X

i=1

xi

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Estimating expected values

  • To estimate the expected value of a variable
  • choose a set of random values based on the probability
  • average their results
  • 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

103

E[X] ≈ 1 N

N

X

i=1

xi

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Estimating expected values

  • To estimate the expected value of a variable
  • choose a set of random values based on the probability
  • average their results
  • 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

104

E[X] ≈ 1 N

N

X

i=1

xi

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SLIDE 105

Realistic Image Synthesis SS2018

Law of large numbers

  • By taking infinitely many samples, the error between the

estimate and the expected value is statistically zero

  • the estimate will converge to the right value

105

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Realistic Image Synthesis SS2018

Variance

106

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Realistic Image Synthesis SS2018

Variance

107

  • Variance: how much different from the average

σ2[X] = E[(X − E[X])2] = E[X2 + E[X]2 − 2XE[X]] = E[X2] + E[E[X]2] − 2E[X]E[E[X]]] = E[X2] + E[X]2 − 2E[X]2 = E[X2] − E[X]2

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SLIDE 108

Realistic Image Synthesis SS2018

Variance

108

  • Variance: how much different from the average

σ2[X] = E[(X − E[X])2] = E[X2 + E[X]2 − 2XE[X]] = E[X2] + E[E[X]2] − 2E[X]E[E[X]]] = E[X2] + E[X]2 − 2E[X]2 = E[X2] − E[X]2

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SLIDE 109

Realistic Image Synthesis SS2018

Variance

109

  • Variance: how much different from the average

σ2[X] = E[(X − E[X])2] = E[X2 + E[X]2 − 2XE[X]] = E[X2] + E[E[X]2] − 2E[X]E[E[X]]] = E[X2] + E[X]2 − 2E[X]2 = E[X2] − E[X]2

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SLIDE 110

Realistic Image Synthesis SS2018

Variance

110

  • Variance: how much different from the average

σ2[X] = E[(X − E[X])2] = E[X2 + E[X]2 − 2XE[X]] = E[X2] + E[E[X]2] − 2E[X]E[E[X]]] = E[X2] + E[X]2 − 2E[X]2 = E[X2] − E[X]2

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SLIDE 111

Realistic Image Synthesis SS2018

Variance

111

  • Variance: how much different from the average

σ2[X] = E[(X − E[X])2] = E[X2 + E[X]2 − 2XE[X]] = E[X2] + E[E[X]2] − 2E[X]E[E[X]]] = E[X2] + E[X]2 − 2E[X]2 = E[X2] − E[X]2

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SLIDE 112

Realistic Image Synthesis SS2018

Variance

112

  • Variance: how much different from the average

σ2[X] = E[(X − E[X])2] = E[X2 + E[X]2 − 2XE[X]] = E[X2] + E[E[X]2] − 2E[X]E[E[X]]] = E[X2] + E[X]2 − 2E[X]2 = E[X2] − E[X]2

σ2[X] = E[X2] − E[X]2

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SLIDE 113

Realistic Image Synthesis SS2018

Variance

113

  • example: Rolling a die
  • variance:

σ2[X] = . . . = 2.917

σ2[X] = E[X2] − E[X]2 E[X] = 1 · 1 6 + 2 · 1 6 + 3 · 1 6 + 4 · 1 6 + 5 · 1 6 + 6 · 1 6 = 3.5

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SLIDE 114

Realistic Image Synthesis SS2018

Variance

114

  • example: Rolling a die
  • variance:

σ2[X] = . . . = 2.917

σ2[X] = E[X2] − E[X]2 E[X] = 1 · 1 6 + 2 · 1 6 + 3 · 1 6 + 4 · 1 6 + 5 · 1 6 + 6 · 1 6 = 3.5

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SLIDE 115

Realistic Image Synthesis SS2018

Variance

115

  • example: Rolling a die
  • variance:

σ2[X] = . . . = 2.917

σ2[X] = E[X2] − E[X]2

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SLIDE 116

Realistic Image Synthesis SS2018

Monte Carlo Integration

116

I = Z

D

f(x) dx Z

Slide after Wojciech Jarosz

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SLIDE 117

Image rendered using PBRT

Questions ?