Normal Normal Posterior Calculator
Analyze prior beliefs with observed normal sample information. See posterior mean, variance, precision, and intervals. Make Bayesian decisions faster using clear tables, graphs, downloads.
Calculator Inputs
Example Data Table
| Scenario | Prior Mean | Prior Variance | Sample Size | Sample Mean | Known Variance | Posterior Mean | Posterior Variance |
|---|---|---|---|---|---|---|---|
| Quality score update | 5.0000 | 4.0000 | 12 | 5.4000 | 9.0000 | 5.2609 | 1.5652 |
| Sensor calibration | 10.0000 | 1.5000 | 20 | 9.6000 | 2.2000 | 9.6255 | 0.1053 |
| Manufacturing fill weight | 50.0000 | 6.0000 | 15 | 51.2000 | 4.5000 | 50.8767 | 0.4478 |
Formula Used
This calculator assumes a normal prior on the unknown mean and a normal sampling model with known observation variance.
μ ~ Normal(μ0, τ0²)
x̄ | μ ~ Normal(μ, σ² / n)
τn² = 1 / (1 / τ0² + n / σ²)
μn = τn² × (μ0 / τ0² + n × x̄ / σ²)
μn ± z × τn, where z is the standard normal critical value for your chosen credible level.
σ² + τn²
How to Use This Calculator
- Choose summary statistics or raw observations mode.
- Enter the prior mean and prior variance.
- Enter the known observation variance for the normal model.
- Provide sample size and sample mean, or paste raw observations.
- Select a credible level, such as 90, 95, or 99.
- Optionally add a threshold to compute P(μ exceeds threshold).
- Press Calculate Posterior to show results above the form.
- Review the metrics, table, graph, and export downloads.
Frequently Asked Questions
1) What does the calculator estimate?
It estimates the posterior distribution of an unknown normal mean. You get the updated mean, posterior variance, interval estimates, predictive values, and optional threshold probability.
2) When should I use the normal-normal model?
Use it when the unknown parameter is a mean, the prior for that mean is normal, and the observation variance is known or treated as fixed.
3) What is the role of prior variance?
Prior variance controls how strongly earlier belief influences the update. Smaller prior variance means stronger prior confidence and more pull toward the prior mean.
4) Why is known observation variance required?
This conjugate form assumes the sampling variance is already known. If variance is unknown, a normal-inverse-gamma or t-based approach is usually more appropriate.
5) Can I use raw observations instead of summary values?
Yes. Raw mode parses comma, space, or line-separated values. The calculator then computes the sample size and sample mean automatically for the posterior update.
6) What does the threshold probability mean?
It returns the posterior probability that the true mean is greater than your chosen threshold. This is useful for decision rules and target checks.
7) What does the graph show?
The chart overlays prior density, data-based likelihood for the mean, and posterior density. It helps you see how the evidence shifts and narrows belief.
8) What is prior equivalent sample size?
It converts prior strength into sample-size units using σ²/τ0². Larger values mean the prior behaves like more observations in the update.