Notation

This appendix collects the notation used throughout the book. Symbols are grouped thematically; the Introduced column indicates the chapter where each symbol first appears.

General

Symbol	Meaning	Domain	Introduced
$N$	Number of persons / models	$\mathbb{N}$	2 Foundations of Measurement
$M$	Number of items / questions	$\mathbb{N}$	2 Foundations of Measurement
$Y_{ij}$	Binary response of model $i$ to item $j$ (0 = incorrect, 1 = correct)	$\{0,1\}$	2 Foundations of Measurement
$S_i = \sum_j Y_{ij}$	Sum score (total correct) for model $i$	$\{0, 1, \ldots, M\}$	2 Foundations of Measurement
$\sigma(x) = \frac{1}{1+e^{-x}}$	Logistic sigmoid function	$(0,1)$	2 Foundations of Measurement
$\Phi(x)$	Standard normal CDF	$(0,1)$	2 Foundations of Measurement

Item Response Theory

Symbol	Meaning	Domain	Introduced
$\theta_i$ or $U_i$	Latent ability of model $i$	$\mathbb{R}$	2 Foundations of Measurement
$\beta_j$ or $V_j$	Difficulty of item $j$	$\mathbb{R}$	2 Foundations of Measurement
$a_j$	Discrimination parameter of item $j$	$\mathbb{R}^+$	2 Foundations of Measurement
$c_j$	Guessing (pseudo-chance) parameter of item $j$	$[0,1]$	2 Foundations of Measurement
$d_j$	Discrimination parameter (alternative notation, as in 1PL/2PL)	$\mathbb{R}^+$	2 Foundations of Measurement
$z_j$	Difficulty parameter (alternative notation)	$\mathbb{R}$	2 Foundations of Measurement
$I_j(\theta)$	Fisher information for item $j$ at ability $\theta$	$\mathbb{R}^+$	4 Efficient Measurement
$\mathcal{I}(\theta)$	Fisher information matrix	$\mathbb{R}^{K \times K}$	4 Efficient Measurement

Learning and Estimation

Symbol	Meaning	Domain	Introduced
$\ell(\theta, \beta)$	Log-likelihood function	$\mathbb{R}$	3 Learning
$\nabla_\theta \ell$	Gradient of log-likelihood w.r.t. ability parameters	$\mathbb{R}^N$	3 Learning
$\pi(\theta)$	Prior distribution over abilities	–	3 Learning
$\pi(\beta)$	Prior distribution over difficulties	–	3 Learning
$\hat{\theta}_{\text{MLE}}$	Maximum likelihood estimate of ability	$\mathbb{R}^N$	3 Learning
$\hat{\theta}_{\text{MAP}}$	Maximum a posteriori estimate of ability	$\mathbb{R}^N$	3 Learning
$\eta$	Learning rate	$\mathbb{R}^+$	3 Learning

Reliability

Symbol	Meaning	Domain	Introduced
$X_{ij}$	Observed score for model $i$ on occasion / item $j$	$\mathbb{R}$	5 Reliability
$T_i$	True score for model $i$ (CTT)	$\mathbb{R}$	5 Reliability
$E_{ij}$	Error component (CTT)	$\mathbb{R}$	5 Reliability
$\rho_{XX'}$	Reliability coefficient	$[0,1]$	5 Reliability
$\alpha$	Cronbach’s alpha	$(-\infty, 1]$	5 Reliability
$\sigma^2_p, \sigma^2_i, \sigma^2_r$	Variance components: person (model), item, rater	$\mathbb{R}^+$	5 Reliability
$G$	Generalizability coefficient	$[0,1]$	5 Reliability
$n_r, n_i$	Number of raters, items in a D-study design	$\mathbb{N}$	5 Reliability
$\kappa$	Cohen’s kappa (inter-rater agreement)	$[-1, 1]$	5 Reliability

Validity

Symbol	Meaning	Domain	Introduced
$g \in \{0,1\}$	Group membership indicator (DIF analysis)	$\{0,1\}$	6 Validity
$\alpha_{MH}$	Mantel-Haenszel odds ratio	$\mathbb{R}^+$	6 Validity
$\lambda_k$	$k$-th eigenvalue of the correlation matrix	$\mathbb{R}$	6 Validity
$\text{MNSQ}_i$	Mean-square fit statistic for item $i$	$\mathbb{R}^+$	6 Validity
$r_{ij}$	Correlation between trait $i$ measured by method $j$ (MTMM)	$[-1,1]$	6 Validity

Causality and Distribution Shift

Symbol	Meaning	Domain	Introduced
$\text{do}(X = x)$	Intervention setting variable $X$ to value $x$	–	7 Causality and Distribution Shift
$P^{(s)}, P^{(t)}$	Source (benchmark) / target (deployment) distribution	–	7 Causality and Distribution Shift
$\pi_0(a \mid x)$	Logging (benchmark) policy	$[0,1]$	7 Causality and Distribution Shift
$\pi(a \mid x)$	Target (deployment) policy	$[0,1]$	7 Causality and Distribution Shift
$w(x)$	Importance weight $P^{(t)}(x)/P^{(s)}(x)$	$\mathbb{R}^+$	7 Causality and Distribution Shift
$\hat{r}(x, a)$	Reward model (e.g., IRT prediction)	$[0,1]$	7 Causality and Distribution Shift
$\hat{V}_{\text{DR}}$	Doubly robust value estimator	$\mathbb{R}$	7 Causality and Distribution Shift
$C_\alpha(x)$	Conformal prediction set at level $\alpha$	subset of $\mathcal{Y}$	7 Causality and Distribution Shift

Information and Mechanism Design

Symbol	Meaning	Domain	Introduced
$F$	Universe of tasks	finite set, $\\|F\\| = N$	8.2 The Evaluation Game
$F_E, F_M$	Evaluator’s / builder’s task sets	$F_E, F_M \subseteq F$	8.2 The Evaluation Game
$\pi_E, \pi_M$	Evaluator’s / builder’s sampling distributions	distributions on $F$	8.2 The Evaluation Game
$f(\theta)$	Task performance function	$[0,1]$	8.2 The Evaluation Game
$u_E(\theta)$	Evaluator’s utility: $\sum_{f \in F} f(\theta)$	$\mathbb{R}^+$	8.2 The Evaluation Game
$k$	Number of sampled evaluation tasks per round	$\mathbb{N}$	8.2 The Evaluation Game
$\rho$	Distribution correction rate	$(0,1]$	8.2 The Evaluation Game
$\gamma$	Gaming penalty weight	$\mathbb{R}^+$	8.2 The Evaluation Game
$\Delta_t$	Residual misalignment at round $t$	$[0,1]$	8.2 The Evaluation Game
$C$	Agent cost (principal-agent model)	$\mathbb{R}^+$	8.2 The Evaluation Game
$b$	Principal’s value from agent effort	$\mathbb{R}^+$	8.2 The Evaluation Game

Red-Teaming and Adversarial Evaluation

Symbol	Meaning	Domain	Introduced
$\theta_j^{(\text{adv})}$	Adversarial robustness of model $j$	$\mathbb{R}$	9 Red-Teaming and Adversarial Evaluation
$\theta_j^{(\text{std})}$	Standard accuracy ability of model $j$	$\mathbb{R}$	9 Red-Teaming and Adversarial Evaluation
$\beta_i^{(\text{atk})}$	Attack strength (difficulty) of adversarial item $i$	$\mathbb{R}$	9 Red-Teaming and Adversarial Evaluation
$\delta$	Perturbation applied to an input	$\mathcal{B}_\epsilon$	9 Red-Teaming and Adversarial Evaluation
$\alpha_{s,\mathcal{D}}$	Attack success probability under criterion $s$ and goal distribution $\mathcal{D}$	$[0,1]$	9 Red-Teaming and Adversarial Evaluation
$J$	Operational judge of attack success	$\{0,1\}$	9 Red-Teaming and Adversarial Evaluation
$s$	Oracle (true) success criterion	$\{0,1\}$	9 Red-Teaming and Adversarial Evaluation
$K$	Number of repeated samples in Top-1 aggregation	$\mathbb{N}$	9 Red-Teaming and Adversarial Evaluation
$\hat{\mu}_{\text{syn}}$	Mean performance estimated from synthetic items	$[0,1]$	9 Red-Teaming and Adversarial Evaluation
$\hat{\mu}_{\text{human}}$	Mean performance estimated from human-authored items	$[0,1]$	9 Red-Teaming and Adversarial Evaluation
$\hat{\mu}_{\text{PPI}}$	Prediction-powered inference estimator	$[0,1]$	9 Red-Teaming and Adversarial Evaluation
$n$	Number of human-evaluated items	$\mathbb{N}$	9 Red-Teaming and Adversarial Evaluation

# Notation {#sec-notation .unnumbered} This appendix collects the notation used throughout the book. Symbols are grouped thematically; the **Introduced** column indicates the chapter where each symbol first appears. ## General | Symbol | Meaning | Domain | Introduced | |--------|---------|--------|------------| | $N$ | Number of persons / models | $\mathbb{N}$ | @sec-foundations | | $M$ | Number of items / questions | $\mathbb{N}$ | @sec-foundations | | $Y_{ij}$ | Binary response of model $i$ to item $j$ (0 = incorrect, 1 = correct) | $\{0,1\}$ | @sec-foundations | | $S_i = \sum_j Y_{ij}$ | Sum score (total correct) for model $i$ | $\{0, 1, \ldots, M\}$ | @sec-foundations | | $\sigma(x) = \frac{1}{1+e^{-x}}$ | Logistic sigmoid function | $(0,1)$ | @sec-foundations | | $\Phi(x)$ | Standard normal CDF | $(0,1)$ | @sec-foundations | ## Item Response Theory | Symbol | Meaning | Domain | Introduced | |--------|---------|--------|------------| | $\theta_i$ or $U_i$ | Latent ability of model $i$ | $\mathbb{R}$ | @sec-foundations | | $\beta_j$ or $V_j$ | Difficulty of item $j$ | $\mathbb{R}$ | @sec-foundations | | $a_j$ | Discrimination parameter of item $j$ | $\mathbb{R}^+$ | @sec-foundations | | $c_j$ | Guessing (pseudo-chance) parameter of item $j$ | $[0,1]$ | @sec-foundations | | $d_j$ | Discrimination parameter (alternative notation, as in 1PL/2PL) | $\mathbb{R}^+$ | @sec-foundations | | $z_j$ | Difficulty parameter (alternative notation) | $\mathbb{R}$ | @sec-foundations | | $I_j(\theta)$ | Fisher information for item $j$ at ability $\theta$ | $\mathbb{R}^+$ | @sec-efficient | | $\mathcal{I}(\theta)$ | Fisher information matrix | $\mathbb{R}^{K \times K}$ | @sec-efficient | ## Learning and Estimation | Symbol | Meaning | Domain | Introduced | |--------|---------|--------|------------| | $\ell(\theta, \beta)$ | Log-likelihood function | $\mathbb{R}$ | @sec-learning | | $\nabla_\theta \ell$ | Gradient of log-likelihood w.r.t. ability parameters | $\mathbb{R}^N$ | @sec-learning | | $\pi(\theta)$ | Prior distribution over abilities | -- | @sec-learning | | $\pi(\beta)$ | Prior distribution over difficulties | -- | @sec-learning | | $\hat{\theta}_{\text{MLE}}$ | Maximum likelihood estimate of ability | $\mathbb{R}^N$ | @sec-learning | | $\hat{\theta}_{\text{MAP}}$ | Maximum a posteriori estimate of ability | $\mathbb{R}^N$ | @sec-learning | | $\eta$ | Learning rate | $\mathbb{R}^+$ | @sec-learning | ## Reliability | Symbol | Meaning | Domain | Introduced | |--------|---------|--------|------------| | $X_{ij}$ | Observed score for model $i$ on occasion / item $j$ | $\mathbb{R}$ | @sec-reliability | | $T_i$ | True score for model $i$ (CTT) | $\mathbb{R}$ | @sec-reliability | | $E_{ij}$ | Error component (CTT) | $\mathbb{R}$ | @sec-reliability | | $\rho_{XX'}$ | Reliability coefficient | $[0,1]$ | @sec-reliability | | $\alpha$ | Cronbach's alpha | $(-\infty, 1]$ | @sec-reliability | | $\sigma^2_p, \sigma^2_i, \sigma^2_r$ | Variance components: person (model), item, rater | $\mathbb{R}^+$ | @sec-reliability | | $G$ | Generalizability coefficient | $[0,1]$ | @sec-reliability | | $n_r, n_i$ | Number of raters, items in a D-study design | $\mathbb{N}$ | @sec-reliability | | $\kappa$ | Cohen's kappa (inter-rater agreement) | $[-1, 1]$ | @sec-reliability | ## Validity | Symbol | Meaning | Domain | Introduced | |--------|---------|--------|------------| | $g \in \{0,1\}$ | Group membership indicator (DIF analysis) | $\{0,1\}$ | @sec-validity | | $\alpha_{MH}$ | Mantel-Haenszel odds ratio | $\mathbb{R}^+$ | @sec-validity | | $\lambda_k$ | $k$-th eigenvalue of the correlation matrix | $\mathbb{R}$ | @sec-validity | | $\text{MNSQ}_i$ | Mean-square fit statistic for item $i$ | $\mathbb{R}^+$ | @sec-validity | | $r_{ij}$ | Correlation between trait $i$ measured by method $j$ (MTMM) | $[-1,1]$ | @sec-validity | ## Causality and Distribution Shift | Symbol | Meaning | Domain | Introduced | |--------|---------|--------|------------| | $\text{do}(X = x)$ | Intervention setting variable $X$ to value $x$ | -- | @sec-causality | | $P^{(s)}, P^{(t)}$ | Source (benchmark) / target (deployment) distribution | -- | @sec-causality | | $\pi_0(a \mid x)$ | Logging (benchmark) policy | $[0,1]$ | @sec-causality | | $\pi(a \mid x)$ | Target (deployment) policy | $[0,1]$ | @sec-causality | | $w(x)$ | Importance weight $P^{(t)}(x)/P^{(s)}(x)$ | $\mathbb{R}^+$ | @sec-causality | | $\hat{r}(x, a)$ | Reward model (e.g., IRT prediction) | $[0,1]$ | @sec-causality | | $\hat{V}_{\text{DR}}$ | Doubly robust value estimator | $\mathbb{R}$ | @sec-causality | | $C_\alpha(x)$ | Conformal prediction set at level $\alpha$ | subset of $\mathcal{Y}$ | @sec-causality | ## Information and Mechanism Design | Symbol | Meaning | Domain | Introduced | |--------|---------|--------|------------| | $F$ | Universe of tasks | finite set, $\|F\| = N$ | @sec-evaluation-game | | $F_E, F_M$ | Evaluator's / builder's task sets | $F_E, F_M \subseteq F$ | @sec-evaluation-game | | $\pi_E, \pi_M$ | Evaluator's / builder's sampling distributions | distributions on $F$ | @sec-evaluation-game | | $f(\theta)$ | Task performance function | $[0,1]$ | @sec-evaluation-game | | $u_E(\theta)$ | Evaluator's utility: $\sum_{f \in F} f(\theta)$ | $\mathbb{R}^+$ | @sec-evaluation-game | | $k$ | Number of sampled evaluation tasks per round | $\mathbb{N}$ | @sec-evaluation-game | | $\rho$ | Distribution correction rate | $(0,1]$ | @sec-evaluation-game | | $\gamma$ | Gaming penalty weight | $\mathbb{R}^+$ | @sec-evaluation-game | | $\Delta_t$ | Residual misalignment at round $t$ | $[0,1]$ | @sec-evaluation-game | | $C$ | Agent cost (principal-agent model) | $\mathbb{R}^+$ | @sec-evaluation-game | | $b$ | Principal's value from agent effort | $\mathbb{R}^+$ | @sec-evaluation-game | ## Red-Teaming and Adversarial Evaluation | Symbol | Meaning | Domain | Introduced | |--------|---------|--------|------------| | $\theta_j^{(\text{adv})}$ | Adversarial robustness of model $j$ | $\mathbb{R}$ | @sec-redteaming | | $\theta_j^{(\text{std})}$ | Standard accuracy ability of model $j$ | $\mathbb{R}$ | @sec-redteaming | | $\beta_i^{(\text{atk})}$ | Attack strength (difficulty) of adversarial item $i$ | $\mathbb{R}$ | @sec-redteaming | | $\delta$ | Perturbation applied to an input | $\mathcal{B}_\epsilon$ | @sec-redteaming | | $\alpha_{s,\mathcal{D}}$ | Attack success probability under criterion $s$ and goal distribution $\mathcal{D}$ | $[0,1]$ | @sec-redteaming | | $J$ | Operational judge of attack success | $\{0,1\}$ | @sec-redteaming | | $s$ | Oracle (true) success criterion | $\{0,1\}$ | @sec-redteaming | | $K$ | Number of repeated samples in Top-1 aggregation | $\mathbb{N}$ | @sec-redteaming | | $\hat{\mu}_{\text{syn}}$ | Mean performance estimated from synthetic items | $[0,1]$ | @sec-redteaming | | $\hat{\mu}_{\text{human}}$ | Mean performance estimated from human-authored items | $[0,1]$ | @sec-redteaming | | $\hat{\mu}_{\text{PPI}}$ | Prediction-powered inference estimator | $[0,1]$ | @sec-redteaming | | $n$ | Number of human-evaluated items | $\mathbb{N}$ | @sec-redteaming |

Symbol	Meaning	Domain	Introduced
\(\theta_i\) or \(U_i\)	Latent ability of model \(i\)	\(\mathbb{R}\)	2 Foundations of Measurement
\(\beta_j\) or \(V_j\)	Difficulty of item \(j\)	\(\mathbb{R}\)	2 Foundations of Measurement
\(a_j\)	Discrimination parameter of item \(j\)	\(\mathbb{R}^+\)	2 Foundations of Measurement
\(c_j\)	Guessing (pseudo-chance) parameter of item \(j\)	\([0,1]\)	2 Foundations of Measurement
\(d_j\)	Discrimination parameter (alternative notation, as in 1PL/2PL)	\(\mathbb{R}^+\)	2 Foundations of Measurement
\(z_j\)	Difficulty parameter (alternative notation)	\(\mathbb{R}\)	2 Foundations of Measurement
\(I_j(\theta)\)	Fisher information for item \(j\) at ability \(\theta\)	\(\mathbb{R}^+\)	4 Efficient Measurement
\(\mathcal{I}(\theta)\)	Fisher information matrix	\(\mathbb{R}^{K \times K}\)	4 Efficient Measurement

Symbol	Meaning	Domain	Introduced
\(g \in \{0,1\}\)	Group membership indicator (DIF analysis)	\(\{0,1\}\)	6 Validity
\(\alpha_{MH}\)	Mantel-Haenszel odds ratio	\(\mathbb{R}^+\)	6 Validity
\(\lambda_k\)	\(k\)-th eigenvalue of the correlation matrix	\(\mathbb{R}\)	6 Validity
\(\text{MNSQ}_i\)	Mean-square fit statistic for item \(i\)	\(\mathbb{R}^+\)	6 Validity
\(r_{ij}\)	Correlation between trait \(i\) measured by method \(j\) (MTMM)	\([-1,1]\)	6 Validity

Symbol	Meaning	Domain	Introduced
\(\theta_j^{(\text{adv})}\)	Adversarial robustness of model \(j\)	\(\mathbb{R}\)	9 Red-Teaming and Adversarial Evaluation
\(\theta_j^{(\text{std})}\)	Standard accuracy ability of model \(j\)	\(\mathbb{R}\)	9 Red-Teaming and Adversarial Evaluation
\(\beta_i^{(\text{atk})}\)	Attack strength (difficulty) of adversarial item \(i\)	\(\mathbb{R}\)	9 Red-Teaming and Adversarial Evaluation
\(\delta\)	Perturbation applied to an input	\(\mathcal{B}_\epsilon\)	9 Red-Teaming and Adversarial Evaluation
\(\alpha_{s,\mathcal{D}}\)	Attack success probability under criterion \(s\) and goal distribution \(\mathcal{D}\)	\([0,1]\)	9 Red-Teaming and Adversarial Evaluation
\(J\)	Operational judge of attack success	\(\{0,1\}\)	9 Red-Teaming and Adversarial Evaluation
\(s\)	Oracle (true) success criterion	\(\{0,1\}\)	9 Red-Teaming and Adversarial Evaluation
\(K\)	Number of repeated samples in Top-1 aggregation	\(\mathbb{N}\)	9 Red-Teaming and Adversarial Evaluation
\(\hat{\mu}_{\text{syn}}\)	Mean performance estimated from synthetic items	\([0,1]\)	9 Red-Teaming and Adversarial Evaluation
\(\hat{\mu}_{\text{human}}\)	Mean performance estimated from human-authored items	\([0,1]\)	9 Red-Teaming and Adversarial Evaluation
\(\hat{\mu}_{\text{PPI}}\)	Prediction-powered inference estimator	\([0,1]\)	9 Red-Teaming and Adversarial Evaluation
\(n\)	Number of human-evaluated items	\(\mathbb{N}\)	9 Red-Teaming and Adversarial Evaluation