The Theta Sketches within the DataSketches Library provide Frequentist Confidence Intervals based on the tails of the Binomial Distribution.
These bounds are only approximate, but we provide a guarantee (backed by experiments, not by proofs) that neither bound is off by more than 1 percent of the estimate.
To avoid numerical issues and high computational cost, the implementation carves up the input space (|S|, p) into multiple regions, and uses different approximations on each.
At a high level, there is a two-way split, between large |S|, and small |S|. When |S| > 120, the continuity-corrected Gaussian approximation to the Binomial Distribution satisfies the 1 percent guarantee, so we just use that.
For small |S| <= 120, the main idea is to fall back to the exact Binomial Distribution. However, that is expensive, and worse yet, subject to numerical issues. These issues are serious, and occur when p is very close to 1.0 or 0.0. Therefore we immediately split off two more cases.
When p is very close to 1.0, we already know (because |S| is small) that the answer is [|S|-1, |S|+1], so we just return that instead of trying to evaluate the incomplete beta function with difficult inputs.
When p is very close to 0.0 (say 1e-6), one could avoid the numerical issues by exploiting the fact that the LB and the UB are very close to being constant multiples of the estimate. The exact multiples would depend on |S|, and delta, but could be pre-computed and stored in tables.
It turns that a slightly fancier version of the idea (7) allows a satisfactory (i.e. 1 percent) approximation to the bounds to be obtained not only for small p, but also for moderate p [but not for large p]. This version of the idea uses the continuity-corrected Gaussian approximation to the Binomial Distribution, but with an “adjusted” number of standard deviations, that is different for the LB and for the UB, and for each value of |S| and of delta. However, these adjusted numbers can be precomputed and stored in a table. This is what the library does.
We have now carved away enough cases so that we only need to compute exact bounds for 0 <= |S| <= 120, for p values that are large (although not super-close to 1.0).
The switchover between scheme (8) and scheme (9) is determined by a threshold on p which depends on the value of |S|. The library’s specific threshold |S|/360 was empirically chosen to support the promise of 1 percent accuracy.
The exact bounds needed by scheme (9) could be computed using binary search on the tails of the Binomial Distribution, with the latter computed using a high-quality implementation of the Incomplete Beta Function obtained from a Scientific Computing library.
We have written and heavily used some auxiliary code which does exactly that: binary search using the Incomplete Beta Function. It was used to pre-compute the tables mentioned in point (7); to choose values for the various empirically determined thresholds such as 120 and |S|/360; and finally in an extensive grid search covering the input parameter space to verify that our overall approximation scheme satisfies the 1 percent accuracy guarantee.
However, it was impractical for the Theta Sketch Library as such to include a high quality implementation of the Incomplete Beta Function; writing one is a task best left to experts, and we didn’t want the Theta Sketch Library to have an external dependence on a Scientific library.
Therefore we developed a more direct approach the computing the exact LB and UB. It is based on an interesting combinatorial identity and exploits the facts that p >= |S|/360 and |S| <= 120. It employs a simple linear search, so there is no tricky bracketing and binary search code, and it takes at most 30 microseconds to compute [the slowest input is |S|=1, p = 1/360].
We mention that UB=LB=|S| when p=1, and LB=0 when |S|=0. Also, the library uses a closed-form expression for the UB when |S|=0, and for the LB when |S|=1.
So far, we have been discussing the Frequentist Confidence Interval based on the tails of the Binomial Distribution. In some cases (for example when p is very close to 1.0) the LB can be smaller than |S|. However, it is obvious that n >= |S|, so the library returns |S| instead of the official Binomial LB in that case.
In addition, just in case it is possible (we are not sure) for the library to compute a confidence interval that doesn’t cover the estimate, we shield the user from these hypothetical outputs by by always checking and if necessary growing the interval to cover the estimate.
Also see the Theta Sketch Framework paper.