General Inference

Chris Gregg

CS109, Stanford University

Summer 2026

Why You Need a Model

2

Chris Piech, CS109, 2021

Why You Need a Model

3

Chris Piech, CS109, 2021

Multiple Random Variables. Start of Digital Revolution

Chris Piech, CS109, 2021

Surprisingly Simple (if you can code)

Code

Probability

Chris Piech, CS109, 2021

Four Prototypical Trajectories

Three Guiding Questions

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1. How do people actually define large models?

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1. How can we do inference in large models?

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1. What data can inform the design process?

Chris Piech, CS109, 2021

Four Prototypical Trajectories

At this point you know inference with two random variables

Last time: Five New Real + Exciting Problems

8

Age from C14

Updated Delivery Prob

Age from Name

Hidden Chambers

Stanford Eye Test

Updating Lidar Belief

Today

Chris Piech, CS109

Chris Piech, CS109

Likelihood function f(X=x | T=t)? (soln)

Chris Piech, CS109

Observe LiDAR measurement of 4m. f(X=4 | T=t)=?

Chris Piech, CS109

We want a new belief in the true dist., f(T=t | X=4)

Chris Piech, CS109

Four Prototypical Trajectories

Many real world problems have way more than two random variables…

Chris Piech, CS109, 2021

Why You Need a Model

14

Chris Piech, CS109, 2021

Why You Need a Model

15

Chris Piech, CS109, 2021

Multiple Random Variables. Start of Digital Revolution

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Chris Piech, CS109, 2021

Challenge #1: Many Inference Questions

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Inference question:

Given the values of some random�variables, what are the conditional�distributions of some other random�variables?

Chris Piech, CS109, 2021

Challenge #1: Many Inference Questions

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One inference question:

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Chris Piech, CS109, 2021

Challenge #1: Many Inference Questions

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Another inference question:

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Chris Piech, CS109, 2021

Challenge #2: Joint is Large

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Chris Piech, CS109, 2021

Challenge #2: Joint is Large

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Naively specifying a joint distribution is, in general, intractable.

Chris Piech, CS109, 2021

N can be large…

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Chris Piech, CS109, 2021

N can be large…

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Chris Piech, CS109, 2021

Four Prototypical Trajectories

Three Guiding Questions

​

1. How do people actually define large models?

​

1. How can we do inference in large models?

​

1. What data can inform the design process?

Chris Piech, CS109, 2021

Four Prototypical Trajectories

Three Guiding Questions

​

1. How do people actually define large models?

​

1. How can we do inference in large models?

​

1. What data can inform the design process?

Chris Piech, CS109, 2021

Why You Need a Model

26

Chris Piech, CS109, 2021

A simpler WebMD

27

Flu

Tired

Fever

Chris Piech, CS109, 2021

A simpler WebMD

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Flu

Tired

Fever

Chris Piech, CS109, 2021

Constructing a Bayesian Network

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✅

Flu

Tired

Fever

Chris Piech, CS109, 2021

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Recall: Probabilistic Model

Chris Piech, CS109, 2021

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Recall: Probabilistic Model

✅

Chris Piech, CS109, 2021

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Recall: Probabilistic Model

✅

Check your Understanding:

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What is P(Fev=0 | Flu = 1)?

Chris Piech, CS109, 2021

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Recall: Probabilistic Model

✅

Chris Piech, CS109, 2021

Four Prototypical Trajectories

Could we write a python program which makes a fake person from this joint?

Chris Piech, CS109, 2021

To the Code

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Can You Sample from the Joint?

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Writing a python program that can sample from the joint, is the same as defining the joint.

Chris Piech, CS109, 2021

Make a Generative Model

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A good probabilistic model is generative. It explains the process through which the joint is created.

Chris Piech, CS109, 2021

Generative Models make Independence Assumptions

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✅

✅

Flu

Tired

Fever

Chris Piech, CS109, 2021

Generative Models make Independence Assumptions

• Each random variable is conditionally independent of its causal non-descendants, given its causal parents.

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Flu

Tired

Fever

Chris Piech, CS109, 2021

Generative Models make Independence Assumptions

• This model assumes that Flu and being an Undergraduate are independent.

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• Advanced: it also assumes that fever and tired are conditionally independent given Flu.

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• You need to tell a generative story. The independence assumptions come for free.

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Flu

Tired

Fever

Chris Piech, CS109, 2021

Bug: Constructing a Bayesian Network

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Must by acyclic!

Flu

Tired

Fever

Chris Piech, CS109, 2021

Four Prototypical Trajectories

Three Guiding Questions

​

1. How do people define large models?

​

1. How can we do inference in large models?

​

1. What data can inform the design process?

Chris Piech, CS109, 2021

Four Prototypical Trajectories

Three Guiding Questions

​

1. How do people define large models?

​

1. How can we do inference in large models?

​

1. What data can inform the design process?

Chris Piech, CS109, 2021

Four Prototypical Trajectories

Chris Piech, CS109, 2021

Algorithm #2: Rejection Sampling

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Chris Piech, CS109, 2021

Algorithm #2: Rejection Sampling

50

Chris Piech, CS109, 2021

Algorithm #2: Rejection Sampling

51

Chris Piech, CS109, 2021

Algorithm #2: Rejection Sampling

52

Chris Piech, CS109, 2021

Algorithm #2: Rejection Sampling

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Chris Piech, CS109, 2021

Algorithm #2: Rejection Sampling

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Chris Piech, CS109, 2021

Algorithm #2: Rejection Sampling

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Chris Piech, CS109, 2021

Algorithm #2: Rejection Sampling

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Chris Piech, CS109, 2021

Lets try it!

Chris Piech, CS109, 2021

Rejection sampling algorithm

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Inference�question:

🤔

Chris Piech, CS109, 2021

Rejection sampling algorithm

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🤔

Why would this definition of approximate probability make sense?

probability ≈

Inference�question:

Chris Piech, CS109, 2021

Why would this approximate probability make sense?

60

Recall our definition of probability as a frequency:

Inference�question:

probability ≈

Chris Piech, CS109, 2021

61

Each one of these is one joint sample

If you can sample enough from the joint distribution, you can answer any probability question

Chris Piech, CS109, 2021

Four Prototypical Trajectories

Lets try another question

Chris Piech, CS109, 2021

63

Cousin 1

Cousin 2

?

You observe that someone has a recessive gene.

What is the probability that their cousin has the same recessive gene?

Each person has a 1/20 chance of having the recessive gene.

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Cousin 1

Cousin 2

?

Parent 1

Parent 2

?

?

Grand Parent 1

?

Grand Parent 2

?

Spouse 1

?

Spouse 2

?

You observe that someone has a recessive gene.

What is the probability that their cousin has the same recessive gene?

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?

?

?

?

?

?

?

You observe that someone has a recessive gene.

What is the probability that their cousin has the same recessive gene?

Cousin 1

Cousin 2

Parent 1

Parent 2

Grand Parent 1

Grand Parent 2

Spouse 1

Spouse 2

Four Prototypical Trajectories

To the code!

Chris Piech, CS109, 2021

Four Prototypical Trajectories

What’s the matter with

rejection sampling?

Chris Piech, CS109, 2021

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Probabilistic Model

Chris Piech, CS109, 2021

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Probabilistic Model

Chris Piech, CS109, 2021

Four Prototypical Trajectories

Back to the code!

Chris Piech, CS109, 2021

71

MCMC

Markov Chain

Monte Carlo

Many Algorithms

Chris Piech, CS109, 2021

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Each one of these is one posterior sample:

[Flu, Undergrad, Fever, Tired]

MCMC is a way to sample with conditioned variables fixed

Many Algorithms

Chris Piech, CS109, 2021

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Many Algorithms

Rejection

Sampling

MCMC

Pyro

Idea2Text

Chris Piech, CS109, 2021

Four Prototypical Trajectories

Three Guiding Questions

​

1. How do people define large models?

​

1. How can we do inference in large models?

​

1. What data can inform the design process?

Chris Piech, CS109, 2021

Four Prototypical Trajectories

Three Guiding Questions

​

1. How do people define large models?

​

1. How can we do inference in large models?

​

1. What data can inform the design process?

Chris Piech, CS109, 2021

Chris Piech, CS109, 2021

From Correlation to Bayes Net!

reggae

rocky

funky

folky

opera

punk

country

dancy

pop

classy

categories

music

Chris Piech, CS109, 2021

Why is it harder to find independences here than for bat DNA expression?

Chris Piech, CS109, 2021

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Bat Data

Chris Piech, CS109, 2021

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Expression Amount

Chris Piech, CS109, 2021

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Spot The Difference

Chris Piech, CS109, 2021

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Spot The Difference

Chris Piech, CS109, 2021

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Vary Together

Chris Piech, CS109, 2021

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Vary Together

Chris Piech, CS109, 2021

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Vary Together

Chris Piech, CS109, 2021

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Understanding Covariance

Chris Piech, CS109, 2021

• Say X and Y are arbitrary random variables
• Covariance of X and Y:

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The Dance of the Covariance

Chris Piech, CS109, 2021

• Say X and Y are arbitrary random variables
• Covariance of X and Y:

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The Dance of the Covariance

Chris Piech, CS109, 2021

• Say X and Y are arbitrary random variables
• Covariance of X and Y:

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The Dance of the Covariance

x

y

(x – E[X])(y – E[Y])p(x,y)

Chris Piech, CS109, 2021

• Say X and Y are arbitrary random variables
• Covariance of X and Y:

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The Dance of the Covariance

x

y

Above mean

Above mean

Positive

(x – E[X])(y – E[Y])p(x,y)

Chris Piech, CS109, 2021

• Say X and Y are arbitrary random variables
• Covariance of X and Y:

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91

The Dance of the Covariance

x

y

Above mean

Above mean

Positive

Bellow mean

Bellow mean

Positive

(x – E[X])(y – E[Y])p(x,y)

Chris Piech, CS109, 2021

• Say X and Y are arbitrary random variables
• Covariance of X and Y:

​

92

The Dance of the Covariance

x

y

(x – E[X])(y – E[Y])p(x,y)

Above mean

Above mean

Positive

Bellow mean

Bellow mean

Positive

Bellow mean

Above mean

Negative

Chris Piech, CS109, 2021

• Say X and Y are arbitrary random variables
• Covariance of X and Y:

​

93

The Dance of the Covariance

x

y

Above mean

Above mean

Positive

Bellow mean

Bellow mean

Positive

Bellow mean

Above mean

Negative

Above mean

Bellow mean

Negative

(x – E[X])(y – E[Y])p(x,y)

Chris Piech, CS109, 2021

94

Covariance

Poll: (a) positive, (b) negative, (c) zero

Chris Piech, CS109, 2021

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Covariance

Is the Covariance: (a) positive, (b) negative, (c) zero

Positive

Chris Piech, CS109, 2021

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Covariance

Is the Covariance: (a) positive, (b) negative, (c) zero

Chris Piech, CS109, 2021

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Covariance

Is the Covariance: (a) positive, (b) negative, (c) zero

Negative

Chris Piech, CS109, 2021

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Covariance

Is the Covariance: (a) positive, (b) negative, (c) zero

Chris Piech, CS109, 2021

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Covariance

Is the Covariance: (a) positive, (b) negative, (c) zero

Zero

Chris Piech, CS109, 2021

• Say X and Y are arbitrary random variables
• Covariance of X and Y:

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• Equivalently:

​

​

​

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• X and Y independent ➔ Cov(X,Y) = 0
• But Cov(X,Y) = 0 does not imply X and Y independent!

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The Dance of the Covariance

Chris Piech, CS109, 2021

• Consider the following data:

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​

​

​

​

​

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101

Cov(W, H) = E[W*H] – E[W]E[H]

= 3355.83 – (62.75)(52.75)

= 45.77

Covariance and Data

Chris Piech, CS109, 2021

Four Prototypical Trajectories

Correlation

Chris Piech, CS109, 2021

• Consider the following data:

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​

​

​

​

​

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Cov(W, H) = E[W*H] – E[W]E[H]

= 3355.83 – (62.75)(52.75)

= 45.77

What is Wrong With This?

Chris Piech, CS109, 2021

• Consider the following data:

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​

​

​

​

​

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Cov(W, H) = E[W*H] – E[W]E[H]

= 3355.83 – (62.75)(52.75)

= 45.77

What is Wrong With This?

Chris Piech, CS109, 2021

Cauchy Schwarz, a great way to normalize!

105

Chris Piech, CS109, 2021

• Say X and Y are arbitrary random variables
• Correlation of X and Y, denoted ρ(X, Y):

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• Note: -1 ≤ ρ(X, Y) ≤ 1

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• ρ(X, Y) = 1 ⇒ perfectly correlated
• ρ(X, Y) = -1 ⇒ perfectly negatively correlated
• ρ(X, Y) = 0 ⇒ absence of linear relationship
• But, X and Y can still be related in some other way!
• If ρ(X, Y) = 0, we say X and Y are “uncorrelated”

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Viva La Correlatión

Chris Piech, CS109, 2021

Recall: It is a useful starting point

107

reggae

rocky

funky

folky

opera

punk

country

dancy

pop

classy

categories

music

Chris Piech, CS109, 2021

108

http://www.aei.org/publication/blog/

Rock Music Vs Oil?

High Correlation

Hubbert Peak Theory

Chris Piech, CS109, 2021

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Tell your friends!

Chris Piech, CS109, 2021

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http://www.bbc.com/news/magazine-27537142

Divorce Vs Butter?

Chris Piech, CS109, 2021

Four Prototypical Trajectories

Three Guiding Questions

​

1. How do people actually define large models?

​

1. How can we do inference in large models?

​

1. What data can inform the design process?

Chris Piech, CS109, 2021

Four Prototypical Trajectories

What haven’t we talked about?

Chris Piech, CS109, 2021

Machine Learning (last section of CS109)

113

Flu

Tired

Fever

1. Learn this from data

2. Learn this from data

Chris Piech, CS109, 2021