Bayesian Sample Size Calculator for AI & Machine Learning

Calculator Inputs

Use this AI and machine learning workflow to estimate how many observations you should complete and recruit.

Prior alpha

Represents prior positive evidence strength.

Prior beta

Represents prior negative evidence strength.

Expected positive rate (%)

Expected model success, conversion, or lift-positive rate.

Credible level (%)

Common choices are 90, 95, and 99.

Desired margin of error (%)

Smaller margins require larger samples.

Design effect

Use values above 1 for clustering or dependence.

Expected dropout or invalid rate (%)

This inflates recruitment above completed observations.

Finite population size (optional)

Leave blank for an effectively unlimited population.

Minimum sample floor

Protects against unrealistically small recommendations.

Reset

Formula Used

This calculator uses a beta-binomial planning model. It assumes a beta prior, an expected positive rate, and a target posterior precision.

Prior: Beta(α₀, β₀) Expected successes: x = n × p Posterior: Beta(α₀ + x, β₀ + n - x) Posterior mean: μ = (α₀ + x) / (α₀ + β₀ + n) Posterior variance: Var = [(α₀ + x)(β₀ + n - x)] / [(α₀ + β₀ + n)²(α₀ + β₀ + n + 1)] Approximate half-width: h = z × √Var Solve for the smallest n where: h ≤ desired margin of error Finite population correction: n_fpc = n / [1 + (n - 1) / N] Design effect adjustment: n_design = ceil(n_fpc × design effect) Dropout adjustment: n_recruit = ceil(n_design / (1 - dropout rate))

The credible interval shown on the page is an approximation based on the posterior mean and posterior variance. It is useful for planning and quick scenario testing.

How to Use This Calculator

Enter prior alpha and beta to reflect previous evidence.
Set the expected positive rate for the target outcome.
Choose the credible level for Bayesian confidence.
Enter the acceptable margin of error in percentage points.
Increase design effect if observations are clustered.
Add expected dropout to protect final usable sample size.
Enter population size only for a limited sampling frame.
Review the recruitment target, posterior interval, and chart.

Example Data Table

Scenario	Prior α	Prior β	Expected Rate	Credible Level	Margin	Design Effect	Dropout	Recommended Recruit
Binary classifier validation	8	4	72%	95%	4%	1.00	10%	231
Human review precision study	12	6	82%	95%	3%	1.10	12%	543
Recommendation click-rate pilot	5	9	38%	90%	5%	1.00	8%	256

Why Bayesian Sample Sizing Helps AI Teams

Bayesian planning lets teams combine prior knowledge with current expectations. That is useful when benchmark data, historical validation results, or pilot labels already exist.

Instead of thinking only about statistical significance, this page focuses on posterior precision. That often matches real AI workflows better, especially when teams want stable estimates for model accuracy, defect rate, acceptance rate, or calibration quality.

Use the calculator during experiment planning, annotation budgeting, offline evaluation design, and launch-readiness review meetings.

FAQs

1. What does this calculator estimate?

It estimates how many observations you should complete and recruit so your posterior interval reaches the chosen precision at the chosen credible level.

2. Why use a beta prior?

A beta prior works naturally for binary outcomes like success, failure, click, no click, pass, or fail. It is simple, interpretable, and computationally efficient.

3. What do alpha and beta mean?

Alpha represents prior positive evidence. Beta represents prior negative evidence. Together they define prior strength and shift the posterior estimate before new data arrives.

4. When should I raise design effect?

Raise it when observations are correlated, grouped, or cluster-sampled. Examples include multiple events from one user, several labels from one document, or batched model outputs.

5. Why is recruitment larger than completed sample?

Recruitment includes expected attrition. Some records may be dropped, invalid, unlabeled, or unusable. The calculator inflates the target so the final valid sample still meets precision needs.

6. What does finite population size change?

If your population is small and fixed, finite correction can reduce the completed sample target. It matters more when the required sample is a meaningful share of the population.

7. Is the displayed interval exact?

It is an approximation using posterior variance and a normal critical value. For planning, it is practical and fast. For reporting, you may also compute exact beta quantiles.

8. Can I use this for model accuracy studies?

Yes. It suits many binary AI outcomes, including accuracy pass rates, moderation acceptance rates, annotation agreement thresholds, and conversion-like machine learning events.