# Variance Reduction - Carnegie Mellon University

## Transcript Of Variance Reduction - Carnegie Mellon University

Variance Reduction

Computer Graphics CMU 15-462/15-662

Last time: Monte Carlo Ray Tracing

Recursive description of incident illumination Diﬃcult to integrate; tour de force of numerical integration Leads to lots of sophisticated integration strategies:

- sampling strategies - variance reduction - Markov chain methods - ...

Today: get a glimpse of these ideas Also valuable outside rendering!

- Monte Carlo one of the “Top 10 Algorithms of the 20th Century”!

CMU 15-462/662

Review: Monte Carlo Integration

any function*

Want to integrate:

any domain

General-purpose hammer: Monte-Carlo integration

()

volume of the domain

*Must of course have a well-defined integral!

uniformly random samples of domain

CMU 15-462/662

Review: Expected Value (DISCRETE)

A discrete random variable X has n possible outcomes xi, occuring w/ probabilities 0≤ pi ≤1, p1 + … + pn=1

probability of event i

expected value

value of event i

(just the “weighted average”!)

E.g., what’s the expected value for a fair coin toss?

p1 = 1/2 x1 = 1

p2 = 1/2 x2 = 0

CMU 15-462/662

Continuous Random Variables

A continuous random variable X takes values x anywhere in a set Ω

Probability density p gives probability x appears in a given region.

E.g., probability you fall asleep at time t in a 15-462 lecture:

()

cool motivating examples

theory

professor is making dumb jokes

more theory class ends

probability you fall asleep exactly at any given time t is ZERO!

can only talk about chance of falling asleep in a given

interval of time

CMU 15-462/662

Review: Expected Value (CONTINUOUS)

Expected value of continuous random variable again just the “weighted average” with respect to probability p:

probability density at point x

expected value

sometimes just use “µ” (for “mean”)

E.g., expected time of falling asleep?

()

µ = 44.9 minutes

(is this result counter-intuitive?)

CMU 15-462/662

Flaw of Averages

CMU 15-462/662

Review: Variance

Expected value is the “average value” Variance is how far we are from the average, on average!

DISCRETE

CONTINUOUS

Standard deviation σ is just the square root of variance

()

µ = 44.9 minutes

σ = 15.8 minutes

(More intuitive perhaps?)

CMU 15-462/662

Variance Reduction in Rendering

higher variance

lower variance

CMU 15-462/662

Q: How do we reduce variance?

CMU 15-462/662

Computer Graphics CMU 15-462/15-662

Last time: Monte Carlo Ray Tracing

Recursive description of incident illumination Diﬃcult to integrate; tour de force of numerical integration Leads to lots of sophisticated integration strategies:

- sampling strategies - variance reduction - Markov chain methods - ...

Today: get a glimpse of these ideas Also valuable outside rendering!

- Monte Carlo one of the “Top 10 Algorithms of the 20th Century”!

CMU 15-462/662

Review: Monte Carlo Integration

any function*

Want to integrate:

any domain

General-purpose hammer: Monte-Carlo integration

()

volume of the domain

*Must of course have a well-defined integral!

uniformly random samples of domain

CMU 15-462/662

Review: Expected Value (DISCRETE)

A discrete random variable X has n possible outcomes xi, occuring w/ probabilities 0≤ pi ≤1, p1 + … + pn=1

probability of event i

expected value

value of event i

(just the “weighted average”!)

E.g., what’s the expected value for a fair coin toss?

p1 = 1/2 x1 = 1

p2 = 1/2 x2 = 0

CMU 15-462/662

Continuous Random Variables

A continuous random variable X takes values x anywhere in a set Ω

Probability density p gives probability x appears in a given region.

E.g., probability you fall asleep at time t in a 15-462 lecture:

()

cool motivating examples

theory

professor is making dumb jokes

more theory class ends

probability you fall asleep exactly at any given time t is ZERO!

can only talk about chance of falling asleep in a given

interval of time

CMU 15-462/662

Review: Expected Value (CONTINUOUS)

Expected value of continuous random variable again just the “weighted average” with respect to probability p:

probability density at point x

expected value

sometimes just use “µ” (for “mean”)

E.g., expected time of falling asleep?

()

µ = 44.9 minutes

(is this result counter-intuitive?)

CMU 15-462/662

Flaw of Averages

CMU 15-462/662

Review: Variance

Expected value is the “average value” Variance is how far we are from the average, on average!

DISCRETE

CONTINUOUS

Standard deviation σ is just the square root of variance

()

µ = 44.9 minutes

σ = 15.8 minutes

(More intuitive perhaps?)

CMU 15-462/662

Variance Reduction in Rendering

higher variance

lower variance

CMU 15-462/662

Q: How do we reduce variance?

CMU 15-462/662