In [1]:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import os
os.makedirs('ipynb_export', exist_ok=True)
In [2]:
RNG = np.random.default_rng(seed=0xC0FFEE)
In [3]:
def sim_var_estimate(p, n):
    x = RNG.binomial(1, p, size=n)
    p_est = x.mean()
    return p_est * (1 - p_est)
In [4]:
def many_var_estimates(p, n):
    return np.array([sim_var_estimate(p, n) for i in range(2048)])
In [5]:
p = 0.5; n = 32
mle = many_var_estimates(p, n)
plt.hist(mle, range=(0,0.3), bins=200)
plt.title("Max Likelihood Estimator")
plt.savefig('ipynb_export/mle_p_half.png')
In [6]:
unbiased = n/(n-1) * mle
plt.hist(unbiased, range=(0,0.3), bins=200)
plt.title("Sample Variance")
plt.savefig('ipynb_export/unbiased_p_half.png')
In [7]:
def report(estimates, truth):
    var = estimates.var()
    ave_error = np.mean(estimates - truth)
    median_error = np.median(estimates - truth)
    pie = (estimates > 0.25).mean()
    return dict(ave_error=ave_error,
                median_error=median_error,
                std_dev=np.sqrt(var),
                sqrt_mse=np.sqrt(var + ave_error**2),
                pie=pie)
In [8]:
def sweep_probs(step, n):
    ret = pd.DataFrame()
    for p in np.arange(0.0, 0.5+step, step):
        mle = many_var_estimates(p, n)
        unbiased = n/(n-1) * mle
        truth = p * (1-p)
        row = dict(prob=p)
        for key, val in report(mle, truth).items():
            row['mle.' + key] = val
        for key, val in report(unbiased, truth).items():
            row['unbiased.' + key] = val
        ret = ret.append(row, ignore_index=True)
    return ret
In [9]:
PERF = dict()
for n in [5, 20, 100]:
    PERF[n] = sweep_probs(1/32, n)

/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
/tmp/ipykernel_147710/1895103947.py:12: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.
  ret = ret.append(row, ignore_index=True)
In [10]:
fig, ax1 = plt.subplots()
ax1.set_xlabel("True p of Bernoulli distribution")
ax1.set_ylabel('Probability of Impossible Estimand (PIE)')
for N, df in PERF.items():
    ax1.plot('prob', 'unbiased.pie', data=df, label="n={}".format(N))
plt.legend()
plt.title('PIE of sample variance')
plt.savefig('ipynb_export/unbiased_pie.png')
In [11]:
def plot_perf(df, fname=None, pie=True, title=None):
    fig, ax1 = plt.subplots()
    ax1.set_xlabel("True p of Bernoulli distribution")
    ax1.set_ylabel('units of estimand')
    ax1.plot('prob', 'mle.ave_error', data=df, linestyle='dashed', color='lightblue')
    ax1.plot('prob', 'mle.sqrt_mse', data=df, linestyle='dashed', color='red')
    ax1.plot('prob', 'unbiased.ave_error', data=df, color='lightblue')
    ax1.plot('prob', 'unbiased.sqrt_mse', data=df, color='red')
    leg1 = plt.legend(bbox_to_anchor=(1.14,1), loc="upper left")
    xtras = (leg1,)
    if pie:
        ax2 = ax1.twinx()
        ax2.set_ylabel('Probability of Impossible Estimand (PIE)')
        ax2.plot('prob', 'unbiased.pie', data=df, color='black')
        leg2 = plt.legend(bbox_to_anchor=(1.14,0.5), loc="center left")
        xtras += (leg2,)
    if title:
        t = plt.title(title)
    if fname:
        plt.savefig(fname, bbox_extra_artists=xtras, bbox_inches='tight')
In [12]:
plot_perf(PERF[5], 'ipynb_export/perf5.png', pie=False, title="Unbiased vs biased at n=5")
In [13]:
plot_perf(PERF[20], 'ipynb_export/perf20.png', title="Unbiased vs biased at n=20")
In [14]:
plot_perf(PERF[100])
In [15]:
p = 0.5; n = 5
true_var = p * (1-p)
mle = many_var_estimates(p, n)
unbiased = n/(n-1) * mle
In [16]:
report(mle, true_var)
Out[16]:
{'ave_error': -0.04968750000000001,
 'median_error': -0.010000000000000009,
 'std_dev': 0.06219849149095177,
 'sqrt_mse': 0.07960841664045329,
 'pie': 0.0}
In [17]:
report(unbiased, true_var)
Out[17]:
{'ave_error': 0.00039062499999999335,
 'median_error': 0.04999999999999999,
 'std_dev': 0.07774811436368971,
 'sqrt_mse': 0.07774909565390455,
 'pie': 0.6220703125}