markov_bound
std.stats.markov_bound · Level L1Markov's inequality on a finite law, for X ≥ 0 and a > 0: the actual tail P(X ≥ a) and the bound E[X]/a. The bound is not the tail, and it does not apply to a variable that takes negative values.
P(X ≥ a) ≤ E[X]/a
Signature
markov_bound(p: f64[n], x: f64[n], a: 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
the mass at or beyond a, and the mean over ain exact rational arithmetic, on all 40 test cases. - All 80 float64 results inside the running error bound; the closest uses 50% 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 53% of it.
Accuracy in detail
- correctly rounded (the float64 nearest the exact value)
- 95%
- bit-equal to the NumPy formula in float64
- 96%
- largest error, in units in the last place
- 1.09
Identity
sha256:297bfec0318e9b1132892b776e8f99c788f3793c97827f1f5b135ae35e5c00dfThe semantic hash of the graph. It changes when the program changes, and never when only its documentation does.
Control handle
- Symbol
- Ω:std.stats.markov_bound · Ω:markov_bound
- Pins
sha256:025c31c29e2570ac7facc1c27e35e4a7cc67c0920cfd35718ee1bd421da587c5this graph and the 1 it reaches through calls- Evidence
sha256:cf52641e45e54c498cec40057ecdba37513ab28fbee36b49d6e3e88d5fe4f205the 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.