Credible IntervalEdit

A credible interval is a probabilistic statement about a parameter, framed in the language of Bayesian statistics. In its simplest form, it is a range of values that contains a parameter with a specified probability given the observed data. Unlike some other kinds of intervals, a credible interval relies on a model that combines the data through the likelihood with beliefs about the parameter encoded in a prior distribution. The result is a posterior distribution from which the interval is derived, and different choices of prior and model can yield different intervals. See Bayesian statistics and posterior distribution for the broader framework, and note how this concept sits alongside other ideas in statistical inference such as confidence interval and likelihood function.

Credible intervals are widely used in science and policy because they provide a direct probability statement about the parameter conditioned on what is known. A 95% credible interval, for example, is a range [a, b] such that P(θ ∈ [a, b] | data) = 0.95, where θ is the parameter of interest. There are several practical ways to construct such an interval, including equal-tailed intervals and highest posterior density (HPD) intervals, both of which can be derived analytically in simple models or approximated numerically in more complex ones. For computation, methods like Markov chain Monte Carlo are common when closed-form solutions are not available.

Definition and Concept

A credible interval is derived from the posterior distribution p(θ | data). The interval is defined so that the posterior probability that the parameter lies in the interval equals the desired level (e.g., 0.95).
Equal-tailed intervals split the posterior probability evenly on both sides and are common in simple cases. HPD intervals give the shortest interval containing the specified probability mass and are often preferred when the posterior is skewed.
The interpretation is conditional: the interval describes what, given the model and the observed data, we believe about θ. This is distinct from a long-run frequency claim about repeated samples, which is the hallmark of frequentist confidence intervals.

Bayesian versus Frequentist perspectives

In the Bayesian view, a credible interval directly expresses uncertainty about θ after observing the data, through the posterior distribution. See Bayesian statistics and posterior distribution.
In the frequentist view, a confidence interval is constructed so that, over numerous repeated samples, a certain proportion of the intervals would cover the true parameter. This does not assign a probability to θ lying in a single observed interval. See confidence interval.
In practice, the two concepts can converge in well-behaved models with noninformative priors, but they can diverge when priors are informative or when the model is misspecified. The choice between them often reflects a balance between transparency about assumptions and the desire for interpretability in decision making.

Priors, models, and interpretation

Priors encode beliefs about plausible parameter values before seeing the current data. They can be informative, reflecting substantive knowledge, or relatively flat/noninformative, aiming to let the data speak more loudly. See prior distribution and noninformative prior.
The posterior, and thus the credible interval, is a product of the prior and the likelihood. Different priors can yield different intervals, which makes sensitivity analysis (checking how conclusions change with alternative priors) an important practice.
Model choice matters. If the likelihood misrepresents the data-generating process, credible intervals can be misleading. Model checking and robustness analysis, including posterior predictive checks, help guard against overconfidence due to model misspecification. See model checking and posterior predictive distribution.

Practical implications and controversies

Advocates argue that credible intervals offer transparent, explicit statements about uncertainty that are naturally aligned with how experts reason about evidence. They are especially useful in contexts with limited data or strong prior information, where Bayesian methods can provide stable inferences and intuitive decision tools.
Critics often point to the subjectivity introduced by priors. They argue that this subjectivity can contaminate the interval with beliefs not warranted by data, particularly in high-stakes policy or public discourse. Proponents respond that all statistical inference involves assumptions; making those assumptions explicit through priors is preferable to hiding them behind a veil of claimed objectivity.
In public-facing reports, credible intervals can be misinterpreted if the audience expects a universal notion of objectivity. The responsible stance is to accompany intervals with clear explanations of the model, priors, and the uncertainty sources, rather than presenting a single number as definitive truth.
Some critics contend that Bayesian intervals can be overly optimistic or conservative depending on the prior and model. Defenders emphasize that frequentist intervals are not inherently objective either and that both frameworks require careful calibration and communication. In practice, cross-checks with alternative methods and sensitivity analyses are common to ensure robust conclusions.

From a decision-making perspective, the key strength of credible intervals is that they quantify uncertainty in a coherent framework tied to a specific model. This makes it easier to propagate uncertainty through calculations and to compare alternatives that hinge on uncertain quantities. In many applications, policymakers and practitioners prefer transparent reasoning about what is known, what is assumed, and how those assumptions shape conclusions, rather than poring over long lists of p-values without context.

Practical computation and extensions

In simple models, closed-form posterior distributions yield straightforward credible intervals. In more complex settings, numerical methods such as Markov chain Monte Carlo or variational techniques are used to approximate the posterior and extract credible intervals.
For multivariate problems, joint credible regions can be constructed, and HPD-like regions generalize to higher dimensions, though interpretation becomes more nuanced.
When reporting results, practitioners often present multiple intervals (e.g., 95% and 90%) or provide a full posterior distribution to help others assess sensitivity to assumptions. They may also report the posterior mean or median as a point estimate alongside the interval.

Examples

A simple normal model with an informative prior: suppose θ represents a treatment effect, data are collected, and the prior reflects prior evidence. The posterior for θ is normal in many conjugate cases, yielding a credible interval around the posterior mean with a width determined by the posterior standard deviation. This interval directly expresses the probability that θ lies within it given the data and prior.
A nonstandard posterior: in models with skewness or heavy tails, the HPD interval may be more informative than an equal-tailed interval, since it concentrates probability mass in the most plausible regions under the posterior.