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Bayesian Experiment Error Rate Estimator

The Bayesian Experiment Error Rate Estimator updates an experiment’s underlying error probability from observed errors and a Beta prior. It is designed for binary quality outcomes such as failed requests, incorrect model outputs, rejected records, or test cases that are marked error versus non-error.

Instead of relying only on the raw observed error percentage, the calculator combines the data with prior alpha and beta parameters to produce a posterior mean error rate. It also shows the raw rate, posterior expected error count per 1,000 trials, and the amount of prior pseudo-count information. This is especially useful when experiments have limited sample sizes and you want a stable Bayesian point estimate while keeping the assumptions visible.

Bayesian error inputs

Result
—
Posterior mean error rate
Raw error rate—
Errors per 1,000—
Posterior alpha—
Posterior beta—

1. Count error outcomes
Enter how many trials were classified as errors and the total number of trials examined.

2. Describe the prior
Use prior alpha for error evidence and prior beta for non-error evidence. Both inputs must be positive.

3. Read the posterior rate
The main result is the posterior mean probability of an error after the prior and observed counts are combined.

4. Compare with the raw rate
The raw observed rate appears separately so you can see how much the prior shifts the estimate.

5. Scale the result
The errors-per-1,000 statistic converts the posterior rate into an operational count that can be easier to communicate.

Posterior alpha = prior alpha + observed errors Posterior beta = prior beta + non-errors Posterior mean error rate = posterior alpha / (posterior alpha + posterior beta) Errors per 1,000 = posterior mean × 1,000

Non-errors equal total trials minus observed errors. Under a Beta-Binomial model, alpha tracks error-side evidence and beta tracks non-error-side evidence. The posterior mean is a point estimate; it does not by itself show uncertainty.

What the result means

The posterior mean is the estimated probability that a new comparable trial will be an error under the Beta-Binomial model.

Use priors that reflect defensible prior information. Strong priors can materially affect the result when the observed sample is small.

Given: 8 observed errors in 200 trials with a Beta(1, 1) prior.

Calculation: Non-errors = 200 − 8 = 192. Posterior alpha = 1 + 8 = 9. Posterior beta = 1 + 192 = 193. Posterior mean = 9 / 202 = 0.044554.

Result: Posterior mean error rate ≈ 4.455%, or about 44.55 errors per 1,000 comparable trials.

The raw sample rate is 4.000%; the weak prior slightly pulls the posterior estimate toward 50% because the sample is finite.

Why does the posterior rate differ from observed errors divided by trials?

The posterior includes prior alpha and beta counts in addition to the observed data. With a weak prior and a large sample, the difference is usually small.

Should alpha represent successes or errors?

On this page alpha is explicitly assigned to the event being estimated: errors. Beta represents non-errors, so keep that convention when entering prior parameters.

Can the prior overwhelm the experiment data?

Yes. Large prior alpha and beta values act like substantial pseudo-counts and can dominate a small observed sample. Use strong priors only when they are justified.

What happens if there are zero observed errors?

The estimator can still produce a nonzero posterior rate when prior alpha is positive. That avoids treating a finite sample with zero observed errors as proof that the true error probability is exactly zero.

When is a credible interval more useful than this point estimate?

Use an interval when you need uncertainty around the error rate, especially for small samples or risk-sensitive decisions. The related Bayesian confidence interval calculator provides an approximate posterior interval.