Appendix F: Notation Guide
This appendix provides a comprehensive reference for the mathematical notation used throughout Learning Prediction Markets — From Concepts to Strategies. Symbols are organized by category. For each entry we give the symbol, its LaTeX representation, its meaning, the chapter where it is first introduced, and an example of its usage.
Probability and Statistics
| Symbol | LaTeX | Meaning | Chapter | Example |
|---|---|---|---|---|
| $P(A)$ | P(A) |
Probability of event $A$ | 1 | $P(\text{Rain}) = 0.3$ |
| $P(A \mid B)$ | P(A \mid B) |
Conditional probability of $A$ given $B$ | 7 | $P(\text{Win} \mid \text{Incumbent}) = 0.6$ |
| $P(A, B)$ | P(A, B) |
Joint probability of $A$ and $B$ | 7 | $P(\text{Rain}, \text{Flood}) = 0.05$ |
| $\mathbb{E}[X]$ | \mathbb{E}[X] |
Expected value of random variable $X$ | 3 | $\mathbb{E}[\text{payoff}] = 0.6 \times 1 + 0.4 \times 0 = 0.6$ |
| $\text{Var}(X)$ | \text{Var}(X) |
Variance of $X$ | 8 | $\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$ |
| $\sigma$ | \sigma |
Standard deviation | 8 | $\sigma = \sqrt{\text{Var}(X)}$ |
| $\sigma^2$ | \sigma^2 |
Variance (alternative notation) | 8 | $\sigma^2 = 0.04$ |
| $\mu$ | \mu |
Mean of a distribution | 8 | $\mu = \mathbb{E}[X]$ |
| $\rho$ | \rho |
Correlation coefficient between two variables | 29 | $\rho_{XY} = \text{Cov}(X,Y) / (\sigma_X \sigma_Y)$ |
| $\text{Cov}(X,Y)$ | \text{Cov}(X,Y) |
Covariance of $X$ and $Y$ | 29 | $\text{Cov}(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]$ |
| $\Phi(\cdot)$ | \Phi(\cdot) |
Standard normal CDF | 15 | $\Phi(1.96) \approx 0.975$ |
| $\phi(\cdot)$ | \phi(\cdot) |
Standard normal PDF | 15 | $\phi(0) = 1/\sqrt{2\pi}$ |
| $\text{Beta}(\alpha, \beta)$ | \text{Beta}(\alpha, \beta) |
Beta distribution with shape parameters $\alpha, \beta$ | 8 | Prior $\sim \text{Beta}(2, 5)$ |
| $\text{Bin}(n, p)$ | \text{Bin}(n, p) |
Binomial distribution | 8 | $X \sim \text{Bin}(10, 0.4)$ |
| $\mathcal{N}(\mu, \sigma^2)$ | \mathcal{N}(\mu, \sigma^2) |
Normal (Gaussian) distribution | 8 | $X \sim \mathcal{N}(0, 1)$ |
| $\hat{p}$ | \hat{p} |
Estimated probability (from data or model) | 9 | $\hat{p} = 0.72$ |
| $p^*$ | p^* |
True underlying probability | 9 | Calibration requires $\hat{p} \approx p^*$ |
| $\mathbf{p}$ | \mathbf{p} |
Vector of probabilities over outcomes | 5 | $\mathbf{p} = (p_1, p_2, \ldots, p_k)$ |
| $n$ | n |
Number of observations or sample size | 8 | $n = 500$ historical forecasts |
| $N$ | N |
Total number of events, forecasts, or contracts | 9 | Brier score averaged over $N$ forecasts |
| $\theta$ | \theta |
Parameter of a statistical model | 8 | Posterior $P(\theta \mid \text{data})$ |
| $\hat{\theta}$ | \hat{\theta} |
Estimated parameter value | 23 | $\hat{\theta}_{\text{MLE}}$ |
Market Mechanics
| Symbol | LaTeX | Meaning | Chapter | Example |
|---|---|---|---|---|
| $\pi_i$ | \pi_i |
Market price (implied probability) of outcome $i$ | 2 | $\pi_{\text{Yes}} = 0.65$ |
| $q_i$ | q_i |
Quantity of outcome-$i$ shares outstanding | 5 | $q_{\text{Yes}} = 1{,}200$ shares |
| $\mathbf{q}$ | \mathbf{q} |
Vector of outstanding shares across outcomes | 5 | $\mathbf{q} = (q_1, q_2, \ldots, q_k)$ |
| $C(\mathbf{q})$ | C(\mathbf{q}) |
Cost function of a market maker | 5 | $C(\mathbf{q}) = b \ln\!\bigl(\sum_i e^{q_i/b}\bigr)$ (LMSR) |
| $b$ | b |
Liquidity parameter of the LMSR | 5 | $b = 100$ controls price sensitivity |
| $\Delta q_i$ | \Delta q_i |
Change in quantity of outcome-$i$ shares | 5 | Cost of trade $= C(\mathbf{q} + \Delta\mathbf{q}) - C(\mathbf{q})$ |
| $p_i^{\text{bid}}$ | p_i^{\text{bid}} |
Bid price for outcome $i$ | 4 | $p_{\text{Yes}}^{\text{bid}} = 0.62$ |
| $p_i^{\text{ask}}$ | p_i^{\text{ask}} |
Ask price for outcome $i$ | 4 | $p_{\text{Yes}}^{\text{ask}} = 0.65$ |
| $s$ | s |
Bid-ask spread | 4 | $s = p^{\text{ask}} - p^{\text{bid}} = 0.03$ |
| $V$ | V |
Trading volume (number of contracts traded) | 14 | $V = 5{,}000$ contracts per day |
| $\lambda$ | \lambda |
Arrival rate of informed traders (Glosten-Milgrom) | 14 | $\lambda = 0.3$ |
| $\Delta$ | \Delta |
Market impact per unit traded | 14 | Price moves by $\Delta$ per contract |
| $x, y$ | x, y |
Token reserves in a CPMM pool | 5 | $x \cdot y = k$ |
| $k$ | k |
Constant product invariant in CPMM | 5 | $x \cdot y = k = 10{,}000$ |
| $\text{fee}$ | \text{fee} |
Transaction fee rate | 6 | $\text{fee} = 0.02$ (2%) |
Scoring Rules and Calibration
| Symbol | LaTeX | Meaning | Chapter | Example |
|---|---|---|---|---|
| $S(\mathbf{p}, \omega)$ | S(\mathbf{p}, \omega) |
Score assigned to forecast $\mathbf{p}$ when outcome $\omega$ is realized | 9 | $S(\mathbf{p}, \omega) = \ln(p_\omega)$ (log score) |
| $\text{BS}$ | \text{BS} |
Brier score | 9 | $\text{BS} = \frac{1}{N}\sum_{i=1}^{N}(f_i - o_i)^2$ |
| $f_i$ | f_i |
Forecast probability for the $i$-th event | 9 | $f_i = 0.8$ |
| $o_i$ | o_i |
Outcome indicator (1 if event occurred, 0 otherwise) | 9 | $o_i = 1$ |
| $\text{LS}$ | \text{LS} |
Logarithmic score | 9 | $\text{LS} = \frac{1}{N}\sum_{i=1}^{N}\ln(f_i \cdot o_i + (1 - f_i)(1 - o_i))$ |
| $\text{ECE}$ | \text{ECE} |
Expected Calibration Error | 10 | $\text{ECE} = \sum_{m=1}^{M}\frac{|B_m|}{N}|\bar{f}_m - \bar{o}_m|$ |
| $B_m$ | B_m |
The $m$-th calibration bin | 10 | Events with $f_i \in [0.3, 0.4)$ |
| $M$ | M |
Number of calibration bins | 10 | $M = 10$ equal-width bins |
| $\bar{f}_m$ | \bar{f}_m |
Mean predicted probability in bin $m$ | 10 | $\bar{f}_3 = 0.35$ |
| $\bar{o}_m$ | \bar{o}_m |
Observed frequency in bin $m$ | 10 | $\bar{o}_3 = 0.33$ |
| $\text{REL}$ | \text{REL} |
Reliability component of Brier score decomposition | 9 | $\text{REL} = \frac{1}{N}\sum_{m}|B_m|(\bar{f}_m - \bar{o}_m)^2$ |
| $\text{RES}$ | \text{RES} |
Resolution component of Brier score decomposition | 9 | Higher RES means better discrimination |
| $\text{UNC}$ | \text{UNC} |
Uncertainty component of Brier score decomposition | 9 | $\text{UNC} = \bar{o}(1 - \bar{o})$ |
Information Theory
| Symbol | LaTeX | Meaning | Chapter | Example |
|---|---|---|---|---|
| $H(X)$ | H(X) |
Shannon entropy of random variable $X$ | 5 | $H(X) = -\sum_i p_i \ln p_i$ |
| $D_{\text{KL}}(P \| Q)$ | D_{\text{KL}}(P \| Q) |
Kullback-Leibler divergence from $Q$ to $P$ | 9 | $D_{\text{KL}} = \sum_i p_i \ln(p_i / q_i)$ |
| $I(X; Y)$ | I(X; Y) |
Mutual information between $X$ and $Y$ | 22 | $I(X;Y) = H(X) - H(X \mid Y)$ |
| $H(X \mid Y)$ | H(X \mid Y) |
Conditional entropy of $X$ given $Y$ | 22 | Remaining uncertainty after observing $Y$ |
Linear Algebra and Vectors
| Symbol | LaTeX | Meaning | Chapter | Example |
|---|---|---|---|---|
| $\mathbf{x}$ | \mathbf{x} |
Feature vector (bold lowercase) | 22 | $\mathbf{x} = (x_1, x_2, \ldots, x_d)$ |
| $\mathbf{w}$ | \mathbf{w} |
Weight vector in a linear model | 22 | $\hat{y} = \mathbf{w}^\top \mathbf{x} + w_0$ |
| $\mathbf{w}^\top$ | \mathbf{w}^\top |
Transpose of vector $\mathbf{w}$ | 22 | Dot product $\mathbf{w}^\top \mathbf{x}$ |
| $\|\mathbf{x}\|$ | \|\mathbf{x}\| |
Euclidean norm of vector $\mathbf{x}$ | 9 | Spherical score uses $\|\mathbf{p}\|$ |
| $\mathbf{I}$ | \mathbf{I} |
Identity matrix | 23 | Ridge penalty: $\lambda \mathbf{I}$ |
| $\boldsymbol{\Sigma}$ | \boldsymbol{\Sigma} |
Covariance matrix | 29 | $\boldsymbol{\Sigma}_{ij} = \text{Cov}(X_i, X_j)$ |
Calculus and Optimization
| Symbol | LaTeX | Meaning | Chapter | Example |
|---|---|---|---|---|
| $\nabla f$ | \nabla f |
Gradient of function $f$ | 23 | $\nabla_{\mathbf{w}} \mathcal{L}$ for gradient descent |
| $\frac{\partial f}{\partial x}$ | \frac{\partial f}{\partial x} |
Partial derivative of $f$ with respect to $x$ | 5 | $\frac{\partial C}{\partial q_i} = \pi_i$ (LMSR price) |
| $\mathcal{L}$ | \mathcal{L} |
Loss function or objective function | 23 | $\mathcal{L}(\theta) = -\sum_i \ln P(y_i \mid \mathbf{x}_i; \theta)$ |
| $\arg\min$ | \arg\min |
Argument that minimizes a function | 23 | $\hat{\theta} = \arg\min_\theta \mathcal{L}(\theta)$ |
| $\arg\max$ | \arg\max |
Argument that maximizes a function | 28 | $f^* = \arg\max_f \mathbb{E}[\ln W]$ |
| $\sum$ | \sum |
Summation | 3 | $\sum_{i=1}^k p_i = 1$ |
| $\prod$ | \prod |
Product | 8 | Likelihood $= \prod_{i=1}^n P(x_i \mid \theta)$ |
| $\ln$ | \ln |
Natural logarithm | 5 | $C(\mathbf{q}) = b \ln(\sum e^{q_i/b})$ |
| $\exp$ | \exp |
Exponential function | 5 | $\pi_i = \exp(q_i/b) / \sum_j \exp(q_j/b)$ |
| $\lambda$ | \lambda |
Regularization parameter (in ML context) | 23 | $\mathcal{L}_{\text{reg}} = \mathcal{L} + \lambda \|\mathbf{w}\|^2$ |
Trading and Portfolio Management
| Symbol | LaTeX | Meaning | Chapter | Example |
|---|---|---|---|---|
| $W_t$ | W_t |
Wealth (bankroll) at time $t$ | 28 | $W_0 = \$1{,}000$ initial capital |
| $f^*$ | f^* |
Optimal Kelly fraction | 28 | $f^* = (bp - q) / b$ for odds $b$ |
| $f$ | f |
Fraction of bankroll wagered | 28 | $f = 0.5 f^*$ (half Kelly) |
| $G$ | G |
Expected geometric growth rate | 28 | $G = \mathbb{E}[\ln(W_{t+1}/W_t)]$ |
| $R_t$ | R_t |
Return in period $t$ | 30 | $R_t = (W_t - W_{t-1}) / W_{t-1}$ |
| $\bar{R}$ | \bar{R} |
Mean return over a period | 30 | $\bar{R} = \frac{1}{T}\sum_{t=1}^T R_t$ |
| $r_f$ | r_f |
Risk-free rate | 30 | $r_f = 0.04$ (4% annually) |
| $\text{SR}$ | \text{SR} |
Sharpe ratio | 30 | $\text{SR} = (\bar{R} - r_f) / \sigma_R$ |
| $\text{MDD}$ | \text{MDD} |
Maximum drawdown | 29 | $\text{MDD} = \max_{t}(W_{\text{peak}} - W_t) / W_{\text{peak}}$ |
| $\text{VaR}_\alpha$ | \text{VaR}_\alpha |
Value at Risk at confidence level $\alpha$ | 29 | $\text{VaR}_{0.95}$: 5% chance of exceeding this loss |
| $\text{PnL}$ | \text{PnL} |
Profit and Loss | 30 | $\text{PnL} = \text{Revenue} - \text{Cost}$ |
| $\alpha$ | \alpha |
Alpha: excess risk-adjusted return (in trading context) | 30 | $\alpha = R_{\text{strategy}} - R_{\text{benchmark}}$ |
| $\text{EV}$ | \text{EV} |
Expected value of a position | 3 | $\text{EV} = p \cdot \text{payoff} - (1-p) \cdot \text{cost}$ |
| $T$ | T |
Number of time periods or trading horizon | 25 | Walk-forward over $T = 52$ weeks |
| $t$ | t |
Time index | 24 | Price at time $t$: $\pi_t$ |
Set Notation and General Conventions
| Symbol | LaTeX | Meaning | Chapter | Example |
|---|---|---|---|---|
| $\Omega$ | \Omega |
Sample space (set of all possible outcomes) | 2 | $\Omega = \{\text{Yes}, \text{No}\}$ |
| $\omega$ | \omega |
A specific realized outcome | 2 | $\omega = \text{Yes}$ |
| $k$ | k |
Number of possible outcomes | 2 | $k = 5$ candidates in an election market |
| $i, j$ | i, j |
Index variables | 2 | Outcome $i$, forecaster $j$ |
| $\mathbb{R}$ | \mathbb{R} |
The set of real numbers | 5 | $C : \mathbb{R}^k \to \mathbb{R}$ |
| $[0, 1]$ | [0, 1] |
The unit interval | 2 | $\pi_i \in [0, 1]$ |
| $\mathbb{1}\{\cdot\}$ | \mathbb{1}\{\cdot\} |
Indicator function (1 if condition holds, 0 otherwise) | 9 | $o_i = \mathbb{1}\{\omega_i = \text{Yes}\}$ |
| $\approx$ | \approx |
Approximately equal to | 2 | $\pi \approx P(\text{event})$ |
| $\propto$ | \propto |
Proportional to | 8 | $P(\theta \mid \text{data}) \propto P(\text{data} \mid \theta) P(\theta)$ |
| $\sim$ | \sim |
Distributed as | 8 | $X \sim \text{Beta}(2, 5)$ |
| $\triangleq$ | \triangleq |
Defined as | 5 | $\pi_i \triangleq \partial C / \partial q_i$ |
Notes on Conventions
-
Bold lowercase letters ($\mathbf{x}, \mathbf{q}, \mathbf{p}, \mathbf{w}$) denote column vectors. Bold uppercase letters ($\mathbf{I}, \boldsymbol{\Sigma}$) denote matrices.
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Greek letters are used for parameters ($\theta, \lambda, \mu, \sigma$), while Latin letters are used for observed quantities and indices ($n, k, i, t$).
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Hats ($\hat{p}, \hat{\theta}$) indicate estimated values. Bars ($\bar{R}, \bar{o}$) indicate sample means. Stars ($f^*, p^*$) indicate optimal or true values.
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Subscripts typically index outcomes ($p_i$), time ($W_t$), or bins ($B_m$). Superscripts denote labels (bid, ask) or powers ($\sigma^2$), depending on context.
-
Throughout the book, $\ln$ denotes the natural logarithm (base $e$) and $\log$ without subscript also refers to the natural logarithm unless stated otherwise. Information-theoretic quantities (entropy, KL divergence) use natural logarithms and are measured in nats.
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When the same symbol serves double duty (e.g., $\lambda$ for both the informed-trader arrival rate in the Glosten-Milgrom model and the regularization parameter in machine learning), the intended meaning is always clear from context. The chapter and section references in this guide identify the primary usage.