Bayesian Structural Time Series

BSTS models explicitly separate a time series into additive structural components, each with a clear operational meaning for supply chain planners:

• Trend Component: Models long-term direction using a local linear trend, capturing gradual shifts in baseline demand without assuming a fixed parametric form
• Seasonality Component: Represents cyclical patterns (weekly, monthly, quarterly) as a sum of trigonometric harmonics, automatically adapting to changing seasonal amplitudes
• Regression Component: Incorporates external covariates such as pricing changes, promotions, or macroeconomic indicators, with coefficients estimated probabilistically
• Irregular Component: Captures residual variation as stochastic noise, preventing overfitting to spurious patterns

This decomposition allows demand planners to isolate the impact of a specific promotion from underlying seasonal effects, providing actionable attribution rather than a black-box forecast.

Unlike point-forecast methods that output a single number, BSTS generates a complete posterior predictive distribution for every future time point, enabling risk-aware inventory decisions:

• Prediction Intervals: Extract credible intervals at any confidence level (e.g., 80%, 95%) directly from posterior samples, providing statistically valid bounds for safety stock calculations
• Distributional Shape: Capture skewness and heavy tails in demand distributions, critical for intermittent demand patterns where normal approximations fail
• Uncertainty Decomposition: Separate epistemic uncertainty (model parameter uncertainty, reducible with more data) from aleatoric uncertainty (inherent demand randomness, irreducible)
• Scenario Generation: Draw thousands of plausible future trajectories for Monte Carlo simulation of inventory policies under the Newsvendor Model

The posterior distribution directly informs safety stock optimization by specifying the exact quantile needed to achieve a target service level.

BSTS employs a spike-and-slab prior for automatic feature selection when incorporating regression components, a critical capability for high-dimensional demand drivers:

• Spike Component: Places a point mass at zero, allowing coefficients to be exactly zero with high probability, effectively removing irrelevant predictors
• Slab Component: A diffuse prior over non-zero coefficients, allowing selected predictors to have substantial effects without shrinkage bias
• Posterior Inclusion Probability: For each covariate (e.g., competitor price, weather, marketing spend), the model outputs the probability that it has a non-zero effect on demand
• Bayesian Model Averaging: Rather than selecting a single best model, BSTS averages predictions over all plausible variable combinations weighted by their posterior probability

This approach prevents overfitting when testing dozens of potential demand drivers while maintaining interpretable feature importance for supply chain analysts.

BSTS combines two complementary algorithms for efficient inference on time series data:

• Kalman Filter: Provides fast, recursive forward filtering of the state-space model, computing the optimal estimate of latent states (trend, seasonality) given observations up to the current time point
• Markov Chain Monte Carlo (MCMC): Samples from the full posterior distribution of model parameters and unobserved states using Gibbs sampling, enabling uncertainty quantification that the Kalman filter alone cannot provide
• Forward Filtering, Backward Sampling: A two-pass algorithm that first runs the Kalman filter forward, then samples state trajectories backward, producing coherent draws from the joint posterior of all latent states
• Conjugate Priors: Where possible, BSTS uses conjugate prior distributions (e.g., inverse-Gamma for variances) to enable efficient Gibbs sampling steps without Metropolis-Hastings rejections

This hybrid approach scales to multi-year daily demand histories while maintaining rigorous Bayesian inference, unlike pure MCMC methods that become computationally prohibitive on long time series.

BSTS natively supports complex calendar effects that are critical for retail and e-commerce demand forecasting:

• Holiday Regression: Model holidays as binary or duration-based regressors with lead and lag effects, capturing pre-holiday demand surges and post-holiday dips
• Day-of-Week Patterns: Incorporate weekly seasonality with separate effects for each day, automatically handling months with five weekends versus four
• Moving Holidays: Handle holidays with shifting dates (Easter, Ramadan, Chinese New Year) by generating dynamic regressor values based on the actual calendar
• Custom Event Windows: Define arbitrary event periods such as promotional campaigns or store openings with user-specified temporal profiles

The model estimates the causal impact of each event type with full posterior distributions, enabling precise measurement of promotion ROI and holiday lift.

BSTS forms the foundation of the CausalImpact methodology, which estimates the counterfactual effect of an intervention on a time series:

• Synthetic Control: Construct a counterfactual baseline using a BSTS model trained on pre-intervention data, then compare post-intervention observations against the predicted counterfactual
• Pointwise Causal Effect: Compute the difference between actual and predicted values at each time point, with credible intervals quantifying the uncertainty of the causal estimate
• Cumulative Impact: Aggregate the pointwise effects over the post-intervention period to estimate the total incremental demand attributable to a supply chain intervention
• Posterior Tail-Area Probability: Calculate the Bayesian probability that the intervention had any non-zero effect, providing a rigorous alternative to frequentist p-values

Supply chain applications include measuring the impact of a new fulfillment center opening, a competitor stockout, or a logistics disruption on regional demand patterns.

BSTS excels at isolating the causal effect of a marketing campaign or price change on demand. By constructing a synthetic counterfactual—a Bayesian forecast of what would have happened without the intervention—planners can precisely measure incremental lift.

• Mechanism: Include a binary regressor for the promotion period; the model estimates its additive effect with full posterior distribution.
• Example: Quantifying the demand uplift from a Black Friday discount, separating it from underlying seasonal trends.
• Key Advantage: Unlike black-box ML, BSTS provides a credible interval for the causal estimate, enabling ROI risk assessment.

Traditional methods like Croston's struggle to incorporate external factors. BSTS naturally handles intermittent demand (frequent zeros) by modeling the latent demand state separately from the observation process, while allowing covariates like equipment utilization rates or weather to influence the non-zero demand size.

• State-Space Decomposition: Trend and seasonality components evolve even during zero-demand periods.
• Regression Component: Link demand size to leading indicators (e.g., flight hours for spare parts).
• Output: A predictive distribution that accurately reflects the probability of zero demand versus a large order.

BSTS directly outputs a full predictive distribution for lead-time demand, not just a point forecast. This is the exact input required for the Newsvendor model to set optimal safety stock levels that minimize the total cost of overstocking and understocking.

• Quantile Extraction: Extract the 95th or 99th percentile of the forecast distribution to achieve a target service level.
• Adaptive Buffers: As the model's epistemic uncertainty narrows with more data, safety stock recommendations automatically adjust downward.
• Supply-Side Integration: Include supplier lead time as a stochastic regressor to model its impact on required buffer size.

BSTS models can be built at multiple levels of a hierarchical time series (SKU, product family, category) and then optimally reconciled. The Bayesian framework provides a natural way to share statistical strength across levels through correlated priors, improving bottom-level forecast accuracy.

• Coherence: Ensures SKU-level forecasts sum exactly to the regional and national totals.
• Information Pooling: A product family's seasonal pattern informs the forecast for a new SKU with limited history.
• Implementation: Use a multivariate BSTS with a hierarchical prior structure on the trend components.

Because BSTS is a generative model, it can simulate future trajectories under different what-if scenarios. Supply chain strategists can condition the forecast on assumed future values of key regressors to stress-test the network.

• Conditional Forecasting: Set future values for a regressor (e.g., a 20% increase in fuel costs) and observe the impact on the demand distribution.
• Risk Mapping: Generate thousands of plausible futures to identify the probability of a stockout at any node in the supply chain.
• Strategic Application: Quantifying the demand impact of a competitor's anticipated market exit or a new tariff policy.

Supply chains undergo sudden, permanent shifts due to events like factory fires or regulatory changes. BSTS can incorporate a local linear trend with a stochastic slope, allowing the model to automatically detect and adapt to structural breaks without manual retraining.

• Spike-and-Slab Priors: Automatically select which regressors are relevant at any given time, turning off spurious correlations.
• Variance Learning: The model learns if the observation noise has increased, signaling a period of heightened volatility.
• Alerting: A sudden, large change in the trend slope posterior signals an operational exception requiring human intervention.