Enter forecast inputs
This tool applies a normal-normal Bayesian update, then extends the forecast with trend, seasonality, and uncertainty growth.
Example data table
Example assumptions: prior mean 120, prior variance 64, observed mean 128, observation variance 100, sample size 24, horizon 6, trend 1.5, process variance growth 2.25, seasonal amplitude 3, and season length 4.
| Period | Seasonal Adjustment | Forecast Mean | 95% Lower Bound | 95% Upper Bound |
|---|---|---|---|---|
| 1 | 3.00 | 132.01 | 111.82 | 152.21 |
| 2 | 0.00 | 130.51 | 110.10 | 150.92 |
| 3 | -3.00 | 129.01 | 108.39 | 149.63 |
| 4 | 0.00 | 133.51 | 112.68 | 154.34 |
| 5 | 3.00 | 138.01 | 116.98 | 159.04 |
| 6 | 0.00 | 136.51 | 115.27 | 157.75 |
Formula used
τ²post = 1 / ( 1 / τ²prior + n / σ² )
μpost = τ²post × ( μprior / τ²prior + n × x̄ / σ² )
μ̂h = μpost + h × trend + seasonalAmplitude × sin( 2πh / seasonLength )
σ²pred,h = τ²post + σ² + h × processVariance
Interval = μ̂h ± z × √σ²pred,h
P(X > T) = 1 − Φ( ( T − μ̂h ) / σpred,h )
In this page, τ² represents variance, σ² represents observation variance, x̄ is the observed sample mean, n is sample size, z comes from the chosen confidence level, and Φ is the standard normal cumulative distribution function.
How to use this calculator
- Enter your prior mean and prior variance to describe your belief before new evidence.
- Fill in the observed sample mean, observation variance, and sample size from recent data.
- Set the forecast horizon, expected trend, seasonality inputs, and process variance growth.
- Choose a confidence level. Add a threshold or actual value when you want decision and accuracy checks.
- Press Calculate Forecast to view the posterior update, interval table, graph, and export options.
FAQs
1) What does this Bayesian forecast tool calculate?
It updates a prior belief with observed evidence, then projects future values using trend, seasonality, and growing uncertainty. It also reports predictive intervals and optional threshold probabilities.
2) When should I use a Bayesian forecast instead of a simple average?
Use it when you want to combine expert expectations with fresh data. Bayesian updating is especially useful when sample sizes are limited or uncertainty needs to be modeled explicitly.
3) What is the difference between prior variance and observation variance?
Prior variance measures uncertainty in your starting belief. Observation variance measures noise in the incoming data. Together, they determine how strongly new evidence shifts the posterior estimate.
4) Why does sample size matter so much?
A larger sample size gives the observed mean more weight. That usually reduces posterior uncertainty and pulls the forecast closer to the data, assuming observation variance remains unchanged.
5) What does process variance growth represent?
It models uncertainty that grows as the forecast moves further into the future. Larger values create wider predictive intervals at later horizons, reflecting more risk over time.
6) How should I interpret the threshold probability?
It estimates the chance that a forecasted future value exceeds your chosen target. This is helpful for risk screening, capacity planning, pricing, and model-based decision rules.
7) Does this tool handle complex Bayesian models?
This page uses a practical normal-normal update with forecast adjustments. It is strong for fast planning, though not a replacement for full hierarchical or simulation-heavy Bayesian pipelines.
8) What does the actual final value check tell me?
It compares the real final outcome with the last horizon forecast. You can quickly see forecast error and whether the actual value stayed inside the predictive interval.