Time Series • Multivariate Time Series Models and Forecasting
Time Series / Bayesian Methods in Forecasting

Bayesian Methods in Forecasting

Notes 6 Multivariate Time Series Models and Forecasting
Quick concept

Bayesian forecasting combines prior knowledge or beliefs with observed data to update the uncertainty about unknown quantities and about future outcomes. The updated belief is the posterior. A forecast can then be read from that posterior (a likely range, not only a single number).

Notes

Bayesian Methods in Forecasting

Definition

Bayesian forecasting combines prior knowledge or beliefs with observed data to update the uncertainty about unknown quantities and about future outcomes. The updated belief is the posterior. A forecast can then be read from that posterior (a likely range, not only a single number).

Example

Historical shop knowledge says weekly demand is usually in a moderate range (prior). This week's new sales counts are extra evidence (likelihood). Bayesian updating mixes the prior with those counts to produce a posterior for next week's demand. If the new weeks are unusually high, the posterior shifts upward, but it need not ignore all past experience.

Prior, Likelihood, Posterior

Prior Knowledge ↓ Observe Data ↓ Update Belief ↓ Posterior Distribution ↓ Forecast
  • Prior — initial belief about a quantity (typical demand, a slope, a seasonal peak) before seeing the current sample, or before adding the latest observations.
  • Likelihood — how plausible the observed data are for different values of that quantity; the evidence coming from the data under a stated model.
  • Posterior — the updated belief after combining prior and likelihood.

Conceptually:

Posterior ∝ Likelihood × Prior

The symbol ∝ means “proportional to”: the posterior is shaped by multiplying the data evidence by the prior (then scaled so that it is a proper distribution). This is the updating idea, not a full probability derivation or a sampling algorithm.

Prior + Observed Data ↓ Posterior ↓ Forecast

Point Forecast vs Uncertainty-Aware Forecast

Simple point forecast Bayesian / uncertainty-aware forecast
What is reported Often one number (next week's sales = 50) A distribution or a range that reflects remaining uncertainty
Prior knowledge May be ignored or used informally Can be written into the prior and updated with data
New observations May require a full re-fit from scratch in spirit Natural story: update the posterior as data arrive

Bayesian forecasting is not always better than classical (frequentist) methods such as OLS or ARIMA. A poor prior or a wrong likelihood can mislead. Classical methods remain standard when a simple fitted model and residual checks are enough.

Advantages and Limitations

Advantages Limitations
Can incorporate prior knowledge (manager experience, earlier studies) Choosing an appropriate prior can be difficult
Uncertainty is represented explicitly (a posterior, not only a point) Computation can be more complex than a single OLS/ARIMA fit
Beliefs can be updated as new observations arrive Results still depend on the model and the assumptions

Exam-Oriented Key Points

  1. Prior = initial belief; likelihood = data evidence; posterior = updated belief.
  2. Posterior ∝ Likelihood × Prior.
  3. Bayesian forecasts can carry a range of uncertainty, not only one point.
  4. Bayesian methods are not automatically superior to classical forecasting.

Unit 5 Quick Revision

Short Exam Definitions

Topic Short definition / exam point
Multivariate time series Several variables recorded over the same times; joint behaviour can inform forecasts.
Multivariate forecasting Produce futures for more than one series while respecting their links.
Multivariate stationarity Stable means/variances and stable cross-variable (including lagged) relationships.
Cross-variable relationship Association at the same time or at a lag; not automatically causation.
Bayesian forecasting Combine prior and data to update uncertainty, then forecast from the posterior.
Prior Initial belief before (or besides) the current observations.
Likelihood Evidence from the observed data under a stated model.
Posterior Updated belief: Posterior ∝ Likelihood × Prior.
Point vs uncertainty-aware One number vs a distribution/range that shows remaining uncertainty.

Key Distinctions

  1. Univariate vs multivariate: one series vs several related series.
  2. Univariate vs multivariate stationarity: own stability vs own plus cross-links.
  3. Correlation vs causation: moving together ≠ one variable caused the other.
  4. Prior vs likelihood vs posterior: initial belief vs data evidence vs updated belief.
  5. Point forecast vs probabilistic forecast: a single value vs an uncertainty-aware range.

Compact Flows

Multivariate Data ↓ Explore Relationships ↓ Check Stationarity ↓ Select Model ↓ Fit ↓ Forecast ↓ Evaluate
Prior + Observed Data ↓ Posterior ↓ Forecast