Bayesian Analysis Fundamentals with Examples

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tl;dr

Bayes enables
updating beliefs
by (fancy) counting

Bayes vs Frequentist

Bayes’ Theorem - What is it good for?

Bayes’ Theorem - What is it good for?

Bayes’ Theorem - What is it good for?

Some Zynga-ish examples
Bayesian A/B testing (T2 subsidiary)
Mixed Media Model (Cathy)
Creative Health Monitoring (Ivy)
Gr...

What to expect from this workshop
What is it?
Why is it true?
When is it useful?
How to use it?
Specifically how to use it...

Who is Steve?
“Steve is very shy and withdrawn,
invariably helpful but with very little
interest in people or in the world...

Kahneman and Tversky
“Steve is very shy and withdrawn,
invariably helpful but with very little
interest in people or in th...

Kahneman and Tversky
“Steve is very shy and withdrawn,
invariably helpful but with very little
interest in people or in th...

20
1

Spoiler Alert
We are going to describe this in pictures
20
1

200
10

200
10
“Steve is very shy and withdrawn,
invariably helpful but with very little
interest in people or in the world of
rea...

200
10
40%

200
10
40%
10%

200
4
40%
10%

4
40%
10%
20

4
40%
10%
20
P(librarian | description) =

Heart of Bayes Theorem
All
Possibilities
All
Possibilities
fitting
evidence
P(Librarian
given the
evidence)

Heart of Bayes Theorem
All
Possibilities
All
Possibilities
fitting
evidence
P(Librarian
given the
evidence)

Heart of Bayes Theorem
For each
possibility
Count the
number of
ways evidence
can happen
Make this
ratio:
P(Librarian
give...

When to use Bayes Rule
We have a
hypothesis
We observe some
evidence
“Steve is very shy and
withdrawn, invariably helpful
...

Heart of Bayes Theorem
All
Possibilities
All
Possibilities
fitting
evidence
P(Librarian
given the
description)
How can we ...

Goal: P(H|E)
P(hypothesis given
evidence)
P(Librarian given
description)

Goal: P(H|E)
P(H) = 1/21

Goal: P(H|E)
Prior → P(H) = 1/21

Goal: P(H|E)
Prior → P(H) = 1/21
P(E|H) = 0.4

Goal: P(H|E)
Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood

Goal: P(H|E)
Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
P(H|E)

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
P(H|E) =

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) =

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) =

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) = =
(# )
210

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) = =
(# )
210

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) = =
(# ) P(H)
10

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) = =
(# ) P(H) P(E|H)
4

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) = =
(# ) P(H) P(E|H)
4
(# ) P(H) P(E|H)
4

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) = =
(# ) P(H) P(E|H)
(# ) P(H) P(E|H) + (# )
4
4 210

P(￢H) = 20/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) = =
(# ) P(H) P(E|H)
(# ) P(H) P(E|H) + (# ) P(￢H)
4
4 200

P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) = =
(# ) P(H) P(E|H)
(# ) P(H) P(E|H) + (# ) P(￢H)P(E|￢H)
4
4 20
P(￢H) = 20...

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) = =
(# ) P(H) P(E|H)
(# ) P(H) P(E|H) + (# ) P(￢H) P(E|...

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) =
P(H) P(E|H)
P(H) P(E|H) + P(￢H) P(E|￢H)

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) =
P(H) P(E|H)
P(H) P(E|H) + P(￢H) P(E|￢H)
Bayes
Theorem

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) =
P(H) P(E|H)
P(H) P(E|H) + P(￢H) P(E|￢H)
Bayes
Theorem...

Prior → P(H) = 1/21
P(E|H) = 0.4
Likelihood
P(E|￢H) = 0.1
+
P(H|E) =
P(H) P(E|H) Bayes
Theorem
P(E)
Posterior

P(H|E) =
P(H) P(E|H) Bayes
Theorem
P(E)

P(H|E) =
P(H) P(E|H) Bayes
Theorem
P(E)
1
1

P(H|E) =
P(H) P(E|H) Bayes
Theorem
P(E)
1
1

P(H|E) =
P(H) P(E|H) Bayes
Theorem
P(E)
1
1
Final Note About Steve

Bayesian Data Analysis
in 3 easy steps!
1. For each possible explanation
of the data
2. Count all the ways data can
happen...

Urns and Marbles - really?

For each possibility

For each possibility

First Possibility

Second
Possibility

Third Possibility

Count ways evidence can occur

Count ways evidence can occur

Count ways evidence can occur

Count ways evidence can occur

Make the ratio

Make the ratio

Make the ratio

Bayesian Updating
in 3 easy steps!
1. Make a model for how observations can happen. Do it
again for each possible explanat...

What
proportion of
the earth is
covered by
water?
demo

1. For each possible
explanation of the
data
2. Count all the ways
data can happen
3. Make a ratio
Get more Evidence and
U...

Bayes enables
updating beliefs
by (fancy) counting

Bayes enables
updating beliefs
using MCMC

When to use Bayesian inference?
● Data is limited
○ Updating Bayesian estimates converge with modest data support
○ In lar...

End of Session 1 - Questions?

Parachute
Manufacturing
Bayesian Workshop

Take-Two: Parachute Division
Strauss just announced that Take-Two is going to pivot it’s Zynga
Mobile division to solely m...

Aerodynamics Overview
Luckily, Take-Two has a resident Rocket
Scientist on retainer. Dr. Ryan has offered
to give some bac...

Aerodynamics Overview
Our manufacturer provided
us the weight measurement.
Equation for Drag force:
Density of air
Referen...

Aerodynamics Overview
Once a ‘Chute is at a terminal
velocity, Drag = Weight. We can
solve for V.
Great, but how can we fi...

Drag Coefficient Experiment
Proposed experiment:
Drop for a known height and record the
glide time. We know:
Frequentist a...

Bayesian Approach
Our experiment generates data. Our unknown
random variable 𝛉 is Cd.
Likelihood:
For a given value of Cd,...

Bayesian Approach
Likelihood:
Prior:
Expert consultant:
Posterior:
PyMC3 package will solve for this, and will give
us pos...

Challenge
Goal: Get Strauss and Legal a Bayesian estimate of Cd!
Split up into groups of two, and conduct your experiment:...

Reference Material
● Bayesian parameter estimation examples:
https://epubs.siam.org/doi/pdf/10.1137/100788604
https://cime...

Appendix

Bayes is
just
counting
computing
the
posterior

MCMC computing
the
posterior

MCMC

MCMC
You can evaluate the function, but you
cannot draw sample from the function

chi-feng.github.io
D E M O

When to use Bayesian inference?
● When data is lacking
○ Frequentist: large error bars, poor estimates
○ By making more as...

Bayes’ Theorem

Bayes’ Theorem

Bayes’ Theorem

Bayes’ Theorem
likelihood evidence
posterior
Observations, data, features
Outcome, label

Bayesian POV
● Experiment → prior notion + data = new (posterior) notion
● Unknown parameters → associated probabilities i...

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