Probability Mass
P(X = 2) = 0.270671 for a Poisson with mean 2. For a continuous variable the same quantity is zero.
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Probability Distribution
Past 4, the normal gives 0.00006334 and the t gives 0.00357283. Same mean, same variance.
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Probabilistic Forecasts
A point forecast is one summary of this. Once you keep the whole distribution, the decision can pick the summary it actually needs.
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Deterministic forecasts
Reported without an interval, a forecast carries no way of telling how accurate it is — which is why a single line reads as certainty.
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Cumulative Distribution Function (CDF)
A discrete CDF is a staircase whose step heights are the masses. A continuous one has no steps, which is P(X = x) = 0.
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M-Competitions (Makridakis Competitions)
An ensemble’s bias is the average of its members’ biases. Opposing biases can cancel; a shared directional bias remains.
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Forecasting Benchmarks
Mean, naive, seasonal naive, drift: the simple methods a model has to beat on the same test set, origins, and horizons before its complexity is justified.
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Average Absolute Error (AAE)
Always 40 units high: mean error -40.0000, AAE 40.0000. Unbiased but scattered: 0.0093 and 48.1484.
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Seasonal Lag
The value from the same season one period ago. Where it appears in the ACF, what seasonal differencing does with it, and why it is expensive in data.
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Naïve Baseline Forecast
Repeating the last value is not a placeholder. For a random walk it is optimal — which is why beating it is the first thing to demonstrate.
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Forecast Error
Minimizing MAE produces forecasts of the median; minimizing RMSE produces forecasts of the mean. The metric is a modeling choice, not a scorecard.
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Forecasting Competitions
Four metrics, three orderings, two winners, from one set of forecasts over the same 3,000 series.
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Predicting Percentiles
A calibrated continuous 90th percentile has 10% exceedance probability, not a guaranteed annual stockout count. Discrete demand can exceed it less often.
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Prediction Intervals (PI)
Interval width depends on the model and horizon. Coverage reflects the uncertainty represented in the model, not every possible change in the world.
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Quantile Regression
Swap squared error for an asymmetric loss and the fitted line becomes a quantile. Three fits give you an interval with no normality assumption.
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Quantile Level
A quantile level selects a position in the outcome distribution. Exact exceedance probabilities require continuity; discrete outcomes can have ties.
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Time Series Forecasting
Predicting the next values of a series. The benchmarks any method has to beat, and the evaluation that decides whether it did.
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Relative accuracy
Removing half the remaining errors reads as 1.5000, 1.2143, 1.0556 and 1.0051 depending only on the majority class.
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R² (R-squared)
Training R² describes fit to observed data. Test R² evaluates predictions on new data; neither establishes causation, and either needs its definition stated.
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Confidence Level
95% is a property of the method, not of the interval in front of you. A higher level targets greater coverage across samples and usually widens the interval; it does not assign a posterior probability to the observed interval.
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Causal Trees
A regression tree whose leaves estimate treatment effects. The criterion that makes the splits, the sample split called honesty, and what it does not fix.
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Uplift Random Forests
Trees that split on the effect of treating rather than on the outcome, averaged. The criteria that make the splits, and the interpretability it costs.
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Uplift Curve
Cumulative incremental outcome against the share targeted. Why its vertical axis scales a difference of response rates into an incremental response count, and where to cut the campaign.
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Likelihood
Likelihood compares parameter values for fixed observations. Ratios, constants, log calculations, and model assumptions determine its interpretation.
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Correlation
Pearson correlation measures linear association only. A coefficient near zero can sit on top of a perfect curved relationship.
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Causal Effect
Causal inference from observational data needs two things: data and identifiability conditions. Choosing a method does not establish the identification assumptions.
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Mean Squared Error (MSE)
Square each error and average. Smooth to optimize and hard on big misses, but in squared units, so it is often reported as RMSE and minimized as MSE.
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Median
The median describes a typical case and cannot be summed. Both facts follow from it being a position rather than a balance point.
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Mean
Means add and medians do not — which is why totals are built from means even when the median describes a typical case better.
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Chi-square (χ²) Test
60% against 56%, scaled from 1,000 rows to 100,000: chi-square goes 1.6420 to 164.2036 and p from 0.2000 to zero.
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Kolmogorov-Smirnov Test (KS Test)
Moving 1% of the data out to six standard deviations gives D = 0.014000 and p = 0.828022. Not detected.
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Kullback-Leibler Divergence
An optimal code for P costs 1 bit; a code for Q costs 1.736966 on samples from P. The 0.736966 gap is the KL.
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Statistical Tests
Which test answers which question, from one-sample t to Fisher’s exact and Kolmogorov–Smirnov, how to choose by question, and the limits of a rejection.
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Seasonality
A repeating pattern tied to the calendar. What separates it from a cycle, and why confusing the two produces forecasts the data never supported.
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Statistical Power
Power is not a property of a test but of a test plus an effect size you name. Without naming one, “underpowered” has no meaning.
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DeLong’s Test
A case-resampling bootstrap gives 0.015711 where DeLong’s closed form gives 0.015541.
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Clopper–Pearson Interval
Exact here means the coverage is guaranteed rather than approximated. The price is an interval that is wider, and not centered on your estimate.
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Wilson Score Interval
The textbook proportion interval can put its lower bound below zero. Inverting the test instead gives bounds that stay inside the range.
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Bootstrap
The bootstrap gives you a sampling distribution for statistics that have no convenient formula. What it cannot give you is a fix for a biased sample.
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Jensen–Shannon (JS) Divergence
JS reaches 0.693147 nats for disjoint supports. A parameter path that keeps them disjoint has a constant objective, unlike one that changes overlap.
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Bayesian Inference
Expected loss from shipping 0.137729, from holding 0.094212. The point estimate of 0.70 says ship.
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Sequential Testing
Looking at a running experiment repeatedly, legitimately. What naive peeking does to the error rate, and how planned boundaries hold it.
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Confidence Intervals (CIs)
95% describes the procedure across repeated samples, not your confidence in this particular interval. The distinction changes how you report it.
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Power Analysis
Sample size, effect size, α and power are four numbers with one equation. Fix any three and the fourth follows — which is the whole method.
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A/B Testing
Randomization makes groups comparable in expectation. Estimate assignment effects with uncertainty and check departures from the intended design.
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Revenue net of treatment cost
Incremental revenue converted to contribution through the margin, with the campaign’s cost taken out, and the interval the result inherits.
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Incremental Conversions
The conversions a campaign caused, as opposed to the ones it was merely present for. What the scaling to a full population quietly assumes.
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Uplift@k
The incremental effect within the top k percent by predicted uplift — the number that matters when the campaign volume is already fixed.
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AUUC (Area Under the Uplift Curve)
The area under the uplift curve, reduced to a single number — by at least three different conventions that do not agree, on the same model and the same data.
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Qini Coefficient
A single number for an uplift model’s ranking, from the area above the random diagonal. The normalization changes it more than most model differences do.