geomean(::AbstractInterval) is not correct

[hyrodium]

julia> using IntervalSets, StatsBase

julia> a, b = 5, 10
(5, 10)

julia> geomean(a..b)  # Should be 7.3575...
7.0710678118654755

julia> geomean(range(a,b,length=10000))
7.357559595510975

julia> exp((b*log(b)-a*log(a)-b+a)/(b-a))
7.357588823428852

geomean(::AbstractInterval) was introduced in #171, but the definition sqrt(leftendpoint(d) * rightendpoint(d)) seems not correct.

$$ \mathop{\text{GM}}[a..b] = \exp\left(\frac{1}{b-a} \int_a^b \log(x) dx \right) = \exp\left(\frac{b\log(b)-a\log(a)-b+a}{b-a}\right) $$

See Geometric mean (wikipedia) for more information.

This correct definition has the following consistency with mean(range(...))

julia> using IntervalSets, StatsBase, Plots

julia> plot([geomean(range(a,b,length=n)) for n in 2:1000], label="geomean of evenly spaced points")

julia> hline!([exp((b*log(b)-a*log(a)-b+a)/(b-a))], label="geomean of an interval set")

@aplavin
Where did you find the original definition in https://github.com/JuliaMath/IntervalSets.jl/pull/171/files#diff-29865909ee404f36758d3fb0d2d549a636b7c5cd4967d9855c245f59b3a0bb67R6?

[aplavin]

My motivation was that:

• mean(interval) is its midpoint on the linear scale, geomean() is its midpoint on the logscale
• mean(interval) = mean(endpoints(interval)), same makes sense for geomean()

I wonder what usecases you have, where the current geomean definition doesn't give the expected result. Don't really understand the connection between mean/geomean of an interval and mean/geomean of a function (that you linked). I don't think there is unambiguous meaning of "(geo)mean of an interval" in maths in general, so here we just choose the most pragmatic one.
Just looked up my usage of geomean(interval) over the past years (I defined it myself quite a few times, and have to continue doing so even now due to #168). All uses are for plotting: find the midpoint of the interval when plotted on the logscale.

[hyrodium]

I see. Thank you for the explanation.

I don't think there is unambiguous meaning of "(geo)mean of an interval" in maths in general, so here we just choose the most pragmatic one.

I think mean(a..b) should be defined by $E[X], (X \sim \text{Uniform}(a,b))$. This definition can be regarded as a centroid of an interval, and it can be generalized to a non-connected subset of $\mathbb{R}$.

Don't really understand the connection between mean/geomean of an interval and mean/geomean of a function (that you linked).

How about this comment (https://stats.stackexchange.com/a/386476)? Similarly to mean(a..b), I think geomean(a..b) should be defined by $\exp(E[\log(X)]), (X \sim \text{Uniform}(a,b))$. The geomean function is owned by StatsBase.jl, so I believe we need statistics theory to add methods to this function.

I wonder what usecases you have, where the current geomean definition doesn't give the expected result.

I actually don't have use cases for geomean for now. I just thought the inconsistent definition discussed above should not be provided from this package.

All uses are for plotting: find the midpoint of the interval when plotted on the logscale.

Then, how about adding a new function mean_logscale(::AbstractInterval) instead of adding a geomean(::AbstractInterval) method?

[aplavin]

How about this comment (https://stats.stackexchange.com/a/386476)?

That again relates to mean/geomean of random variables, not of intervals. Basically, that's how [geo]mean(Uniform(interval)) or [geo]mean(Uniform(union_of_intervals)) should work.
Actually, I proposed to add Uniform(interval) constructors, but somehow Distributions.jl devs don't want them (JuliaStats/Distributions.jl#1809).

Similarly to mean(a..b), I think geomean(a..b) should be defined by <...>.

That's definitely a non-starter. I'm sure many potential users (a large fraction, not a large absolute number :) ) would be surprised if geomean(1..100) works (doesn't error) and returns something other than 10.

With geomean and collections, it has a nice property that exp(mean(X)) == geomean(exp.(X)). The current definition of geomean(interval) also follows this property (conceptually, because literal exp.(X) isn't supported for intervals). Just remember that "interval" is not a "probability distribution".

I just thought the inconsistent definition discussed above should not be provided from this package.

I wouldn't call it inconsistent, but agree that it's probably best not to define it here. As this thread shows, there are at least two different meanings geomean(interval) can have, and both are consistent with other functions in different ways.
So, just reverting 6074974 should be fine, and is the obvious solution.

[timholy]

I agree that the notion proposed by @hyrodium is the more mathematically correct one. Of course, for both mean and geomean, there's an implicit sense of measure, i.e., that we're computing, e.g.,

$$ \frac{\int x dx}{\int dx} $$

rather than

$$ \frac{\int x w(x) dx}{\int w(x) dx} $$

Strictly speaking that's not encoded by just knowing the endpoints of the interval, and it's not invariant to changes of coordinates, i.e., mean(f.(a..b)) != f(mean(a..b)).

[aplavin]

From the discussion in this thread, it seems clear that there are at least two different conflicting meanings one could assign to geomean(interval). And there is no definite meaning assigned to "geomean of an interval" in maths in general.
These are totally reasonable arguments to revert 6074974, and recommend geomean(endpoints(interval)) or geomean(Uniform(interval)) instead, depending on which meaning is relevant in each case. (the latter is now geomean(Uniform(endpoints(interval)...)), but I'm hopeful that at some point Distributions will support intervals).

mean(interval) technically shares the same concerns, but both definitions happen to give the same answer there – so there are no issues with that method staying.