Probabilistic Models

Chris Gregg

CS109, Stanford University

Summer 2026

Multiple Random Variables. Start of Digital Revolution

Multiple Random Variables. Start of Digital Revolution

Bayesian Carbon Dating

4

5

First, some review

Chris Piech, CS109

Normal Distribution

5

0

-5

1

CDF of a Normal

F(x)

PDF of a Normal

f(x)

f(x) = derivative of probability

F(x) = P(X < x)

Chris Piech, CS109

Cumulative Distribution Function (CDF)

Table of Φ(z) values are precomputed

CDF of Standard Normal: A function that has been solved for numerically

The cumulative density function (CDF) of any normal

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Probability Density Function (PDF)

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Probability Density Function (PDF)

probability density at x

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Probability Density Function (PDF)

probability density at x

a constant

Chris Piech, CS109

Probability Density Function (PDF)

probability density at x

a constant

“exponential”

Chris Piech, CS109

Probability Density Function (PDF)

probability density at x

the distance to the mean

a constant

“exponential”

Chris Piech, CS109

Probability Density Function (PDF)

probability density at x

the distance to the mean

a constant

“exponential”

sigma shows up twice

Chris Piech, CS109

14

How does python sample from a Gaussian?

Chris Piech, CS109

Some code

import random

​

for i in range(10):

mean = 5

std = 1

sample = random.gauss(mean, std)

print(sample)

Some code

import random

​

for i in range(10):

mean = 5

std = 1

sample = random.gauss(mean, std)

print(sample)

4.903977966179095

4.618730222918121

6.677086972180218

4.495202488284518

4.389647328597387

3.2682120965975368

5.495799099102823

6.264671749470066

5.283570354163004

5.117329500833404

Some code

import random

​

for i in range(10):

mean = 5

std = 1

sample = random.gauss(mean, std)

print(sample)

4.903977966179095

4.618730222918121

6.677086972180218

4.495202488284518

4.389647328597387

3.2682120965975368

5.495799099102823

6.264671749470066

5.283570354163004

5.117329500833404

How does this work?

How Does a Computer Sample a Normal?

5

0

-5

1

CDF of a Normal

F(x)

PDF of a Normal

f(x)

f(x) = derivative of probability

F(x) = P(X < x)

Chris Piech, CS109

How Does a Computer Sample a Normal?

5

0

-5

1

CDF of a Normal

F(x)

PDF of a Normal

f(x)

f(x) = derivative of probability

F(x) = P(X < x)

Inverse Transform Sampling

Chris Piech, CS109

How Does a Computer Sample a Normal?

5

0

-5

1

CDF of a Normal

𝚽(x)

PDF of a Normal

f(x)

Inverse Transform Sampling

Step 1: pick a uniform number y between 0 and 1

Step 2: Find the x such that 𝚽(x) = y

Chris Piech, CS109

How Does a Computer Sample a Normal?

5

0

-5

1

CDF of a Normal

𝚽(x)

Inverse Transform Sampling

Step 1: pick a uniform number y between 0 and 1

Step 2: Find the x such that 𝚽(x) = y

x = 𝚽^-1(y)

y=0.855

Chris Piech, CS109

How Does a Computer Sample a Normal?

5

0

-5

1

CDF of a Normal

𝚽(x)

Inverse Transform Sampling

Step 1: pick a uniform number y between 0 and 1

Step 2: Find the x such that 𝚽(x) = y

x = 𝚽^-1(y)

y=0.855

Sample 1: 1.201234

Chris Piech, CS109

How Does a Computer Sample a Normal?

5

0

-5

1

CDF of a Normal

𝚽(x)

Inverse Transform Sampling

Step 1: pick a uniform number y between 0 and 1

Step 2: Find the x such that 𝚽(x) = y

x = 𝚽^-1(y)

y=0.855

Sample 1: 1.201234

y=0.3259

Chris Piech, CS109

How Does a Computer Sample a Normal?

5

0

-5

1

CDF of a Normal

𝚽(x)

Inverse Transform Sampling

Step 1: pick a uniform number y between 0 and 1

Step 2: Find the x such that 𝚽(x) = y

x = 𝚽^-1(y)

y=0.855

Sample 1: 1.201234

y=0.3259

Sample 2:

-0.45123

Chris Piech, CS109

25

End of review

Chris Piech, CS109

Epsilon: Useful perspective

* In the limit as ϵ_x goes to zero

Relative Probability of Continuous Variables

How much more likely are you to complete pset 3 in 10 hours than in 5?

X = time to finish pset 3

X ~ N(μ = 10, σ² = 2)

Relative Probability of Continuous Variables

How much more likely are you to complete pset 3 in 10 hours than in 5?

X = time to finish pset 3

X ~ N(μ = 10, σ² = 2)

Relative Probability of Continuous Variables

How much more likely are you to complete pset 3 in 10 hours than in 5?

X = time to finish pset 3

X ~ N(μ = 10, σ² = 2)

Relative Probability of Continuous Variables

How much more likely are you to complete pset 3 in 10 hours than in 5?

X = time to finish pset 3

X ~ N(μ = 10, σ² = 2)

Relative Probability of Continuous Variables

How much more likely are you to complete pset 3 in 10 hours than in 5?

X = time to finish pset 3

X ~ N(μ = 10, σ² = 2)

Where are we in CS109?

Chris Piech, CS109

CS109

Probability Fundamentals

Single Random Variables

Probabilistic Models

Uncertainty Theory

Machine Learning

Piech + Cain, CS109, Stanford University

Discrete Probabilistic Models

34

11b_discrete_joint

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The world is full of interesting probability problems

Have multiple random variables interacting with one another

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Joint probability mass functions

36

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Joint probability mass functions

37

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Joint probability mass functions

38

probability of�an event

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Joint probability mass functions

39

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Joint probability mass functions

40

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Joint probability mass functions

41

probability of the intersection�of two events

recall: the comma

random variables

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Joint probability mass functions

42

probability of the intersection�of two events

recall: the comma

random variables

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Two dice

43

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Two dice

44

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Two dice

45

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Two dice

46

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Dating at Stanford. Data from a few years ago

47

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Joint is Complete Information!

48

A joint distribution is complete information. It can be used to answer any probability question.

Chris Piech, CS109

Joint table: mutually exclusive and covers sample space.

49

X is dating status.

Y is year.

Each combination is mutually exclusive, and they span the sample space

Chris Piech, CS109

Joint table: mutually exclusive and covers sample space.

50

X is dating status.

Y is year.

Each combination is mutually exclusive, and they span the sample space

Chris Piech, CS109

Joint table: mutually exclusive and covers sample space.

51

X is dating status.

Y is year.

Each combination is mutually exclusive, and they span the sample space

Chris Piech, CS109

Joint table: mutually exclusive and covers sample space.

52

X is dating status.

Y is year.

Each combination is mutually exclusive, and they span the sample space

Chris Piech, CS109

Joint table: mutually exclusive and covers sample space.

53

X is dating status.

Y is year.

Each combination is mutually exclusive, and they span the sample space

Chris Piech, CS109

Joint table: mutually exclusive and covers sample space.

54

X is dating status.

Y is year.

Each combination is mutually exclusive, and they span the sample space

Chris Piech, CS109

Joint table: mutually exclusive and covers sample space.

55

X is dating status.

Y is year.

Each combination is mutually exclusive, and they span the sample space

Chris Piech, CS109

What is the probability someone is in a relationship?

56

We can use the law of total probability!

X is dating status. Y is year.

Frosh

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What is the probability someone is in a relationship?

57

We can use the law of total probability!

X is dating status. Y is year.

Frosh

Chris Piech, CS109

What is the probability someone is in a relationship?

58

We can use the law of total probability!

X is dating status. Y is year.

Frosh

Chris Piech, CS109

What is the probability someone is in a relationship?

59

We can use the law of total probability!

X is dating status. Y is year.

Chris Piech, CS109

What is the probability someone is in a relationship?

60

We can use the law of total probability!

X is dating status. Y is year.

Chris Piech, CS109

Four Prototypical Trajectories

Welcome the marginal

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Marginal Distribution

62

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Marginal Distribution

63

Use marginal distributions to get a 1-D RV from a joint PMF.

Chris Piech, CS109

Four Prototypical Trajectories

Why is that called the marginal?

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Quick note on independence

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If A and B are independent:

66

Joint

For all a, b

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Four Prototypical Trajectories

Biggest Game Changer this half of CS109

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Belief in Vision Given User Responses

68

P(Ability to See | Observed Responses)

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Belief in Age Given Observed C14

69

P(Age| Observed C14)

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Belief in Age Given Name

70

P(Age| Name)

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Belief in Author Given Text

71

P(Author| Text)

Author A

Author B

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Today: Inference

Inference noun

​

Updating one’s belief about a random variable (or multiple) based on conditional knowledge regarding another random variable (or multiple) in a probabilistic model.

​

TLDR: conditional probability with random variables.

Chris Piech, CS109

Bayes Theorem

P(B|E) =

P(E|B)

P(B)

P(E)

Posterior belief

Prior belief

Likelihood of

evidence

Normalization constant

Chris Piech, CS109

Bayes with Discrete Random Variables

Let M be a discrete random variable

Let N be a discrete random variable

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Bayes with Discrete Random Variables

Let M be a discrete random variable

Let N be a discrete random variable

More generally

Chris Piech, CS109

Bayes with Discrete Random Variables

Let M be a discrete random variable

Let N be a discrete random variable

More generally

Shorthand

notation

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Chris Piech, CS109

I Heard That

78

Let X be the change in gaze (measured in degrees) over 3 seconds after a sound is played

You observe X = 0. What is the probability the baby can hear the sound?

Chris Piech, CS109

Question: Have I Been Given the Joint?

79

Let X be the change in gaze (measured in degrees) over 3 seconds after a sound is played

You observe X = 0. What is the probability the baby can hear the sound?

Chris Piech, CS109

I Heard That

80

You observe X = 0. What is the probability the baby can hear the sound?

Y = 1 means the child can hear the sound

Chris Piech, CS109

I Heard That

81

You observe X = 0. What is the probability the baby can hear the sound?

Y = 1 means the child can hear the sound

Chris Piech, CS109

I Heard That

82

You observe X = 0. What is the probability the baby can hear the sound?

Y = 1 means the child can hear the sound

Chris Piech, CS109

I Heard That with Continuous

83

Normal Assumption:

​

For babies who can hear sounds, we approximate their gaze movement after the sound is played as: N(µ = 15, σ² = 25).

​

For babies who can not hear sounds, we approximate gaze movement as N(µ = 8, σ² = 25).

​

For a new baby we observe a 14 degree movement after the sound is played. What is your belief that a baby can hear, under The Normal Assumption?

​

Chris Piech, CS109

Four Prototypical Trajectories

How do you handle observing a continuous value?

Chris Piech, CS109

Four Prototypical Trajectories

Aside

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All the Bayes Belong to Us

M,N are discrete. X, Y are continuous

OG Bayes

Mix Bayes #1

Mix Bayes #2

TLDR:

​

If random variable is discrete: use PMF

​

If random variable is continuous: use PDF

Chris Piech, CS109

Mixing Discrete and Continuous

Let X be a continuous random variable

Let N be a discrete random variable

Chris Piech, CS109

Mixing Discrete and Continuous

Let X be a continuous random variable

Let N be a discrete random variable

Chris Piech, CS109

Mixing Discrete and Continuous

Let X be a continuous random variable

Let N be a discrete random variable

Chris Piech, CS109

Mixing Discrete and Continuous

Let X be a continuous random variable

Let N be a discrete random variable

Chris Piech, CS109

Mixing Discrete and Continuous

Change notation

Let X be a continuous random variable

Let N be a discrete random variable

Chris Piech, CS109

All the Bayes Belong to Us

M,N are discrete. X, Y are continuous

OG Bayes

Mix Bayes #1

Mix Bayes #2

TLDR:

​

If random variable is discrete: use PMF

​

If random variable is continuous: use PDF

Chris Piech, CS109

LOTP? Chain Rule? You can play too!

N is discrete. X is continuous

Law of total probability

Chain Rule

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Four Prototypical Trajectories

End Aside

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I Heard That with Continuous

95

Normal Assumption:

​

For babies who can hear sounds, we approximate their gaze movement after the sound is played as: N(µ = 15, σ² = 25).

​

For babies who can not hear sounds, we approximate gaze movement as N(µ = 8, σ² = 25).

​

For a new baby we observe a 14 degree movement after the sound is played. What is your belief that a baby can hear, under The Normal Assumption?

​

Chris Piech, CS109

Equivalently

96

def sample ():

# bernoulli sample

can_hear = rand_bern(0.75)

if can_hear == 1:

# gaussian sample

return rand_gauss (mu = 15 , std = 5)

else:

# gaussian sample

return rand_gauss (mu = 8, std = 5)

The function sample returned the value 14.

What is the probability that can_hear was 1?

Normal Assumption: For babies who can hear sounds, we approximate their gaze movement after the sound is played as: N(µ = 15, σ² = 25).

​

For babies who can not hear sounds, we approximate gaze movement as N(µ = 8, σ² = 25).

​

For a new baby we observe a 14 degree movement after the sound is played. What is your belief that a baby can hear, under The Normal Assumption?

​

Chris Piech, CS109

All the Bayes Belong to Us

M,N are discrete. X, Y are continuous

OG Bayes

Mix Bayes #1

Mix Bayes #2

TLDR:

​

If random variable is discrete: use PMF

​

If random variable is continuous: use PDF

Chris Piech, CS109

Goal: Inference

98

Change your belief distribution

(Joint, PMF, or PDF) of random variables, based on observations

*Note in the earlier examples, we were updating Bernoulli Random Variables

Chris Piech, CS109

Four Prototypical Trajectories

Lets Play Number or Function!

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Number or Function?

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Number

Number or Function?

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Number or Function?

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Function

(a probability mass function

if discrete)

Number or Function?

Chris Piech, CS109

104

Chris Piech, CS109

105

Baby Delivery

Its 17 days before the due date and still no baby

Chris Piech, CS109

106

Baby Delivery

Its 17 days before the due date and still no baby

Chris Piech, CS109

107

Baby Delivery

Its 17 days before the due date and still no baby

Probability baby is born on due date increases by 12%

Chris Piech, CS109

Bayesian Carbon Dating

108

Take a fixed size sample from a dead thing

Count C14 in Sample

Know the probability distribution for when it died

1

2

3

4

Profit

Chris Piech, CS109

Bayesian Carbon Dating

Before observation

After

Observation

Chris Piech, CS109

Warmup with Code

110

def update_belief(m = 900):

"""

Returns a dictionary A, where A[i] contains the

corresponding probability, P(A = i| M = 900).

m is the number of C14 molecules remaining and i

is age in years. i is in the range 100 to 10000

"""

pr_A = {}

n_years = 9901

for i in range(100,10000+1):

prior = 1 / n_years # P(A = i)

likelihood = calc_likelihood(m, i) # P(M=m | A=i)

pr_A[i] = prior * likelihood

# implicitly computes the normalization constant

normalize(pr_A)

return pr_A

Chris Piech, CS109

Warmup with Code

111

def update_belief(m = 900):

"""

Returns a dictionary A, where A[i] contains the

corresponding probability, P(A = i| M = 900).

m is the number of C14 molecules remaining and i

is age in years. i is in the range 100 to 10000

"""

pr_A = {}

n_years = 9901

for i in range(100,10000+1):

prior = 1 / n_years # P(A = i)

likelihood = calc_likelihood(m, i) # P(M=m | A=i)

pr_A[i] = prior * likelihood

# implicitly computes the normalization constant

normalize(pr_A)

return pr_A

Chris Piech, CS109

Warmup with Code

112

def update_belief(m = 900):

"""

Returns a dictionary A, where A[i] contains the

corresponding probability, P(A = i| M = 900).

m is the number of C14 molecules remaining and i

is age in years. i is in the range 100 to 10000

"""

pr_A = {}

n_years = 9901

for i in range(100,10000+1):

prior = 1 / n_years # P(A = i)

likelihood = calc_likelihood(m, i) # P(M=m | A=i)

pr_A[i] = prior * likelihood

# implicitly computes the normalization constant

normalize(pr_A)

return pr_A

Chris Piech, CS109

Normalize in Python

113

# list normalization

def normalize_list(data_list):

total_sum = np.sum(data_list)

return np.array(data_list) / total_sum

​

>>> norm = normalize_list([10, 20, 30, 40])

>>> np.sum(norm) # 1.0, always (within floating point error)

​

​

# dictionary normalization

def normalize_dict(data_dict):

total_sum = sum(data_dict.values())

normalized = {}

for key, value in data_dict.items():

normalized[key] = value / total_sum

return normalized

​

>>> norm = normalize_dict({'a': 100, 'b': 200, 'c': 300})

>>> np.sum(norm.values()) # 1.0, always (within floating point error)

​

​

return

[0.1 0.2 0.3 0.4]

return

{'a': 0.166, 'b': 0.333, 'c': 0.5}

Chris Piech, CS109

Bayesian Carbon Dating: Inference Overview

114

Let A be how many years old the sample is (A = 100 means the sample is 100 years old)

Let M be the observed amount of C14 left in the sample

Such that

Chris Piech, CS109

Understanding why Denom. is a Constant

Notice which term doesn’t change as i changes (four example calculations).

Changes with i

Doesn’t change

A = age

M = measured C14

Chris Piech, CS109

Bayesian Carbon Dating: Inference Overview

116

Let A be how many years old the sample is (A = 100 means the sample is 100 years old)

Let M be the observed amount of C14 left in the sample

Such that

Chris Piech, CS109

Understanding Through Code

117

def update_belief(m = 900):

"""

Returns a dictionary A, where A[i] contains the

corresponding probability, P(A = i| M = 900).

m is the number of C14 molecules remaining and i

is age in years. i is in the range 100 to 10000

"""

pr_A = {}

n_years = 9901

for i in range(100,10000+1):

prior = 1 / n_years # P(A = i)

likelihood = calc_likelihood(m, i) # P(M=m | A=i)

pr_A[i] = prior * likelihood

# implicitly computes the normalization constant

normalize(pr_A)

return pr_A

Chris Piech, CS109

Understanding Through Code

118

def update_belief(m = 900):

"""

Returns a dictionary A, where A[i] contains the

corresponding probability, P(A = i| M = 900).

m is the number of C14 molecules remaining and i

is age in years. i is in the range 100 to 10000

"""

pr_A = {}

n_years = 9901

for i in range(100,10000+1):

prior = 1 / n_years # P(A = i)

likelihood = calc_likelihood(m, i) # P(M=m | A=i)

pr_A[i] = prior * likelihood

# implicitly computes the normalization constant

normalize(pr_A)

return pr_A

Chris Piech, CS109

Understanding Through Code

119

def update_belief(m = 900):

"""

Returns a dictionary A, where A[i] contains the

corresponding probability, P(A = i| M = 900).

m is the number of C14 molecules remaining and i

is age in years. i is in the range 100 to 10000

"""

pr_A = {}

n_years = 9901

for i in range(100,10000+1):

prior = 1 / n_years # P(A = i)

likelihood = calc_likelihood(m, i) # P(M=m | A=i)

pr_A[i] = prior * likelihood

# implicitly computes the normalization constant

normalize(pr_A)

return pr_A

Chris Piech, CS109

Four Prototypical Trajectories

Carbon Dating Likelihood Math

Chris Piech, CS109

Probability of Having 900 Remain

121

There were originally 1000 C14 molecules.

Each molecule remains independently with equal probability p_i

What is the probability that 900 remain?

Chris Piech, CS109

Probability of Having 900 Remain

122

There were originally 1000 C14 molecules.

Each molecule remains independently with equal probability p_i

What is the probability that 900 remain?

Chris Piech, CS109

Probability of Having 900 Remain

123

There were originally 1000 C14 molecules.

Each molecule remains independently with equal probability p_i

What is the probability that 900 remain?

Each molecules’ time to live is exponential with λ = 1/8267

Chris Piech, CS109

Probability of Having 900 Remain

124

There were originally 1000 C14 molecules.

Each molecule remains independently with equal probability p_i

What is the probability that 900 remain?

Let T be the time to decay for any one molecule

Each molecules’ time to live is exponential with λ = 1/8267

Chris Piech, CS109

Probability of Having 900 Remain

125

There were originally 1000 C14 molecules.

Each molecule remains independently with equal probability p_i

What is the probability that 900 remain?

Let T be the time to decay for any one molecule

Each molecules’ time to live is exponential with λ = 1/8267

Chris Piech, CS109

Probability of Having 900 Remain

126

There were originally 1000 C14 molecules.

Each molecule remains independently with equal probability p_i

What is the probability that 900 remain?

Let T be the time to decay for any one molecule

Each molecules’ time to live is exponential with λ = 1/8267

Chris Piech, CS109

Probability of Having 900 Remain

127

def calc_likelihood(m = 900, age):

"""

Computes P(M = m | A = age), the probability of

having m molecules left given the sample is age

years old. Uses the exponential decay of C14

"""

n_original = 1000

p_remain = math.exp(-age/C14_MEAN_LIFE)

return stats.binom.pmf(m, n_original, p_remain)

Chris Piech, CS109

Probability of Having 900 Remain

128

def calc_likelihood(m = 900, age):

"""

Computes P(M = m | A = age), the probability of

having m molecules left given the sample is age

years old. Uses the exponential decay of C14

"""

n_original = 1000 + delta_start(age)

p_remain = math.exp(-age/C14_MEAN_LIFE)

return stats.binom.pmf(m, n_original, p_remain)

Chris Piech, CS109

Posterior Belief in Age

Chris Piech, CS109