Hypothesis Testing
Term | Definition |
H0 | Null hypothesis, the default position that there is no change |
H1 | Alternative hypothesis, the alternative position that there is some change |
Type I error | Rejection of H0, when H0 is true |
Type II error | Failure to reject H0, when H1 is true |
Error | False rejection of H0 |
α, Significance level, false positive rate | Pr(reject H0 | H0 is true); probability of type I error |
β, false negative rate | 1 – Pr(reject H0 | H1 is true); probability of type II error |
Power | Pr(reject H0 | H1 is true); 1 – probability of type II error; 1 – β |
P-value | Probability of observing a result at least as extreme as the observed result, given that H0 and all other model assumptions are true |
Confidence Interval | The set of effect sizes whose test produced P > 0.05 define a 1 – 0.05 = 0.95 or 95 % confidence interval. |
PCE | Per comparison error rate |
PFE | Error rate per family (expected number of false rejections per family) |
FWE | Family wise error rate (probability of at least one error in the family) |
FDR | False discovery rate (expected number of false significances / number of significances) |
Classification
Term | Definition |
True positive rate; TPR; Recall; Sensitivity | True positives / All positives, i.e. the fraction of positives that are correctly classified by the model |
False positive rate; FPR; Type I error rate | False positives / All negatives, i.e. the fraction of all negatives that are incorrectly classified by the model |
False negative rate; FNR; Type II error rate; Miss rate | False negatives / All positives, i.e. the fraction of all positives that are incorrectly classified by the model |
True negative rate; Selectivity; Specificity | True negatives / All negatives, i.e. the fraction of all negatives that are correctly classified by the model |
Accuracy | True positives + True negatives / Total population, i.e. the fraction of population that is correctly classified by the model |
Precision; PPV; Positive predictive value | True positives / Predicted positives, i.e. the fraction of true positives amongst all predicted positives |
FDR; False discovery rate | False positives / Predicted positives, i.e the fraction of false positives amongst all predicted positives |
Population prevalence | Positives / Population, i.e. the fraction of population that is positive |
F1 score | 2 * Precision * Recall / (Precision + Recall), i.e. the harmonic mean of precision and recall |
Reinforcement Learning
Multi-armed Bandits
Term | Definition |
Action taken by the agent at time step t | |
Estimate of the expected reward from action a at time step t | |
The number of times the action ‘a’ has been played till time step t | |
Reward obtained on the i_th play of action ‘a’ | |
UCB | Upper-Confidence-Bound algorithm for action selection in bandits |
Markov Decision Processes
Term | Definition |
MDP | Markov Decision Process |
State, reward, and action at time step t | |
Set of all valid states, rewards, and actions (in state ‘s’) | |
Finite MDP | A MDP in which the set of all states, rewards and actions is finite |
Trajectory | The sequence of state, action, and rewards starting at some given state, ![]() |
State dynamics | Also known as the dynamics of a MPD, given by the four argument function ![]() |
Limit Theorems
Weak law of large numbers
The sample average converges in probability to its expected value.
Strong law of large numbers
The sample average converges almost surely to its expected value.
Central limit theorem
The average of a large number of IID random variables, when shifted by the population average and scaled by the square root of the number of variables, approaches a normal distribution. More precisely, let {x1, x2, …, xn} be n IID random variables such that E(xi) = μ, Var(xi) = σ2. Let Sn = (x1 + x2 + … + xn)/n, then √n(Sn – μ) → N(0, σ2)
Extreme value theorem
The maximum (minimum) of a large number of IID random variables, when centered and scaled properly, converges to a generalized extreme value distribution. The Weibull, Gumbel, and Frechet distributions are members of this family. The centering and scaling here are not as universal as the central limit theorem.
Bootstrap theorem (for sample mean)
Let {x1, x2, …, xn} be an i.i.d with sample mean X̅, sample variance S, and a defined population variance. Let X̅B denote the mean of a bootstrap sample drawn from {x1, x2, …, xn}. Then, √n(X̅B – X̅)/S approaches the standard normal distribution in the limit of large n.