Manuscript En10-A@1 · En10-A.tex · section · 0a1d7743

An explicit rational instance

The finite Enter theorem and estimation contract › An explicit rational instance · lines 972–1036

En10-A@1#eq:benchmark, En10-A@1#eq:rational-law, En10-A@1#eq:rational-estimator, En10-A@1#eq:rational-shots

Freeze

ϕ = \frac{π}{6}, γ = \frac{1}{4} .

Then

t = \frac{1}{4}, h = \frac{3}{4}, d = \frac{1}{2}, a = \frac{7}{16}, b = \frac{3}{8} .

The complete probabilities become

r p_{r} (θ) 00 \frac{7 + 32 θ}{64} 01 \frac{9}{64} 10 \frac{21 - 8 θ}{64} 11 \frac{27 - 24 θ}{64} .

They are nonnegative and sum to one for every $θ \in [0, 1]$ .

The inference map is

θ = Π_{[0, 1]} (\frac{16 K / n - 7}{6}), R_{n} (α) = min {1, \frac{8}{3} \frac{lo g ( 2/ α )}{2 n}} .

Its estimate values are rational at every finite $n$ . A sufficient shot count is

n \geq ⌈ \frac{32}{9 ε ^{2}} lo g \frac{2}{α} ⌉ .

The information ledger is

F_{A} (θ) = \frac{36}{( 7 + 6 θ ) ( 9 - 6 θ )},

and

F_{C} (θ) = \frac{16}{7 + 32 θ} + \frac{1}{21 - 8 θ} + \frac{9}{27 - 24 θ} .

These are analytic formulas, not reported numerical execution.

Displayed formulas in this passage

En10-A@1#eq:benchmark · line 975
$ϕ = \frac{π}{6}, γ = \frac{1}{4} .$
En10-A@1#eq:rational-law · line 986
$r p_{r} (θ) 00 \frac{7 + 32 θ}{64} 01 \frac{9}{64} 10 \frac{21 - 8 θ}{64} 11 \frac{27 - 24 θ}{64} .$
En10-A@1#eq:rational-estimator · line 1001
$θ = Π_{[0, 1]} (\frac{16 K / n - 7}{6}), R_{n} (α) = min {1, \frac{8}{3} \frac{lo g ( 2/ α )}{2 n}} .$
En10-A@1#eq:rational-shots · line 1014
$n \geq ⌈ \frac{32}{9 ε ^{2}} lo g \frac{2}{α} ⌉ .$
unlabeled · line 981
$t = \frac{1}{4}, h = \frac{3}{4}, d = \frac{1}{2}, a = \frac{7}{16}, b = \frac{3}{8} .$
unlabeled · line 1022
$F_{A} (θ) = \frac{36}{( 7 + 6 θ ) ( 9 - 6 θ )},$
unlabeled · line 1028
$F_{C} (θ) = \frac{16}{7 + 32 θ} + \frac{1}{21 - 8 θ} + \frac{9}{27 - 24 θ} .$