Notation glossary
The symbols and conventions this course commits to
So a symbol never means two things, the course fixes one convention wherever conventions compete. These are the choices the weekly notes inherit. (This mirrors the instructor’s private notation ledger; it is an original reference, not reproduced from any source.)
Core symbols
| Symbol | Reads as | Convention |
|---|---|---|
| \(P(A)\) | probability of the event \(A\) | capital \(P\) for events |
| \(f(y)\), \(f(\theta)\) | a probability mass/density function | lowercase \(f\) for pmf/pdf |
| \(\theta\) | the unknown parameter | Greek letters for parameters |
| \(y\) | the observed data | lowercase Latin for data |
| \(f(\theta)\) | the prior | the prior is written \(f(\theta)\) |
| \(L(\theta \mid y)\) | the likelihood | a function of \(\theta\) — not a distribution over \(\theta\) |
| \(f(\theta \mid y)\) | the posterior | the prior updated by the data |
| \(f(y)\) (marginal) | the evidence / marginal likelihood | the normalizing constant, never “a likelihood” |
| \(\propto\) | “proportional to” | drops constants that do not depend on \(\theta\) |
The one equation everything rests on
\[ f(\theta \mid y) \;=\; \frac{L(\theta \mid y)\, f(\theta)}{f(y)} \;\propto\; L(\theta \mid y)\, f(\theta). \]
In words: the posterior is proportional to the likelihood times the prior. The evidence \(f(y)\) is just the constant that makes the posterior integrate to one. When we write \(\propto\), we are deliberately dropping that constant.
Model parameterizations we fix
- Beta prior: \(\text{Beta}(\alpha, \beta)\) in the shape–shape parameterization (never mean/precision).
- Gamma prior: \(\text{Gamma}(s, r)\) in the shape–rate parameterization (\(s\) = shape, \(r\) = rate; never shape–scale).
- Normal: \(N(\mu, \sigma^2)\) in the mean–variance parameterization (\(\sigma^2\), not \(\sigma\), not precision).
Words we keep distinct
- A credible interval is a posterior statement: “given the data, there is a 95% probability the parameter lies in \([L, U]\).” We always say credible for posteriors.
- A confidence interval is a classical statement about a procedure’s long-run coverage. It is not a probability about the parameter. We contrast the two carefully in Week 12.
- A posterior mean is one number; it is never “the answer” by itself — always report it with a credible interval.
Public vs. graded
A public reference page. Graded work and its keys are authoritative in the LMS (Blackboard).