Probabilistic Forecasting - Index

Routing Summary

Modern probabilistic / ML forecasting, complementing the vault’s classical time-series coverage (ARIMA, transfer functions, VAR/cointegration, state-space/Kalman, BSTS). Anchored by four papers: Gneiting & Raftery (2007) on proper scoring rules; Salinas, Flunkert & Gasthaus (2017) on DeepAR; Ansari et al. (2024) on Chronos; Wickramasuriya, Athanasopoulos & Hyndman (2019) on MinT reconciliation.

• Need the big picture / where to start? → Probabilistic Forecasting - Overview
• Need to score a predictive distribution, quantile or interval (CRPS, log score, pinball, interval score)? → Proper Scoring Rules (CRPS, Log Score, Pinball Loss)
• Need to train one neural model across thousands of series (counts, cold start, sample paths)? → DeepAR and Global Autoregressive Neural Forecasters
• Need a zero-shot pretrained forecaster, or to understand tokenised time series? → Time-Series Foundation Models (Chronos)
• Need to decide between per-series models, a pooled model, or a foundation model? → Local vs Global Forecasting Models
• Need forecasts that add up across a product/geo hierarchy? → Hierarchical Forecast Reconciliation (MinT)
• Need to design a backtest (rolling origin, MASE, WQL, coverage, aggregation)? → Forecast Evaluation and Backtesting
• Need the MinT closed form? → Theorem 1 (MinT)
• Need the CRPS definition / Gaussian closed form? → CRPS, Gaussian CRPS

Concept Map

Concept	Note	Type	Depends On	Key Result
Probabilistic forecasting framing	Probabilistic Forecasting - Overview	overview	Single Marketing Time Series; Linear-Gaussian State-Space Models; BSTS; Model Comparison	Forecast = predictive distribution; maximise sharpness subject to calibration; local / task-specific / pretrained taxonomy; coherence
Proper scoring rules	Proper Scoring Rules (CRPS, Log Score, Pinball Loss)	concept	Overview; Model Comparison; Quantile Regression	$S(Q,Q)\ge S(P,Q)$; proper ⇔ convex $G$ (Thm 1); log score ↔ KL/Bayes factor; CRPS $=\mathbb{E}|X-x|-\frac{1}{2}\mathbb{E}|X-X^{\prime }|$; pinball proper for quantiles; interval score; improper scores mislead
DeepAR	DeepAR and Global Autoregressive Neural Forecasters	method	Overview; Proper Scoring Rules	Autoregressive LSTM → likelihood parameters (Gaussian / neg-binomial); max-likelihood over pooled windows; ancestral sample paths; scale ${\nu }_i$ + weighted sampling; ≈15% better quantile risk
Chronos	Time-Series Foundation Models (Chronos)	method	DeepAR; Transformers and LLM Foundations - Overview	Mean-scale + 4094-bin quantisation; T5 + cross-entropy; TSMixup + KernelSynth; zero-shot agg. rel. WQL 0.645 / MASE 0.823 vs Seasonal Naive 1.0
Local vs global	Local vs Global Forecasting Models	concept	DeepAR; Chronos; Hierarchical Models	No-pooling vs complete pooling of parameters; variance reduction, cold start; power-law scale obstacle; task-specific > local, pretrained ≈ task-specific
MinT reconciliation	Hierarchical Forecast Reconciliation (MinT)	method	Overview	$\tilde{y}=S(S^{\top }{W}_h^{-1}S{)}^{-1}{S}^{\top }{W}_h^{-1}\hat{y}$ minimises trace of reconciled error covariance s.t. $SPS=S$; never worse than base; MinT(Shrink) best empirically
Evaluation and backtesting	Forecast Evaluation and Backtesting	method	Proper Scoring Rules; Cross Validation Checking	Rolling origin; MASE; WQL / $\rho$-risk ≈ CRPS; coverage curves incl. span sums; geometric mean of relative scores; leakage cautions

Notes

• Probabilistic Forecasting - Overview — CONTAINS: definition of probabilistic forecast (conditioning/prediction range, context/horizon), calibration vs sharpness, local/task-specific/pretrained taxonomy, coherence, cluster routing table, relevance to MMM / geo experiments / counterfactual baselines, end-to-end workflow, sample-based CRPS snippet.
• Proper Scoring Rules (CRPS, Log Score, Pinball Loss) — CONTAINS: proper/strictly proper definition, Theorem 1 (convex characterisation), entropy & divergence, log/quadratic/spherical scores, impropriety of the linear score, CRPS (integral + kernel form, Gaussian closed form), energy score, Theorem 6 quantile scores and pinball loss, interval score, Bayes-factor/prequential/BIC links, random-fold CV, skill-score warning, Pacific Northwest ensemble and bilinear-process case studies, code.
• DeepAR and Global Autoregressive Neural Forecasters — CONTAINS: model factorisation and LSTM recursion, Gaussian and negative-binomial heads, training objective and windowing, ancestral sampling algorithm, power-law scale handling (${\nu }_i$, weighted sampling), features, missing-data treatment, architecture table, Table 1-2 results and ablations, uncertainty-growth and shuffled-sample calibration findings, lead-time quantile example, GluonTS sketch.
• Time-Series Foundation Models (Chronos) — CONTAINS: mean scaling, uniform quantisation $q/d$, vocabulary and special tokens, cross-entropy objective (regression via classification), zero-shot forecasting algorithm, TSMixup and KernelSynth algorithms, training setup and cost, Benchmark I/II aggregate scores by model size, fine-tuning result, ablations (size, LLM init, augmentation, context, vocabulary), limitations (range overflow, precision, no covariates, leakage), synthetic diagnostics, tokenisation arithmetic, code.
• Local vs Global Forecasting Models — CONTAINS: formal definitions of local/global/pretrained, five reasons pooling helps, obstacles (power-law scales, heterogeneity bias, likelihood mismatch, covariates, leakage), evidence table from both papers, pooling analogy with hierarchical Bayes, geo-panel decision example.
• Hierarchical Forecast Reconciliation (MinT) — CONTAINS: summing matrix, $\tilde{y}=SP\hat{y}$, bottom-up/top-down as special $P$, unbiasedness $SPS=S$, non-identifiability of the 2011 GLS covariance, Lemma 1, Theorem 1 (both forms), projection interpretation, Pythagorean “never hurts” inequality, five $W_h$ estimators (OLS, WLS${}_v$, WLS${}_s$, Sample, Shrink), simulation and Australian-tourism results table, hand-worked 3-series example, code.
• Forecast Evaluation and Backtesting — CONTAINS: prequential justification, fixed-origin vs rolling-origin algorithms (sliding/expanding, refit vs re-condition), leakage and tuning warnings, MASE, pinball/WQL/$\rho$-risk, ND/NRMSE, coverage curves, relative-score geometric-mean aggregation, skill-score caution, checklist table, geo-test placebo backtest example, code.

External / Cross-Folder Links

• Single Marketing Time Series, Transfer Function Model, Multivariate Persistence and Cointegration — classical (local) time-series models in Market Response Models.
• Linear-Gaussian State-Space Models, The Kalman Filter, Bayesian Structural Time-Series Model, Local Linear Trend and Seasonality, Counterfactual Impact Estimation — Bayesian state-space forecasting and counterfactuals.
• Hilbert Space Gaussian Processes, Gaussian Process Regression, Model Building - Time-Series Decomposition for Birthdays — GP kernels for time series (KernelSynth connection).
• Cross Validation Checking, Model Comparison, Overfitting and Information Criteria, Posterior Predictive Checking — predictive model assessment.
• Hierarchical Models, Empirical Bayes Interpretation of Shrinkage — pooling and shrinkage.
• Quantile Regression — pinball loss as an estimator.
• Optimal Marketing Decisions and Forecasting, Bayesian Media Mix Modeling - Overview, Geo-Experiment Methodology - Overview, Time-Based Regression Estimator for Geo Experiments — applied marketing context.
• Transformers and LLM Foundations - Overview, Conformal Prediction - Overview — sibling clusters in Machine Learning and AI.

Sources

• Gneiting Raftery 2007 - Strictly Proper Scoring Rules Prediction and Estimation — Gneiting, T. & Raftery, A. E. (2007), “Strictly Proper Scoring Rules, Prediction, and Estimation,” Journal of the American Statistical Association 102(477), 359-378.
• Salinas 2017 - DeepAR Probabilistic Forecasting with Autoregressive Recurrent Networks — Salinas, D., Flunkert, V. & Gasthaus, J. (2017), “DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks,” arXiv:1704.04110 (v3, 2019; later International Journal of Forecasting 2020 with Januschowski).
• Ansari 2024 - Chronos Learning the Language of Time Series — Ansari, A. F. et al. (2024), “Chronos: Learning the Language of Time Series,” Transactions on Machine Learning Research, arXiv:2403.07815.
• Wickramasuriya 2019 - Optimal Forecast Reconciliation MinT — Wickramasuriya, S. L., Athanasopoulos, G. & Hyndman, R. J. (2019), “Optimal Forecast Reconciliation for Hierarchical and Grouped Time Series Through Trace Minimization,” JASA 114(526), 804-819 (Monash working paper 22/17 version).