AI · PROBABILISTIC REASONING

15 · Probabilistic Programming

Represent uncertainty with ordinary program structure: random choices describe possible worlds, and evidence narrows which worlds remain plausible.

15.1 Relational Probability Models

Relational models share uncertainty across entities and their links. A property of one person can depend on the properties of connected people, not just on a fixed list of variables.

15.2 Open-Universe Probability Models

In an open universe, the number and identity of objects are uncertain too. A model can generate a different set of people or events in each possible world.

15.3 Keeping Track of a Complex World

A world contains many dependent random choices. Structured representations and inference algorithms make it practical to reason about only the parts relevant to a query.

15.4 Programs as Probability Models

A probabilistic program combines deterministic computation, random sampling, and observations. Running it samples worlds; conditioning on evidence focuses inference on worlds consistent with what is observed.

Key terms and a world trace

Possible world: One complete assignment to the model's random choices: here, one weather value, one population size, and one health outcome per resident.
Conditioning / evidence: Conditioning incorporates an observed fact (for example, “it is rainy”) so inference considers worlds consistent with that fact. This demo samples worlds; it does not perform posterior inference.
Open universe: The set or count of entities is itself random, rather than fixed in advance.
Shared dependency: Weather is sampled once, then influences every resident's health chance; resident outcomes are conditionally independent given that shared weather in this toy model.

Pseudocode / trace

rainy ~ Bernoulli(0.35); n ~ Uniform{1,…,6}
repeat n times: ill[i] ~ Bernoulli(rainy ? 0.58 : 0.14)
Example: rainy → n=3 → outcomes (ill, well, ill) → 2 ill residents

The numbers illustrate the assumed generator, not evidence that real weather causes illness at these rates.

Interactive model

Sample a changing population

Each run samples shared weather, an uncertain population size, then each resident's health conditional on that weather.

World summary

Not sampled

Residents: —

Ill: —

Ready. Sample a world to see its random choices.

Demo performance

Let n be the sampled resident count. Sampling uses O(1) time for weather and population count plus O(n) health draws, so total time is O(n) worst case; here n ≤ 6, making it bounded O(1) in this fixed demo. Algorithm auxiliary space is O(1) counters beyond the result; the displayed world needs O(n) DOM elements. Reset removes the displayed elements in O(n) time.