total_variance

std.stats.total_variance · Level L3

The law of total variance on a finite probability space, with the conditioning given by a partition of the outcomes into cells (g[s] the cell of s): the expected conditional variance E[Var(X | G)], the variance of the conditional mean Var(E[X | G]), and Var(X), which is their sum. A zero-mass cell contributes nothing; its conditional values are determined only almost surely, so it is never divided by. The two parts are not interchangeable, and the cells need not be independent.

Var X = E[Var(X | G)] + Var(E[X | G])

Signature

total_variance(p: f64[n], x: f64[n], g: i64[n], cells: f64[k]) → f64[], f64[], f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference cell masses, conditional means and variances, in exact arithmetic in exact rational arithmetic, on all 40 test cases.
All 120 float64 results inside the running error bound; the closest uses 22% of it.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them inside the running error bound at u = 2⁻²⁴; the closest uses 11% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 70%
bit-equal to the NumPy formula in float64: 79%
largest error, in units in the last place: 171

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Identity

Calls

dot

Called by

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sha256:b1e53e785848a25f7f7e002a313a11040062ad85e638174938b9440b850568e2

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.stats.total_variance · Ω:total_variance
Pins: sha256:5442887cb74be492db3cfae7a45f8ea54be2b441849d66719076b853a600226cthis graph and the 1 it reaches through calls
Evidence: sha256:40f15d97dc38a8e464eb3d4accc4f8d58592e0eb5f1b0a8c23f2e093bd9bf921the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.