#!/usr/bin/env python3 """Reproduce displayed calculations for Volume II.""" from __future__ import annotations from fractions import Fraction from math import atan, log, pi, sin, sqrt def chapter_01() -> None: joint = { (1, 1): Fraction(3, 8), (1, 0): Fraction(1, 8), (-1, 1): Fraction(1, 8), (-1, 0): Fraction(3, 8), } assert sum(joint.values()) == 1 selected_probability = sum(probability for (z, s), probability in joint.items() if s == 1) selected_mean = sum(z * probability for (z, s), probability in joint.items() if s == 1) / selected_probability unselected_mean = sum(z * probability for (z, s), probability in joint.items() if s == 0) / (1 - selected_probability) selected_variance = 1 - selected_mean**2 product_mean = sum(z * s * probability for (z, s), probability in joint.items()) assert (selected_probability, selected_mean, unselected_mean, selected_variance, product_mean) == ( Fraction(1, 2), Fraction(1, 2), Fraction(-1, 2), Fraction(3, 4), Fraction(1, 4) ) assert selected_probability * selected_mean + (1 - selected_probability) * unselected_mean == 0 assert selected_variance + Fraction(1, 4) == 1 print("II-01 selection probability, conditional means, variance, E[ZS]:", selected_probability, selected_mean, unselected_mean, selected_variance, product_mean) def chapter_02() -> None: # For M, the median of three Uniform(-1,1) draws, f(m)=3(1-m^2)/4. expected_absolute_median = Fraction(3, 8) expected_squared_median = Fraction(1, 5) selected_mean = -expected_absolute_median selected_variance = expected_squared_median - expected_absolute_median**2 assert selected_variance == Fraction(19, 320) print("II-02 E|median|, selected mean and variance:", expected_absolute_median, selected_mean, selected_variance) def chapter_03() -> None: def outcome(a: int, b: int) -> int: return 1 + 2 * a + 3 * b + 4 * a * b potential = {(a, b): outcome(a, b) for a in (0, 1) for b in (0, 1)} main_effect = sum(Fraction(outcome(1, b) - outcome(0, b), 2) for b in (0, 1)) interaction = outcome(1, 1) - outcome(1, 0) - outcome(0, 1) + outcome(0, 0) ht_expectation = sum( Fraction(1, 4) * (2 * a * outcome(a, b) - 2 * (1 - a) * outcome(a, b)) for a in (0, 1) for b in (0, 1) ) assert potential == {(0, 0): 1, (0, 1): 4, (1, 0): 3, (1, 1): 10} assert main_effect == interaction == ht_expectation == 4 print("II-03 outcomes, main effect, interaction, HT expectation:", potential, main_effect, interaction, ht_expectation) def chapter_04() -> None: lower = (Fraction(3, 5), Fraction(0), Fraction(2, 5)) upper = (Fraction(0), Fraction(3, 5), Fraction(2, 5)) def target(probabilities: tuple[Fraction, Fraction, Fraction]) -> Fraction: return probabilities[1] + 2 * probabilities[2] assert target(lower) == Fraction(4, 5) assert target(upper) == Fraction(7, 5) projected_lower = 2 * (1 - Fraction(13, 20)) projected_upper = 2 - Fraction(11, 20) assert projected_lower == Fraction(7, 10) assert projected_upper == Fraction(29, 20) print("II-04 sharp and projected bounds:", (target(lower), target(upper)), (projected_lower, projected_upper)) def chapter_05() -> None: for m in (10, 100, 1000): n, n_i, n_s = m**4, m, m**3 bias_ratio = Fraction(n, n_s**2) centered_ratio = Fraction(n, n_i * n_s) assert bias_ratio == Fraction(1, m**2) assert centered_ratio == 1 print("II-05 ratios at m=10,100,1000:", [(Fraction(1, m**2), Fraction(1)) for m in (10, 100, 1000)]) def chapter_06() -> None: sample_mean, nuisance, theta_zero = Fraction(21, 20), Fraction(101, 50), Fraction(3) theta_hat = sample_mean + nuisance sampling_component = sample_mean - 1 nuisance_component = nuisance - 2 assert theta_hat == Fraction(307, 100) assert theta_hat - theta_zero == sampling_component + nuisance_component == Fraction(7, 100) assert theta_hat + Fraction(1, 100) == Fraction(77, 25) assert theta_hat - Fraction(1, 100) == Fraction(153, 50) print("II-06 estimate and error components:", theta_hat, sampling_component, nuisance_component) def chapter_07() -> None: n_i, n_s = 10, 100 independent_variance = Fraction(1, n_i * n_s) shared_variance = Fraction(1, n_s) assert independent_variance == Fraction(1, 1000) assert shared_variance == Fraction(1, 100) assert shared_variance / independent_variance == n_i print("II-07 independent/shared variances:", independent_variance, shared_variance) def chapter_08() -> None: score_distribution = {Fraction(0): Fraction(3, 4), Fraction(2): Fraction(1, 4)} mean = sum(value * probability for value, probability in score_distribution.items()) second_moment = sum(value**2 * probability for value, probability in score_distribution.items()) variance = second_moment - mean**2 assert mean == Fraction(1, 2) and variance == Fraction(3, 4) one_draw_mean = Fraction(1) three_draw_mean = 2 * (Fraction(1, 3) * Fraction(3, 8) + Fraction(1) * Fraction(1, 8)) assert three_draw_mean == Fraction(1, 2) asymptotic_scaled_mean = sqrt(2 / pi) assert 0.797 < asymptotic_scaled_mean < 0.799 print("II-08 two-draw mean/variance, one/three-draw means, asymptotic constant:", mean, variance, one_draw_mean, three_draw_mean, asymptotic_scaled_mean) def chapter_09() -> None: derivative = (Fraction(1), Fraction(2)) error = (Fraction(1), Fraction(-1)) def outer(left: tuple[Fraction, Fraction], right: tuple[Fraction, Fraction]) -> tuple[tuple[Fraction, Fraction], tuple[Fraction, Fraction]]: return tuple(tuple(a * b for b in right) for a in left) # type: ignore[return-value] true_gram = outer(derivative, derivative) error_gram = outer(error, error) naive = tuple(tuple(true_gram[i][j] + error_gram[i][j] for j in range(2)) for i in range(2)) assert true_gram == ((1, 2), (2, 4)) assert naive == ((2, 1), (1, 5)) eta = Fraction(1, 10) squared_distance = eta**4 / (1 + eta**4) assert squared_distance == Fraction(1, 10001) print("II-09 true/naive Gram and squared range distance at eta=.1:", true_gram, naive, squared_distance) def chapter_10() -> None: epsilon = 0.1 high = 2 + sqrt(1 + epsilon**2) low = 2 - sqrt(1 + epsilon**2) angle = 0.5 * atan(epsilon) sine_angle = sin(angle) assert abs(high - 3.004987562112089) < 1.0e-14 assert abs(low - 0.995012437887911) < 1.0e-14 assert sine_angle <= epsilon / 2 + 1.0e-15 print("II-10 eigenvalues, angle, sine and bound:", high, low, angle, sine_angle, epsilon / 2) def chapter_11() -> None: coefficient, standard_error, scale = Fraction(2, 5), Fraction(1, 10), Fraction(100) scaled_coefficient = coefficient / scale scaled_standard_error = standard_error / scale assert scaled_coefficient == Fraction(1, 250) assert scaled_standard_error == Fraction(1, 1000) assert coefficient / standard_error == scaled_coefficient / scaled_standard_error == 4 print("II-11 original/scaled coefficient, standard error, t-statistic:", coefficient, standard_error, scaled_coefficient, scaled_standard_error, coefficient / standard_error) def chapter_12() -> None: directions = ((1, -2), (-2, 1), (1, 1), (0, 3)) derivatives = tuple(min(direction) for direction in directions) assert derivatives == (-2, -2, 1, 0) print("II-12 directional derivatives:", tuple(zip(directions, derivatives))) def chapter_13() -> None: candidate_count, sample_size, delta = 10, 1000, 0.05 epsilon = sqrt(log(2 * candidate_count / delta) / (2 * sample_size)) excess = 2 * epsilon assert abs(epsilon - 0.054733283051119734) < 1.0e-14 assert abs(excess - 0.10946656610223947) < 1.0e-14 print("II-13 uniform and excess-risk bounds:", epsilon, excess) def main() -> None: for check in (chapter_01, chapter_02, chapter_03, chapter_04, chapter_05, chapter_06, chapter_07, chapter_08, chapter_09, chapter_10, chapter_11, chapter_12, chapter_13): check() print("PASS: 13 Volume II example groups verified") if __name__ == "__main__": main()