# BIC for a two-dimensional discrete distribution

> ML preparation item.

- Stable ID: `pml-book1-ex4.5`
- Area: Machine Learning
- Topic: Theory
- Origin: external
- Expected time: 20 minutes
- Skills: generalization, overfitting, model-complexity
- Item JSON: https://mlprep.iwase.dev/data/items/pml-book1-ex4.5.json

This site does not reproduce the third-party problem.

- Source: Probabilistic Machine Learning: An Introduction — online draft dated 2025-04-18
- Document: PML Book 1 online draft (2025-04-18), Chapter 4 exercises
- Locator: Exercise 4.5, book p. 165 (PDF p. 195)
- Official answer PDF page(s): 11–12
- Original problem: https://github.com/probml/pml-book/releases/download/2025-04-18/book1.pdf#page=195
- Official answer: https://probml.github.io/pml-book/solns-public.pdf#page=11
- Answer status: official-solution
- Source caveat: The public solution manual predates the dated book draft; the exercise title and task align, but notation may differ.


## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `pml-book1-ex4.5`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.