Today

• Who you are, what you're hoping to get out of the course
• Why this course exists — inductive inference under uncertainty
• Basic probability and Bayes' rule
• Two worked examples: sick friend, Kahneman-Tversky cab problem
• What is GenJAX, and how will we use it?
• Logistics and homework

Deduction vs. induction

The problems that people excel at — where we outperform machines — are inductive.

They feel easy. They are not.

1. Perception

Which square is darker — A or B?

The visual system solves
square color + shadow = intensity
for square color, using priors about how retinal images are generated.

You are literally looking at an inductive inference right now.

2. Mental states

Data: other people's behavior (motion, speech, gaze).
Hypotheses: their goals and beliefs.

Heider & Simmel (1944) — animated geometric shapes.

Everyone spontaneously narrates them as agents with intentions, grudges, fears.

Those intentions were never in the video.

(Watch ~90s: https://www.youtube.com/watch?v=VTNmLt7QX8E)

3. Word learning

You're in the Australian outback. Someone points at a hopping animal and says "jumbuck."

What does jumbuck mean?

• the animal itself?
• undetached kangaroo-parts?
• kangaroo temporal-stages?
• any mammal?
• dinner?

All consistent with the data. Children pick one — fast.
They bring strong prior expectations about what words mean.

Outcomes, events, event space

Flip two coins.

• Outcome set: $S = \{HH, HT, TH, TT\}$     $|S| = 4$
• Event: any subset of $S$.
  • e.g., "at least one head" $= \{HH, HT, TH\}$
• Event space: the set of events (the powerset of $S$, when $S$ is finite).

For continuous $S$, the math gets trickier — we'll come back to it next week.

Random variables

A random variable is a function from the outcome space to some value space.

$$Y: S \to \{T, F\}, \quad Y(s) = \text{"does } s \text{ have at least one H?"}$$

$$Y(HH) = T, \quad Y(HT) = T, \quad Y(TH) = T, \quad Y(TT) = F.$$

The word random is about the input, not the mapping. The function itself is deterministic.

Probability

$$P(Y = T) = \frac{|\{s \in S : Y(s) = T\}|}{|S|} = \frac{3}{4}$$

Count the outcomes where $Y$ takes the value you care about. Divide by total outcomes.

Joint probability

$$P(\text{first} = H,\, \text{second} = T) = P(\{HT\}) = \tfrac{1}{4}$$

Joint = "both of these at once."

Conditional probability — set restriction

The ratio definition

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$

Same answer:

$$P(\text{first}=H \mid \geq 1\,H) = \frac{P(\{HH\})}{P(\{HH, HT, TH\})} = \frac{2/4}{3/4} = \frac{2}{3} \checkmark$$

Marginalization

$$P(X) = \sum_c P(X, C=c)$$

Sum the joint over the values of whatever you want to get rid of.

Bayes' rule — one line

From the product rule, both directions:

$$P(A \mid B)\,P(B) = P(A, B) = P(B \mid A)\,P(A)$$

Divide:

$$\boxed{\,P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}\,}$$

That's it. The rest of the course is understanding what this means.

Sick friend

Your 50-year-old friend has been a chain smoker since 18.
There's a nasty cold going around.
You go over to their house. You hear them cough.

Three possibilities:

• Cold     ·     Stomach virus     ·     Lung cancer

How likely is each?

First pass — qualitatively

Before any math: cold is most likely.

With numbers

Sum of numerators: $0.405 + 0.045 + 0.090 = 0.540$.
Divide each numerator by the sum to normalize.

The pedagogical punchline

$$P(\text{cold} \mid \text{cough}) = \frac{0.9 \cdot 0.45}{0.9 \cdot 0.45 + 0.45 \cdot 0.10 + 0.9 \cdot 0.10} = \frac{0.405}{0.540} = 0.75$$

Even with a chain-smoking friend, $P(\text{lung cancer} \mid \text{cough}) \approx 17\%$ — not dominant.

The cold wins because:

1. Priors matter: colds are going around; lung cancer is rare in any given week, even for smokers. Cold has a ~4.5× higher prior.
2. Likelihoods matter: a cough is more likely from a cold (0.9) than a stomach virus (0.45).
3. Neither alone is enough. You need the whole rule.

Setup

• A city has two cab companies: Blue and Green.
• 85% of cabs are Green. 15% are Blue.
• At night, a hit-and-run happens. A witness says the cab was Blue.
• The witness correctly identifies cab colors at night 80% of the time.

What is the probability the cab was actually Blue?

(Commit to a number before we compute.)

Pass 2 — the area diagram

Imagine 100 cabs.

• Blue cabs: 15. Witness correctly says "blue" on 80% of them → 12.
• Green cabs: 85. Witness incorrectly says "blue" on 20% of them → 17.

Total "blue" reports: $12 + 17 = 29$.

$$P(\text{Blue} \mid \text{witness says Blue}) = \frac{12}{29} \approx 0.41$$

Below 50%. The witness is more likely wrong than right, despite being 80% accurate.

Pass 3 — formal Bayes

Hypotheses $h \in \{B, G\}$.   Observation $o$ = "witness says Blue."

$$P(B \mid o) = \frac{P(o \mid B)\,P(B)}{P(o \mid B)\,P(B) + P(o \mid G)\,P(G)}$$

$$= \frac{0.80 \cdot 0.15}{0.80 \cdot 0.15 + 0.20 \cdot 0.85} = \frac{0.12}{0.12 + 0.17} = \frac{0.12}{0.29} \approx 0.41$$

Same number as the area diagram. The equation and the counting are the same calculation.

Base-rate neglect

Most people report something close to 80%.
They anchor on the likelihood and ignore the prior.

Kahneman & Tversky (1972); Bar-Hillel (1980).

Medical doctors fail this too, even for diagnoses (Casscells, Schoenberger, Grayboys, 1978).

Heuristics and biases

Synthesis — the 5-step recipe

This is the course's method.

1. Formalize the problem people face.
2. Formalize the knowledge they bring to it.
3. Apply Bayes' rule — compute what an ideal learner would infer.
4. Characterize the ideal learner's behavior.
5. Identify what knowledge and constraints must be assumed for model and human behavior to match.

Every week we pick a different cognitive domain and run this loop.

What is GenJAX?

Bayes' rule is easy to write. Enumerating hypothesis spaces by hand is not.

For continuous distributions you can't enumerate at all.

GenJAX is a probabilistic programming language.
You write the generative process as code. The machine does the enumeration and counting.

GenJAX by example

import genjax
from genjax import gen, flip

@gen
def chibany_day():
    lunch = flip(0.5) @ "lunch"
    dinner = flip(0.5) @ "dinner"
    return (lunch, dinner)

• @gen — this is a generative model, not a regular function.
• flip(0.5) — a Bernoulli random variable.
• @ "lunch" — name the random choice so we can condition on it later.

This function IS the outcome space $\Omega$ from Block 3. Every call samples a point.

Preemptive callouts

• Use flip(p), not bernoulli(p). bernoulli takes a logit, not a probability.
• The @ operator here is not matrix multiplication. GenJAX overloads it.
• Output displays as Array(0, dtype=int32). Treat these as 0s and 1s.
• Everything runs in Google Colab. No local installation.

Admin — grading

All four assignments are completed in GenJAX. Weekly reflections replace the SP25 paper-presentation format (doesn't fit a 2-4 person seminar).

Admin — the rest

• Textbook: A Narrative Introduction to Probability — https://josephausterweil.github.io/probintro/
• Tooling: Google Colab, GenJAX. No local setup.
• Office hours: TBD — I'll poll.
• Email: best-effort 36h response (48h on weekends).
• AI tools: welcome as a technical resource; must cite; not for quiz/assignment answers.
• Full syllabus on the course website (coming up).

Homework for Week 2

1. Textbook T1 Ch 1-3 — reinforces everything from Block 3.
   intro/01_goals.md · 02_hungry.md · 03_prob_count.md

2. Textbook T2 Ch 0-1 — gets GenJAX running in Colab.
   genjax/00_getting_started.md · 01_python_basics.md

3. Start thinking about a final-project topic. We'll talk Week 2.

Week 2 has a short optional self-check quiz (Intro Probability Theory 1).