[#fpr_estimation]
= Appendix A: FPR Estimation

:idprefix: fpr_estimation_

For a classical Bloom filter, the theoretical false positive rate, under some simplifying assumptions,
is given by

[.formula-center]
{small}stem:[\text{FPR}(n,m,k)=\left(1 - \left(1 - \displaystyle\frac{1}{m}\right)^{kn}\right)^k \approx \left(1 - e^{-kn/m}\right)^k]{small-end} for large {small}stem:[m]{small-end},

where {small}stem:[n]{small-end} is the number of elements inserted in the filter, {small}stem:[m]{small-end} its capacity in bits and {small}stem:[k]{small-end} the
number of bits set per insertion (see a https://en.wikipedia.org/wiki/Bloom_filter#Probability_of_false_positives[derivation^]
of this formula). For a fixed inverse load factor {small}stem:[c=m/n]{small-end}, 
the expression reaches at

[.formula-center]
{small}stem:[k_{\text{opt}}=c\cdot\ln2]{small-end}

its minimum value
{small}stem:[1/2^{k_{\text{opt}}} \approx 0.6185^{c}]{small-end}.
The optimum {small}stem:[k]{small-end}, which must be an integer,
is either
{small}stem:[\lfloor k_{\text{opt}}\rfloor]{small-end} or
{small}stem:[\lceil k_{\text{opt}}\rceil]{small-end}.

In the case of filter of the form `boost::bloom::filter<T, K, block<Block, K'>>`, we can extend
the approach from https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=f376ff09a64b388bfcde2f5353e9ddb44033aac8[Putze et al.^]
to derive the (approximate but very precise) formula:

[.formula-center]
{small}stem:[\text{FPR}_{\text{block}}(n,m,b,k,k')=\left(\displaystyle\sum_{i=0}^{\infty} \text{Pois}(i,nbk/m) \cdot \text{FPR}(i,b,k')\right)^{k},]{small-end}

where

[.formula-center]
{small}stem:[\text{Pois}(i,\lambda)=\displaystyle\frac{\lambda^i e^{-\lambda}}{i!}]{small-end}

is the probability mass function of a https://en.wikipedia.org/wiki/Poisson_distribution[Poisson distribution^]
with mean {small}stem:[\lambda]{small-end}, and {small}stem:[b]{small-end} is the size of `Block` in bits. If we're using `multiblock<Block,K'>`, we have

[.formula-center]
{small}stem:[\text{FPR}_\text{multiblock}(n,m,b,k,k')=\left(\displaystyle\sum_{i=0}^{\infty} \text{Pois}(i,nbkk'/m) \cdot \text{FPR}(i,b,1)^{k'}\right)^{k}.]{small-end}

As we have commented xref:primer_multiblock_filters[before], in general 

[.formula-center]
{small}stem:[\text{FPR}_\text{block}(n,m,b,k,k') \geq \text{FPR}_\text{multiblock}(n,m,b,k,k') \geq \text{FPR}(n,m,kk'),]{small-end}

that is, block and multiblock filters have worse FPR than the classical filter for the same number of bits
set per insertion, but they will be faster. We have the particular case

[.formula-center]
{small}stem:[\text{FPR}_{\text{block}}(n,m,b,k,1)=\text{FPR}_{\text{multiblock}}(n,m,b,k,1)=\text{FPR}(n,m,k),]{small-end}

which follows simply from the observation that using `{block|multiblock}<Block, 1>` behaves exactly as
a classical Bloom filter.

We don't know of any closed, simple formula for the FPR of block and multiblock filters when
`Stride` is not its "natural" size `xref:subfilters_used_value_size[_used-value-size_]<Subfilter>`,
that is, when subfilter subarrays overlap.
We can use the following approximations ({small}stem:[s]{small-end} = `Stride` in bits):

[.formula-center]
{small}stem:[\text{FPR}_{\text{block}}(n,m,b,s,k,k')=\left(\displaystyle\sum_{i=0}^{\infty} \text{Pois}\left(i,\frac{n(2b-s)k}{m}\right) \cdot \text{FPR}(i,2b-s,k')\right)^{k},]{small-end} +
{small}stem:[\text{FPR}_\text{multiblock}(n,m,b,s,k,k')=\left(\displaystyle\sum_{i=0}^{\infty} \text{Pois}\left(i,\frac{n(2bk'-s)k}{m}\right) \cdot \text{FPR}\left(i,\frac{2bk'-s}{k'},1\right)^{k'}\right)^{k},]{small-end}

where the replacement of {small}stem:[b]{small-end} with {small}stem:[2b-s]{small-end} 
(or {small}stem:[bk']{small-end} with {small}stem:[2bk'-s]{small-end} for multiblock filters) accounts
for the fact that the window of hashing positions affecting a particular bit spreads due to
overlapping. Note that the formulas reduce to the non-overlapping case when {small}stem:[s]{small-end} takes its
default value (stem:[b] for block, stem:[bk'] for multiblock). These approximations are acceptable for
low values of {small}stem:[k']{small-end} but tend to underestimate the actual FPR as {small}stem:[k']{small-end} grows.
In general, the use of overlapping improves (decreases) FPR by a factor ranging from
0.6 to 0.9 for typical filter configurations.

{small}stem:[\text{FPR}_{\text{block}}(n,m,b,s,k,k')]{small-end} and {small}stem:[\text{FPR}_\text{multiblock}(n,m,b,s,k,k')]{small-end}
are the formulas used by the implementation of
`xref:filter_fpr_estimation[boost::filter::fpr_for]`.
