我们常常需要检验某个假设是否成立。有时，假设是诸如“这枚硬币是均匀的”“数据科学家喜欢 Python 胜过 R”或“如果人们点开某个突然弹出的小广告，广告的关闭按钮又小又难找，那么大家更倾向于离开这个页面，压根不会阅读”等可以被翻译成统计数据的断言。在各种各样的假设之下，这些统计数据可以理解为从某种已知分布中抽取的随机变量观测值，这可以让我们对这些假设是否成立做出论断。
典型的步骤是这样的，首先我们有一个零假设 H 0 ，它代表一个默认的立场，而替代假设
H 1 代表我们希望与零假设对比的立场。我们通过统计来决定我们是否可以拒绝 H 0 ，即判断它是否错误。通过举例能更直观地说明这个过程。
from probability import normal_cdf, inverse_normal_cdf
import math, random
def normal_approximation_to_binomial(n, p):
"""finds mu and sigma corresponding to a Binomial(n, p)"""
mu = p * n
sigma = math.sqrt(p * (1 - p) * n)
return mu, sigma
#####
#
# probabilities a normal lies in an interval
#
######
# the normal cdf _is_ the probability the variable is below a threshold
normal_probability_below = normal_cdf
# it's above the threshold if it's not below the threshold
def normal_probability_above(lo, mu=0, sigma=1):
return 1 - normal_cdf(lo, mu, sigma)
# it's between if it's less than hi, but not less than lo
def normal_probability_between(lo, hi, mu=0, sigma=1):
return normal_cdf(hi, mu, sigma) - normal_cdf(lo, mu, sigma)
# it's outside if it's not between
def normal_probability_outside(lo, hi, mu=0, sigma=1):
return 1 - normal_probability_between(lo, hi, mu, sigma)
######
#
# normal bounds
#
######
def normal_upper_bound(probability, mu=0, sigma=1):
"""returns the z for which P(Z <= z) = probability"""
return inverse_normal_cdf(probability, mu, sigma)
def normal_lower_bound(probability, mu=0, sigma=1):
"""returns the z for which P(Z >= z) = probability"""
return inverse_normal_cdf(1 - probability, mu, sigma)
def normal_two_sided_bounds(probability, mu=0, sigma=1):
"""returns the symmetric (about the mean) bounds
that contain the specified probability"""
tail_probability = (1 - probability) / 2
# upper bound should have tail_probability above it
upper_bound = normal_lower_bound(tail_probability, mu, sigma)
# lower bound should have tail_probability below it
lower_bound = normal_upper_bound(tail_probability, mu, sigma)
return lower_bound, upper_bound
def two_sided_p_value(x, mu=0, sigma=1):
if x >= mu:
# if x is greater than the mean, the tail is above x
return 2 * normal_probability_above(x, mu, sigma)
else:
# if x is less than the mean, the tail is below x
return 2 * normal_probability_below(x, mu, sigma)
def count_extreme_values():
extreme_value_count = 0
for _ in range(100000):
num_heads = sum(1 if random.random() < 0.5 else 0 # count # of heads
for _ in range(1000)) # in 1000 flips
if num_heads >= 530 or num_heads <= 470: # and count how often
extreme_value_count += 1 # the # is 'extreme'
return extreme_value_count / 100000
upper_p_value = normal_probability_above
lower_p_value = normal_probability_below
##
#
# P-hacking
#
##
def run_experiment():
"""flip a fair coin 1000 times, True = heads, False = tails"""
return [random.random() < 0.5 for _ in range(1000)]
def reject_fairness(experiment):
"""using the 5% significance levels"""
num_heads = len([flip for flip in experiment if flip])
return num_heads < 469 or num_heads > 531
##
#
# running an A/B test
#
##
def estimated_parameters(N, n):
p = n / N
sigma = math.sqrt(p * (1 - p) / N)
return p, sigma
def a_b_test_statistic(N_A, n_A, N_B, n_B):
p_A, sigma_A = estimated_parameters(N_A, n_A)
p_B, sigma_B = estimated_parameters(N_B, n_B)
return (p_B - p_A) / math.sqrt(sigma_A ** 2 + sigma_B ** 2)
##
#
# Bayesian Inference
#
##
def B(alpha, beta):
"""a normalizing constant so that the total probability is 1"""
return math.gamma(alpha) * math.gamma(beta) / math.gamma(alpha + beta)
def beta_pdf(x, alpha, beta):
if x < 0 or x > 1: # no weight outside of [0, 1]
return 0
return x ** (alpha - 1) * (1 - x) ** (beta - 1) / B(alpha, beta)
if __name__ == "__main__":
mu_0, sigma_0 = normal_approximation_to_binomial(1000, 0.5)
print("mu_0", mu_0)
print("sigma_0", sigma_0)
print("normal_two_sided_bounds(0.95, mu_0, sigma_0)", normal_two_sided_bounds(0.95, mu_0, sigma_0))
print()
print("power of a test")
print("95% bounds based on assumption p is 0.5")
lo, hi = normal_two_sided_bounds(0.95, mu_0, sigma_0)
print("lo", lo)
print("hi", hi)
print("actual mu and sigma based on p = 0.55")
mu_1, sigma_1 = normal_approximation_to_binomial(1000, 0.55)
print("mu_1", mu_1)
print("sigma_1", sigma_1)
# a type 2 error means we fail to reject the null hypothesis
# which will happen when X is still in our original interval
type_2_probability = normal_probability_between(lo, hi, mu_1, sigma_1)
power = 1 - type_2_probability # 0.887
print("type 2 probability", type_2_probability)
print("power", power)
print
print("one-sided test")
hi = normal_upper_bound(0.95, mu_0, sigma_0)
print("hi", hi) # is 526 (< 531, since we need more probability in the upper tail)
type_2_probability = normal_probability_below(hi, mu_1, sigma_1)
power = 1 - type_2_probability # = 0.936
print("type 2 probability", type_2_probability)
print("power", power)
print()
print("two_sided_p_value(529.5, mu_0, sigma_0)", two_sided_p_value(529.5, mu_0, sigma_0))
print("two_sided_p_value(531.5, mu_0, sigma_0)", two_sided_p_value(531.5, mu_0, sigma_0))
print("upper_p_value(525, mu_0, sigma_0)", upper_p_value(525, mu_0, sigma_0))
print("upper_p_value(527, mu_0, sigma_0)", upper_p_value(527, mu_0, sigma_0))
print()
print("P-hacking")
random.seed(0)
experiments = [run_experiment() for _ in range(1000)]
num_rejections = len([experiment
for experiment in experiments
if reject_fairness(experiment)])
print(num_rejections, "rejections out of 1000")
print()
print("A/B testing")
z = a_b_test_statistic(1000, 200, 1000, 180)
print("a_b_test_statistic(1000, 200, 1000, 180)", z)
print("p-value", two_sided_p_value(z))
z = a_b_test_statistic(1000, 200, 1000, 150)
print("a_b_test_statistic(1000, 200, 1000, 150)", z)
print("p-value", two_sided_p_value(z))
执行结果
mu_0 500.0
sigma_0 15.811388300841896
normal_two_sided_bounds(0.95, mu_0, sigma_0) (469.01026640487555, 530.9897335951244)
power of a test
95% bounds based on assumption p is 0.5
lo 469.01026640487555
hi 530.9897335951244
actual mu and sigma based on p = 0.55
mu_1 550.0
sigma_1 15.732132722552274
type 2 probability 0.11345199870463285
power 0.8865480012953671
one-sided test
hi 526.0073585242053
type 2 probability 0.06362051966928273
power 0.9363794803307173
two_sided_p_value(529.5, mu_0, sigma_0) 0.06207721579598857
two_sided_p_value(531.5, mu_0, sigma_0) 0.046345287837786575
upper_p_value(525, mu_0, sigma_0) 0.056923149003329065
upper_p_value(527, mu_0, sigma_0) 0.04385251499101195
P-hacking
46 rejections out of 1000
A/B testing
a_b_test_statistic(1000, 200, 1000, 180) -1.1403464899034472
p-value 0.254141976542236
a_b_test_statistic(1000, 200, 1000, 150) -2.948839123097944
p-value 0.003189699706216853
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