Credible Intervals and Posterior Summaries

A Bayesian analysis ends with a full posterior distribution — a curve describing every possible value of the parameter and how plausible each one is, given your data and your prior beliefs. That's rich, but it's not something you can put in a one-line trading memo. You need to compress it: a single best-guess number, and a range that communicates how sure you are. A credible interval is that range, and a posterior summary is the compression. The appeal is that, unlike a frequentist confidence interval, a Bayesian credible interval means exactly the sentence people intuitively want to say: "there is a 95% chance the true value is in here" — because in the Bayesian framework, the parameter itself is treated as a random quantity with a distribution, so that sentence is literally, mathematically true.

An analogy: summarizing a crowd's guesses, not a coin flip

Imagine you ask a thousand well-informed people to guess a stock's fair value, and you collect a histogram of their answers. That histogram — spread out, maybe skewed, maybe with a fat tail — is your posterior. If someone asks "so what's the number?", you don't hand them the histogram; you say "most people guessed around $52, and 95% of the guesses fell between $48 and $57." That's a posterior summary and a credible interval in one sentence. Crucially, the statement "95% of guesses were in that range" is a direct, literal description of the crowd you actually have in front of you — not a statement about some imagined repetition of the guessing exercise. That directness is the whole appeal of Bayesian summaries over their frequentist counterparts.

The summaries, one symbol at a time

Let $\theta$ ("theta") be the unknown parameter and $p(\theta \mid \text{data})$ the posterior distribution — everything you believe about $\theta$ after combining your prior with the observed data. The most common point summaries are the posterior mean, $E[\theta \mid \text{data}]$, and the posterior mode (the single most probable value, called the MAP estimate). A credible interval at level $1-\alpha$ is any range $[L, U]$ satisfying

$$\Pr(L \le \theta \le U \mid \text{data}) = 1 - \alpha .$$

In words: integrate the posterior curve between $L$ and $U$, and the area under it equals your chosen probability, $1-\alpha$ — a direct probability statement about where $\theta$ itself lies, unlike a frequentist confidence interval, whose $1-\alpha$ refers to how often the procedure would capture the true value across imagined repeated samples, not a probability about this one interval.

Two common constructions give different intervals from the same posterior. The equal-tailed interval takes the $\alpha/2$ and $1-\alpha/2$ percentiles — trim equal area off both tails. The highest posterior density (HPD) interval instead finds the shortest interval containing $1-\alpha$ of the area, which for a skewed posterior does not sit symmetrically around the mean:

$$\text{HPD: the shortest } [L,U] \text{ such that } \Pr(L\le\theta\le U) = 1-\alpha .$$

In words: instead of trimming equal probability from each tail, HPD trims wherever the posterior is lowest, guaranteeing every point inside the interval is more probable than every point outside it — which the equal-tailed interval does not guarantee for a skewed posterior.

Worked example 1: Beta posterior for a win rate

A strategy wins $k=18$ out of $n=40$ trades. Using a flat (uniform) prior, $\text{Beta}(1,1)$, the posterior for the win probability $p$ is $\text{Beta}(1+18,\ 1+40-18) = \text{Beta}(19, 23)$ — a standard conjugate-prior update. The posterior mean is $19/(19+23) = 19/42 = 0.452$. Using standard Beta percentile tables (or a quick numerical lookup), the 2.5th and 97.5th percentiles of $\text{Beta}(19,23)$ are approximately $0.304$ and $0.605$. So the equal-tailed 95% credible interval is $[0.304,\ 0.605]$, and the correct sentence is: "given this data and a flat prior, there is a 95% probability the true win rate is between 30.4% and 60.5%." Note the interval is nearly symmetric here because $\text{Beta}(19,23)$ is close to bell-shaped — not a coincidence for moderate sample sizes, but not guaranteed in general.

Worked example 2: HPD versus equal-tailed on a skewed posterior

Suppose instead only $k=2$ wins out of $n=40$ trades — a rare-event setting. Posterior: $\text{Beta}(3, 39)$, strongly right-skewed since it's bunched near zero with a long tail. Posterior mean $= 3/42 = 0.0714$. The equal-tailed 95% interval works out to roughly $[0.0150,\ 0.174]$ — trimming 2.5% probability off each tail. But because the distribution is so skewed, that lower bound of $0.0150$ sits in a region where the density is actually still quite high (near the peak), while cutting the interval there instead of a bit further out wastes some of your probability budget on a low-density stretch near the top. The HPD interval for this same posterior comes out closer to $[0.004,\ 0.155]$ — shifted left, and about 15% narrower in total width, because it hunts for the shortest span achieving 95% coverage rather than insisting on equal tail mass. For a skewed posterior like this one, HPD is the interval that actually answers "give me the smallest range I can be 95% confident contains the truth."

Feed in a small number of trials with a low win rate in the explorer above and watch the posterior pile up near zero with a long right tail — that skew is exactly the shape that makes equal-tailed and HPD intervals disagree, as in worked example 2.

What this means in practice

• Report the interval type you used. "95% credible interval" without saying equal-tailed or HPD is ambiguous whenever the posterior is skewed — for a strategy Sharpe ratio, hit rate near an extreme, or any bounded quantity, the two methods can meaningfully disagree, as shown above.
• The prior matters most with little data. With $n=40$ trades in example 1, a flat prior barely mattered; with $k=2$ wins in example 2, the choice of prior would visibly shift both the mean and the interval — always state the prior alongside the summary.
• Credible intervals compose naturally through simulation. Because MCMC or grid-based posteriors are just collections of samples, propagating a credible interval through a further calculation (say, expected P&L from the win rate) is as simple as transforming each posterior sample and re-summarizing — no delta method required.
• Communicating to non-Bayesians. A credible interval and a confidence interval often land at similar numbers with weak priors and large samples, but only the credible interval licenses the "95% probability" sentence — say so explicitly when the audience might conflate the two.