martingale_step

std.stats.martingale_step · Level L1

One time step of a martingale check on a finite probability space: on each block of the partition F_t, E[X_{t+1} | F_t] − X_t (0 on a zero-mass block, which is determined only almost surely), and whether X_t is constant on the blocks of F_t (adapted). The body martingale_check scans.

E[X_{t+1} | F_t] − X_t

Signature

martingale_step(adapted: f64[], gap: f64[n], a: f64[n], xt: f64[n], xn: f64[n], p: f64[n]) → f64[], f64[n]

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 block masses and block means of X_{t+1}, in exact arithmetic in exact rational arithmetic, on all 40 test cases.
All 193 float64 results inside the running error bound; the closest uses 30% 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 41% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 87%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 11

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

—

Called by

martingale_check

sha256:746c42be288c72103efea63d001dd1e044a80e8566e75a4c648bdb425b526dcb

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.martingale_step · Ω:martingale_step
Pins: sha256:49f191b88862c63dac0ba5c2f39cbc38ed3d36ba2d6560053583b7b557a3a41athis graph alone
Evidence: sha256:e0c8a53115d4edd9e8245724db59f58f23f237e082d46fe3262f6f4ba1f4a64bthe 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.