Glossary

Modeling Terms

Adstock

A parametric transformation that models the carry-over effect of advertising: today's media exposure influences sales today and over a decaying window of subsequent periods. Common forms include geometric (single decay rate) and Weibull (more flexible curve). See the Technical Guide for the math and the transforms/adstock.py module for the implementation.

Saturation

A parametric transformation capturing diminishing marginal returns to media spend — each additional dollar at high spend levels adds less than each additional dollar at low spend levels. Common forms include Hill, logistic, and Michaelis-Menten curves. Implemented in transforms/saturation.py.

MMM (Marketing Mix Model)

A regression-style model that decomposes an outcome metric (revenue, conversions) into contributions from media spend, price, distribution, seasonality, and other drivers. The framework's MMMs are Bayesian and use saturation + adstock transforms per channel. See: Modeling Guide, Technical Guide

MFF (Master Flat File)

The framework's input data format — a single long-form table with columns for date, outcome, media spend per channel, controls, and (optionally) geo. Loaded via MFFLoader; configuration is validated against a schema before fitting.

Hierarchical model

A model where parameters are themselves drawn from a higher-level distribution — for example, channel-level coefficients drawn from a category-level prior. Pooling tightens estimates for data-poor channels by borrowing strength from data-rich siblings. Critical for portfolio MMMs with many channels.

ROI / mROI

Return on investment — outcome generated per unit of spend (e.g., $1.40 of revenue per $1 of media spend). Marginal ROI (mROI) is the return on the next dollar at the current spend level — the relevant quantity for budget allocation under saturation.

Counterfactual

The unobserved outcome that would have occurred under a different action — e.g., "what would sales have been if we had spent zero on TV last quarter?" Computing counterfactuals is the central task of causal MMM analysis. See analysis.py and the Causal Inference guide.

Decomposition

Attributing each unit of outcome to a specific driver (channel, baseline, seasonality). Bayesian decomposition is a posterior distribution over contributions, not a single number; report intervals, not just means.

Cross-effects

When a marketing action on one product or channel affects outcomes for another (cannibalisation, halo effects, multi-pack pull-through). Ignoring cross-effects inflates per-channel ROI. Modeled via the multivariate extension; see Technical Guide.

Bayesian Inference

Bayes' rule

$p(\theta \mid y) \propto p(y \mid \theta)\, p(\theta)$. Posterior beliefs about an unknown parameter $\theta$ are proportional to the likelihood of the observed data times the prior. The mathematical engine behind every model in the framework.

Prior

$p(\theta)$ — a probability distribution encoding what we believe about a parameter before seeing the current data. Priors are how we regularise; defensible priors beat flat priors. See the Bayesian Workflow guide.

Likelihood

$p(y \mid \theta)$ — the probability of observing the data given a candidate value of the parameter. Combined with the prior to form the posterior.

Posterior

$p(\theta \mid y)$ — the updated belief about a parameter after seeing the data. The posterior is what every Bayesian inference produces. Summarised by mean, median, and credible interval.

Conjugate prior

A prior whose posterior is in the same family. The Gaussian-Gaussian conjugate update (precisions add; means are precision-weighted) is the workhorse of the calibration loop.

Credible interval (CI)

An interval that contains the parameter with a stated posterior probability (e.g., 95% CI). Distinct from a frequentist confidence interval — a credible interval is a direct probabilistic statement about the parameter. The default reporting unit in this framework.

HDI (Highest Density Interval)

The narrowest credible interval at a given probability mass — the set of most probable parameter values. Useful when the posterior is skewed or multi-modal.

ROPE (Region of Practical Equivalence)

A pre-specified range around a null value within which differences are considered practically meaningless. Decision rule: if the HDI lies entirely outside the ROPE, accept the effect; if entirely inside, reject; if overlapping, undecided.

Posterior predictive check (PPC)

Simulating new data from the posterior and comparing it to the observed data. The basic smoke test for whether a Bayesian model is even capable of producing the kind of data we see. ArviZ provides standard plots; see Bayesian Workflow.

$\hat{R}$ (R-hat)

A convergence diagnostic for MCMC chains. $\hat{R} \approx 1.0$ means chains have mixed; $\hat{R} > 1.01$ is a red flag. Always check before interpreting any posterior.

ESS (Effective Sample Size)

The number of independent posterior samples your MCMC chain is effectively worth. Low ESS means high autocorrelation and unreliable summary statistics. Aim for ESS in the thousands for any quantity you'll report.

Divergence

A NUTS-sampler failure mode signalling that the posterior geometry has features the sampler can't navigate (typically funnels in hierarchical models). Even one divergence invalidates inference; reparameterise rather than ignore.

Measurement & Experiments

EIG (Expected Information Gain)

The expected reduction in posterior entropy from running an experiment. Measures how much we'd learn. Defined as the expected KL divergence from prior to posterior. Under Gaussian-Gaussian: $\mathrm{EIG} = \tfrac{1}{2}\log(1 + \sigma_{\text{prior}}^2 / \sigma_{\text{exp}}^2)$. Used to prioritise the experimental portfolio. See Calibration Loop.

EVOI (Expected Value of Information)

The expected dollar improvement in budget decision quality from running an experiment. Measures how much an answer would change what we do. EIG and EVOI together drive the 2x2 priority map. EVPI (Expected Value of Perfect Information) is the upper bound.

Calibration

Using experimental causal estimates to anchor MMM parameters so that model-implied ROIs are consistent with experimental truth. Three mechanisms: soft prior, likelihood augmentation, and hierarchical pooling. See Calibration Loop.

Geo-lift / matched-market test

An experiment that withholds (or boosts) media spend in a set of geographic markets and compares outcomes against matched control markets. Produces a causal estimate for the tested channel that's independent of the rest of the marketing plan.

TWFE / DiD (Difference-in-Differences)

A panel-data estimator that identifies treatment effects from differential changes between treated and control units across pre/post periods. The standard estimator for geo-lift readouts; outputs a point estimate and standard error that plug directly into the Bayesian update via the bridge equation.

SCM (Synthetic Control Method)

An estimator that constructs a counterfactual control unit as a weighted combination of donor pool units, then attributes the post-treatment gap to the intervention. Useful when no clean matched control exists. Augmented Synthetic Control (ASC) extends with outcome model bias correction.

MDE (Minimum Detectable Effect)

The smallest effect size an experiment can detect at a given power and significance level. Computed before launch from sample size, baseline variance, and design choices. Drives geo selection and test duration.

ITT (Intent-to-Treat)

Analyse subjects according to their assigned arm regardless of compliance. The default and pre-registered analysis for geo-lift to avoid post-hoc adjustments biased toward results that look good.

Pre-registration

Locking in the experimental design, estimator, and primary outcome before the test runs. Eliminates the temptation to choose the analysis that produces the most favourable answer (a major source of bias).

Submodularity

A property of set functions: marginal value diminishes as the set grows. The EIG of a portfolio of experiments is submodular — meaning greedy selection is provably within $1 - 1/e \approx 63\%$ of optimum, and the test program naturally caps at 2-3 experiments per cycle.

Causal Inference

DAG (Directed Acyclic Graph)

A diagram of cause-effect relationships among variables. Edges point from cause to effect; no cycles. The basic representation of a causal model. The framework's dag_model_builder validates DAGs before allowing model fits. See Causal Inference.

Confounder

A variable that causes both the treatment and the outcome — e.g., promo periods that drive both media spend and sales. Confounders must be included in an MMM; omitting them biases the media coefficient.

Mediator

A variable on the causal path from media to outcome (e.g., brand awareness mediates between TV and purchase). Including a mediator as a control absorbs the very effect you're trying to measure. Use a NestedMMM to model mediation explicitly.

Collider

A variable caused by both the treatment and the outcome. Conditioning on a collider opens a spurious path and biases estimates — the opposite mistake from omitting a confounder.

do-operator

$\mathrm{do}(X = x)$ denotes the interventional distribution: the world in which $X$ is set to $x$ by external intervention, breaking incoming causal arrows. Distinct from conditioning $X = x$ (which observes without intervening).

Backdoor criterion

A graphical condition for selecting a control set that blocks all non-causal paths between treatment and outcome. The standard tool for identifying causal effects from observational data.

Specification shopping

Re-running models with different choices (variables, transforms, lags) until one produces the desired result. Inflates apparent precision and biases conclusions. The framework's emphasis on pre-specified analyses is a direct response.

Data & Operations

Frequentist

The school of statistics built around long-run frequency: probabilities are limits of repeated trials, parameters are fixed, and inference is via confidence intervals and p-values. Most production geo-lift estimators are frequentist; their outputs feed Bayesian calibration through the bridge equation.

NumPyro

A JAX-backed probabilistic programming library used by the framework as a fast alternative sampler to native PyMC. Pass nuts_sampler="numpyro" to BayesianMMM.fit() for a 5-10x speedup on most models.

NutPie

A Rust-backed NUTS sampler available as another fast PyMC backend. Sometimes faster than NumPyro for small-to-medium models; benchmark on your data.

MAPE (Mean Absolute Percentage Error)

Standard predictive accuracy metric: average of $|y_t - \hat{y}_t| / y_t$. Reported on a held-out test window; used as the gate for whether a new model spec is at least as good as the baseline.

ARQ

Async job queue used by the API service for long-running model fits. Workers run via arq worker.WorkerSettings against a Redis backend.

Pre-period

The historical window used to estimate baseline geo-week variance and calibrate experiment designs before a test launches. Powers the noise estimate $\sigma_{\text{exp}}^2$ used in EIG/EVOI.

OLS (Ordinary Least Squares)

The classical frequentist regression estimator. Many incumbent MMMs are OLS-based; the framework's recommended migration path is to fit a Bayesian MMM in parallel and only decommission OLS after two cycles of comparable answers under audit.

Uncertainty

The width of the posterior. Critical to report alongside any point estimate — a narrow interval is information; a wide interval is also information. See Interpreting Results for how to communicate it.

Variable selection

The process of deciding which controls to include. The framework's stance: confounders always, mediators never, colliders never. Apply selection only to precision controls (variables that don't bias but improve fit). See Variable Selection.

Mathematical Symbols

Notation used throughout the technical guides: