Confidence interval vs credible interval

 

1. Confidence interval (frequentist)

Definition
A 95% confidence interval is a procedure that, if repeated many times on new data generated under the same conditions, would contain the true parameter 95% of the time.

Key point
The parameter is fixed but unknown; the interval is random.

Correct interpretation

“If we were to repeat this study infinitely many times and compute a 95% confidence interval each time, 95% of those intervals would contain the true parameter.”

Incorrect (but common) interpretation

“There is a 95% probability that the true parameter lies within this interval.” ❌

That statement is not valid in frequentist statistics.

Example
You estimate a mean nest temperature and obtain a 95% CI of [28.1, 29.3] °C.
You cannot assign a probability to the true mean being inside this specific interval—either it is or it isn’t.

2. Credible interval (Bayesian)

Definition
A 95% credible interval is an interval within which the parameter lies with 95% probability, given the data and the prior.

Key point
The parameter is treated as a random variable; the interval is fixed.

Correct interpretation

“Given the data and the prior, there is a 95% probability that the parameter lies within this interval.”

This is a direct probability statement about the parameter.

Example
Using a Bayesian model of nest temperature (with a prior on the mean), you obtain a 95% credible interval of [28.0, 29.4] °C.
You can legitimately say there is a 95% probability that the true mean lies in that range.

3. Why they can be numerically similar

  • With large sample sizes

  • With weak or non-informative priors

  • For simple models (e.g., normal errors)

a confidence interval and a credible interval may be almost identical numerically—but their interpretations remain different.

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