Bayesian Forecasting

Bayesian forecasting treats predictions as probability distributions that update as new evidence arrives. Rather than producing a single point estimate, the Bayesian approach maintains a full belief distribution — a prior — and systematically refines it with each new piece of data using Bayes' theorem. This produces calibrated uncertainty estimates, naturally handles sparse data, and provides a principled framework for sequential updating that is essential for real-time forecasting.

## Key Points

- **Prior**: Your initial forecast before new data
- **Likelihood**: How probable the observed data is under different scenarios
- **Posterior**: Your updated forecast after incorporating new data
1. Bayesian forecasting maintains full probability distributions, not just point estimates, giving calibrated uncertainty
2. Conjugate priors (Beta-Binomial, Normal-Normal, Gamma-Poisson) provide analytical updates without computation
3. Prior selection encodes domain knowledge; uninformative priors let data dominate, informative priors regularize with sparse data
4. Sequential updating via Kalman-filter-style methods enables real-time forecast refinement as evidence arrives
5. Bayesian networks model causal relationships, enabling "what if" reasoning for prediction
6. Calibration tracking is essential: regularly check whether your 80% confidence intervals contain the truth 80% of the time
7. PyMC and similar tools make complex Bayesian models accessible without deriving posteriors analytically
8. The information value calculation helps decide which data to collect next — prioritize observations that maximally reduce uncertainty