Notation and assumptions

Core symbols

Symbol	Meaning
\(y, y_{1:n}\)	observed data
\(\tilde{y}\)	future or replicated data
\(\theta\)	parameter vector
\(\phi\)	hyperparameter or nuisance parameter
\(\pi(\theta)\)	prior density or mass function
\(p(y \mid \theta)\)	likelihood
\(p(\theta \mid y)\)	posterior
\(p(\tilde{y} \mid y)\)	posterior predictive distribution
\(\ell(\theta; y)\)	log-likelihood
\(U(\theta)\)	score or potential energy, depending on context
\(\mathcal{L}(q)\)	evidence lower bound (ELBO)
\(\hat{R}\)	rank-normalized convergence diagnostic
ESS	effective sample size

Probability conventions

Densities and probability mass functions are both written with \(p(\cdot)\) unless distinction is essential.
Priors are often written as \(\pi(\theta)\) to distinguish them visually from sampling densities.
Expectations under a distribution \(p\) are written \(\mathbb{E}_p[\cdot]\) when ambiguity is possible.
Conditional independence is denoted by \(\perp\!\!\!\perp\).

Modeling conventions

Scalar parameters use lower-case Greek letters.
Vectors use bold Roman letters when clarity matters, for example \(\mathbf{x}_i\).
Matrices use upper-case Roman letters, for example \(X\).
Hierarchical indices follow the pattern \(y_{ij}\) for unit \(i\) in group \(j\).

Computational conventions

Unnormalized log posteriors are used frequently because MCMC and optimization often need only ratios or gradients.
All MCMC examples assume the target density is known up to proportionality.
Diagnostics are interpreted jointly rather than in isolation.

Standing assumptions

Unless otherwise stated:

parameter spaces are measurable;
posterior normalizing constants exist when posteriors are claimed proper;
gradients and Hessians are assumed to exist where optimization or HMC derivations need them;
independence claims are always model assumptions, not universal truths.

---
title: "Notation and assumptions"
number-sections: false
---

## Core symbols

| Symbol | Meaning |
|---|---|
| $y, y_{1:n}$ | observed data |
| $\tilde{y}$ | future or replicated data |
| $\theta$ | parameter vector |
| $\phi$ | hyperparameter or nuisance parameter |
| $\pi(\theta)$ | prior density or mass function |
| $p(y \mid \theta)$ | likelihood |
| $p(\theta \mid y)$ | posterior |
| $p(\tilde{y} \mid y)$ | posterior predictive distribution |
| $\ell(\theta; y)$ | log-likelihood |
| $U(\theta)$ | score or potential energy, depending on context |
| $\mathcal{L}(q)$ | evidence lower bound (ELBO) |
| $\hat{R}$ | rank-normalized convergence diagnostic |
| ESS | effective sample size |

## Probability conventions

- Densities and probability mass functions are both written with $p(\cdot)$ unless distinction is essential.
- Priors are often written as $\pi(\theta)$ to distinguish them visually from sampling densities.
- Expectations under a distribution $p$ are written $\mathbb{E}_p[\cdot]$ when ambiguity is possible.
- Conditional independence is denoted by $\perp\!\!\!\perp$.

## Modeling conventions

- Scalar parameters use lower-case Greek letters.
- Vectors use bold Roman letters when clarity matters, for example $\mathbf{x}_i$.
- Matrices use upper-case Roman letters, for example $X$.
- Hierarchical indices follow the pattern $y_{ij}$ for unit $i$ in group $j$.

## Computational conventions

- Unnormalized log posteriors are used frequently because MCMC and optimization often need only ratios or gradients.
- All MCMC examples assume the target density is known up to proportionality.
- Diagnostics are interpreted jointly rather than in isolation.

## Standing assumptions

Unless otherwise stated:

- parameter spaces are measurable;
- posterior normalizing constants exist when posteriors are claimed proper;
- gradients and Hessians are assumed to exist where optimization or HMC derivations need them;
- independence claims are always model assumptions, not universal truths.