A Beta(38, 164) posterior — 37 of 200 under a flat prior — forecasting the next hundred trials. The predictive standard deviation is 4.7667. A binomial with the same mean but a known rate gives 3.9081, so treating the estimated rate as though it were exact understates the spread by 21.97%. That is the usual reason a forecast interval turns out too narrow in practice.
Beta(38.00, 164.00) over 100 future trials
18.812 expected, 95% between 10 and 29
The predictive standard deviation is 4.7667. A binomial with the same mean but a rate treated as known gives 3.9081 — 21.97% narrower. That gap is the uncertainty in the rate itself, and a forecast interval built from a point estimate discards it entirely.
Expected successes
18.812
rate 0.18812
95% predictive
10 to 29
width 19
Predictive SD
4.7667
binomial 3.9081
Extra spread
21.97%
from rate uncertainty
Binomial randomness grows as √m and the contribution of rate uncertainty grows as m, so the gap widens with the horizon. At 100 trials it is 21.97%; the same posterior over ten times as many trials would show a larger one.
The predictive distribution
Probability of each number of future successes, around the interval
Successes
Probability
Cumulative
Bar
8
0.004652
0.00845
█
9
0.008579
0.01703
██
10
0.014446
0.03148
███
11
0.022424
0.05390
█████
12
0.032339
0.08624
███████
13
0.043607
0.12985
█████████
14
0.055281
0.18513
████████████
15
0.066190
0.25132
██████████████
16
0.075147
0.32647
████████████████
17
0.081178
0.40765
█████████████████
18
0.083690
0.49134
██████████████████
19
0.082557
0.57389
██████████████████
20
0.078108
0.65200
█████████████████
21
0.071021
0.72302
███████████████
22
0.062177
0.78520
█████████████
23
0.052496
0.83770
███████████
24
0.042808
0.88050
█████████
25
0.033759
0.91426
███████
26
0.025778
0.94004
██████
27
0.019079
0.95912
████
28
0.013700
0.97282
███
29
0.009553
0.98237
██
30
0.006473
0.98885
█
31
0.004266
0.99311
█
Rows in bold fall inside the 95% interval. The probabilities across the whole range sum to exactly 1, which the verification suite asserts on 150 generated forecasts.
Where the spread comes from
The two sources of variance in a predictive forecast
Source
Variance
Share
Trial randomness
15.27301
67.22%
Uncertainty in the rate
7.44841
32.78%
Total
22.72143
100%
A binomial forecast keeps the first row and discards the second. When the second is 32.8% of the variance, that is not a rounding decision.
Carries rate uncertainty Full distribution shown Exact, not simulated Still assumes a fixed rate
What this tool shows
A Beta(38, 164) posterior forecasting the next hundred trials has a standard deviation of 4.7667. A binomial with the same mean and a known rate gives 3.9081. The forecast is 21.97% wider, and 32.78% of its variance is uncertainty about the rate rather than randomness in the trials. Over a thousand trials that becomes 143.33% wider. Treating an estimated rate as exact is the usual reason a forecast interval turns out too narrow.
The exact Beta-Binomial predictive distribution, every outcome probability computed
The variance split into trial randomness and uncertainty in the rate itself
A binomial forecast on the same mean, so the gap is measured rather than described
A 95% predictive interval on the count of future successes
Presets from barely-any-data to a well-known rate, and from a short horizon to a long one
Why the gap grows with the horizon rather than shrinking
Carries rate uncertainty Full distribution Exact, not simulated Binomial for contrast
It still assumes the rate is fixed. Only that you do not know which value.
Updated 13 September 2026 · Works in any browser, no installation
The Beta-Binomial predictive distribution says how many successes to expect in a future batch, carrying the uncertainty in the rate rather than fixing it at a point estimate. A binomial forecast asks “if the rate is 18.8%, how many of the next hundred will convert?”. This asks “given everything observed so far, how many of the next hundred will convert?” — and the second question has a wider answer, because the rate is not known to be 18.8%.
At a glance
Formula shown
P(k successes in m future trials) = C(m, k)·B(α + k, β + m − k)/B(α, β), where Beta(α, β) is the current posterior. Its mean is m·α/(α+β), the same as a binomial at the posterior mean, and its variance is m·αβ(α+β+m) / [(α+β)²(α+β+1)] — which contains the binomial variance plus an extra term that grows with m² rather than m. That is why the gap between the two widens with the forecast horizon.
Scenario support
Forecasting conversions, defects or failures in a coming period, capacity planning from an estimated failure rate, setting expectations for a batch before it runs, and any prediction where the rate was estimated from data rather than known in advance.
Educational estimate
Planning support from the values you enter — not professional advice.
The forecast is 22% wider, and the gap grows
Binomial forecasting is the default because it is easy, and it is wrong in a specific, quantifiable direction.
Beta(38, 164) over a hundred trials: predictive SD 4.7667, binomial SD 3.9081. 21.97% wider.
Over a thousand trials: 30.0723 against 12.3584. 143.33% wider — more than double.
The reason is how the two sources scale. Trial randomness grows as √m; the contribution from not knowing the rate grows as m. Forecast far enough ahead and the second dominates completely.
Which is why the failure is worst exactly where forecasts matter. A one-week forecast from an estimated rate is nearly fine; a quarter-ahead forecast from the same rate is not.
Two sources, and the table splits them
The variance decomposition on this page is the whole argument in one row, and it is exact rather than approximate.
Trial randomness is what a binomial captures: even with a perfectly known rate, a hundred trials do not give exactly the expected count.
Rate uncertainty is what it discards: the rate could be 16% or 22%, and that moves the whole forecast rather than jittering it.
On the shipped preset the split is 67.22% to 32.78%. A third of the forecast uncertainty is simply not knowing the rate.
With eight observations behind the estimate it becomes 36.67% to 63.33%. With a thousand it drops to 91.00% and 9.00% — which is why the distinction only matters when the rate was estimated from little data or the horizon is long.
The same distribution, seen from the other side
This distribution turns up under a second name in a completely different context, and recognising them as one thing is useful.
As a model for observed data, it is the standard fix for overdispersed counts. When proportions vary more than a binomial allows — different batches, different operators, different days — a Beta-Binomial fits.
As a forecast, it is the same distribution pointed forward. The rate varies because you do not know it, rather than because it genuinely differs.
The arithmetic does not distinguish the two, which is why a p chart on overdispersed data and a binomial forecast from an estimated rate fail in exactly the same way: limits too narrow, everything flagged.
The interpretation does distinguish them. Rate uncertainty shrinks with more data; genuine batch-to-batch variation does not.
A predictive interval is not a credible interval
The two intervals in Bayesian work look similar and answer different questions, and mixing them is a common and consequential error.
A credible interval is about the rate. Beta(38, 164) gives 0.1354 to 0.2425 — where the underlying rate is.
A predictive interval is about the next batch. Over a hundred trials it is 10 to 29 successes, which is 10% to 29% — far wider.
The predictive one is always wider, because it adds the randomness of the future trials on top of the uncertainty about the rate.
Quoting the credible interval as a forecast range is the error to avoid. It is the same mistake as quoting a confidence interval on a mean when a prediction interval was wanted.
Computed, not simulated
Predictive distributions are usually obtained by sampling, and there is no need to here.
Every outcome probability has a closed form, so the distribution is computed term by term rather than approximated by draws.
Which means the same input always gives the same answer, with no seed and no Monte Carlo error to explain.
The suite checks the closed form against itself from two directions: the reported mean and standard deviation are compared against those computed by summing k·P(k) and k²·P(k) over the whole distribution, at three configurations.
And the probabilities sum to exactly 1 on 150 generated forecasts, which is the cheapest possible check that nothing has been dropped or double-counted.
What it still assumes
Carrying uncertainty about the rate fixes one thing. Three assumptions survive untouched.
The rate is fixed, just unknown. If it is drifting, this forecast is wrong in a way no amount of extra spread repairs.
Future trials are like past ones. A different season, audience or process is not what the posterior was built from.
Trials are independent given the rate. Clustering — the same visitor returning, the same batch of material — breaks that and widens the true distribution further.
And the prior still matters at the far end. With very little data the predictive distribution is dominated by the prior, which is visible on the second preset and worth checking before quoting a forecast from eight observations.
Reporting a forecast
Four items, and the second is the one that distinguishes this from a point forecast with a plausible range attached.
Give the horizon. Everything here depends on m, and the gap against a binomial forecast grows with it.
Say the interval is predictive, not credible. They differ by a factor of several and answer different questions.
Give the posterior it came from, and the data behind that. A forecast from Beta(3, 7) and one from Beta(500, 500) deserve very different trust.
And say what fraction of the spread is rate uncertainty. It is the number that tells a reader whether more data would help or whether the trials themselves are the limit.
Method. Every outcome probability is computed from the closed form rather than sampled, so the distribution is exact and reproducible with no seed. The suite checks that closed form from the other direction: the reported mean and standard deviation are compared against those obtained by summing k·P(k) and k²·P(k) across the whole distribution, at three configurations, which would catch an error in either the moment formulas or the probabilities. It also asserts on 150 generated forecasts that the probabilities sum to exactly 1, that the predictive spread is never below the binomial spread with the same mean, and that the mean is exactly m·α/(α+β). The variance decomposition printed on the page is the difference between those two variances, which is why it is exact rather than an attribution. Horizons above 5,000 are refused rather than computed, since the term-by-term sum is linear in the horizon. That engine is verified on every change against 106 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Credible IntervalBayesian credible intervals for a proportion with both the highest-density and equal-tailed versions and the prior's weight reported.
Binomial DistributionExact binomial probabilities at any n — including thousands, where a factorial overflows — with the normal approximation beside them and its error measured, which is 0.6% at the centre and 261% in the tail.
Posterior ProbabilitySequential Bayesian updating with the posterior after every batch, the prior's weight reported, and grouping-independence demonstrated.
Beta DistributionTakes raw successes and failures and runs the conjugate update, so 5 out of 5 returns 6/7 rather than the 100% a plain proportion claims — with a credible interval that stays honest at zero.
p Chartp and np charts with limits computed per point when the sample sizes differ, and the cost of flat limits measured on your own chart.
Bayesian A/B TestP(B beats A) from a closed form rather than sampling, with expected loss as the stopping rule and a margin of practical interest.
An educational tool. The predictive distribution carries uncertainty about the rate but still assumes that rate is fixed and that future trials resemble the ones behind the posterior — a drifting rate or a changed population is not something extra spread can repair. With very little data the forecast is dominated by the prior rather than by the observations.