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c2b9f11 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | #!/usr/bin/env python3
"""Finite exact-arithmetic regression for the target-aware selection pilot.
For n = 2,...,6 and q in {3/16, 1/4, 5/16}, enumerate each position's
accepted-zero, accepted-one, or rejected outcome. Then enumerate every
uniformly selected subset of m = floor(n/2) accepted positions. Check the
unconditional joint mass of the selected labels and retained values against
the teacher product law times the exact success probability.
This is a bounded regression of the selection mechanism, not a proof of the
asymptotic three-round theorem or its final-batch discrepancy bound.
Only Python's standard library is required; all checks use rational arithmetic.
"""
from collections import defaultdict
from fractions import Fraction
from itertools import combinations, product
from math import comb, prod
TAU = Fraction(1, 4)
REJECTED = 2
def verify_case(n: int, q: Fraction) -> tuple[int, int, Fraction]:
"""Return enumerated mark vectors, checked output cells, and shortage mass."""
m = n // 2
alpha = min(q / TAU, (1 - q) / (1 - TAU))
raw_probabilities = (1 - q, q)
teacher_probabilities = (1 - TAU, TAU)
acceptance_probabilities = tuple(
alpha * teacher_probabilities[e] / raw_probabilities[e]
for e in (0, 1)
)
assert all(0 <= probability <= 1 for probability in acceptance_probabilities)
# Compute the three mark probabilities from the original Bernoulli proposal
# and its value-dependent acceptance coin, rather than starting with a
# teacher-distributed retained value.
mark_probabilities = (
raw_probabilities[0] * acceptance_probabilities[0],
raw_probabilities[1] * acceptance_probabilities[1],
sum(
raw_probabilities[e] * (1 - acceptance_probabilities[e])
for e in (0, 1)
),
)
assert sum(mark_probabilities) == 1
assert sum(mark_probabilities[:2]) == alpha
joint = defaultdict(Fraction)
success_mass = Fraction(0)
shortage_mass = Fraction(0)
enumerated = 0
for marks in product((0, 1, REJECTED), repeat=n):
enumerated += 1
mark_mass = prod(mark_probabilities[mark] for mark in marks)
if mark_mass == 0:
continue
accepted = tuple(i for i, mark in enumerate(marks) if mark != REJECTED)
if len(accepted) < m:
shortage_mass += mark_mass
committed_in_round_two = 0
else:
success_mass += mark_mass
committed_in_round_two = m
choice_mass = mark_mass / comb(len(accepted), m)
for selected in combinations(accepted, m):
retained_values = tuple(marks[i] for i in selected)
joint[(selected, retained_values)] += choice_mass
# On a shortage, commit nothing. Round three fills every remaining
# B-coordinate, so shortage is not a failure to finish sampling.
remaining_for_round_three = n - committed_in_round_two
assert committed_in_round_two + remaining_for_round_three == n
assert 2 * n + remaining_for_round_three <= 3 * n
exact_binomial_shortage = sum(
Fraction(comb(n, k)) * alpha**k * (1 - alpha) ** (n - k)
for k in range(m)
)
assert shortage_mass == exact_binomial_shortage
assert success_mass + shortage_mass == 1
assert sum(joint.values()) == success_mass
# Compare every possible selected-label/value pair, including any cells
# that an erroneous selection implementation might have omitted.
checked_cells = 0
for selected in combinations(range(n), m):
selected_mass = Fraction(0)
for retained_values in product((0, 1), repeat=m):
teacher_mass = prod(teacher_probabilities[e] for e in retained_values)
expected = success_mass * teacher_mass / comb(n, m)
actual = joint[(selected, retained_values)]
assert actual == expected, (n, q, selected, retained_values, actual, expected)
selected_mass += actual
checked_cells += 1
assert selected_mass == success_mass / comb(n, m)
return enumerated, checked_cells, shortage_mass
def main() -> None:
cases = 0
enumerated = 0
checked_cells = 0
for n in range(2, 7):
for q in (Fraction(3, 16), Fraction(1, 4), Fraction(5, 16)):
states, cells, shortage = verify_case(n, q)
cases += 1
enumerated += states
checked_cells += cells
print(f"PASS n={n}, q={q}, exact pilot-shortage probability={shortage}")
print(
f"PASS: {cases} cases; {enumerated} mark vectors; "
f"{checked_cells} selected-label/value cells."
)
print("Finite rational-arithmetic regression only; not a proof of the theorem.")
if __name__ == "__main__":
main()
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