What is a Credible Interval 🌿

A credible interval is the Bayesian equivalent of the confidence interval where the estimated interval is dependent on the prior distribution. See Confidence Intervals for Known Distributions and Pivotal Values to learn about what a confidence interval is.

• In confidence intervals we treat the parameter as a fixed value and the bounds as random variables
• In credible intervals, the estimated parameter is treated as a random variable while the bounds are considered fixed

Any random variable following the posterior distribution $\pi(\theta \mid y_1,\ldots, y_n)$ will fall in the credible interval with probability $1-\alpha$

• Conceptually, probability comes into play in a frequentist confidence interval before collecting the data.
  • Ex: there is a 95% probability that we will collect data that produces an interval that contains the true parameter value.
• Probability comes into play in a credible interval after collecting the data
  • Ex: based on the data, we now think there is a 95% probability that the true parameter value is in the interval.

Here is an Example Problem:

Interested in the proportion of the population of American college students that sleep at least eight hours each night $\theta$.

• Suppose a random sample of 27 students from UF, where 11 students recorded they slept at least eight hours each night
• Suppose that $X \sim \text{Binomial}(27,\theta)$
• Let’s assume we have a prior on $\theta$ where $\theta \sim \text{Beta}(3.3,7.2)$.

Thus, the posterior distribution (which we know from Why are Conjugate Priors Useful) is:

• $\theta \mid (11,27) ∼ \text{Beta}(11 + 3.3, 27 − 11 + 7.2) = \text{Beta}(14.3, 23.2)$

Suppose now we would like to find a 90% credible interval for $\theta$ with equal probability in each tail

• So, we want to find $a$ and $b$ such that $P(\theta < a

  x) = 0.05$ and $P(\theta > b

  x) = 0.05$

  • We can solve this by taking the cdf of the Beta posterior up to the lower bound:
    • $\int_0^a \text{Beta}(14.3, 23.2) d\theta = 0.05$
  • Then for the upper bound
    • $\int_d^1 \text{Beta}(14.3, 23.2) d\theta = 0.05$
• These values are extremely difficult to calculate by hand (who knows how to integrate the Beta distribution?) so we can just plug our values into the beta cdf in R to find what values $\theta$ value will give us a cdf value (area under pdf) of 0.05 (for $a$) and 0.95 (for $b$).

  a = 3.3
  b = 7.2
  n = 27
  x = 11
  a.star = x+a
  b.star = n-x+b
  c = qbeta(0.05,a.star,b.star)
  d = qbeta(1-0.05,a.star,b.star)
  

This gives us $c = 0.256$ and $d = 0.514$ so our credible interval is $(0.256, 0.514)$

Credible intervals are not unique on a posterior distribution. Methods for defining a suitable credible interval include:

• Choosing the narrowest interval, which for a unimodal distribution will involve choosing those values of highest probability density including the mode (the maximum a posteriori). This is sometimes called the highest posterior density interval (HPDI)
• Choosing the interval where the probability of being below the interval is as likely as being above it. This interval will include the median. This is sometimes called the equal-tailed interval.
• Assuming that the mean exists, choosing the interval for which the mean is the central point.