Add error bars to CE and CW likelihood examples

These are also Bernoulli yes/no trials that produce binomial
distributions.

Update results images

examples/merrill_1984_table_1_fig_1.py

@@ -27,6 +27,7 @@
 import time
 from collections import Counter
 import numpy as np
+from scipy.stats import binomtest
 import matplotlib.pyplot as plt
 from tabulate import tabulate
 from elsim.methods import (fptp, runoff, irv, approval, borda, coombs,
@@ -113,7 +114,14 @@
                'Black'):
     x, y = zip(*sorted(count[method].items()))
     CE = np.array(y)/y_cw
-    plt.plot(x, CE*100, '-', label=method)
+
+    # Add 95% confidence interval error bars (Clopper-Pearson exact method)
+    ci = np.empty((2, len(y)))
+    for i in range(len(y)):
+        ci[:, i] = binomtest(y[i], y_cw[i]).proportion_ci()
+    yerr = ci - CE
+    yerr[0] = -yerr[0]
+    plt.errorbar(x, CE*100, yerr*100, fmt='-', label=method)
     table.append([method, *np.array(y)/y_cw*100])
 
 # Likelihood that social utility maximizer is Condorcet Winner

examples/wikipedia_condorcet_paradox_likelihood.py

@@ -22,6 +22,7 @@
 """
 from collections import Counter
 import numpy as np
+from scipy.stats import binomtest
 import matplotlib.pyplot as plt
 from tabulate import tabulate
 from joblib import Parallel, delayed
@@ -68,7 +69,14 @@ def func():
 
 x, y = zip(*sorted(is_CP.items()))
 CP = np.asarray(y) / iterations  # Likelihood of paradox
-plt.plot(x, CP*100, '-', label='Simulation')
+
+# Add 95% confidence interval error bars (Clopper-Pearson exact method)
+ci = np.empty((2, len(y)))
+for i in range(len(y)):
+    ci[:, i] = binomtest(y[i], iterations).proportion_ci()
+yerr = ci - CP
+yerr[0] = -yerr[0]
+plt.errorbar(x, CP*100, yerr*100, fmt='-', label='Simulation')
 
 plt.legend()
 plt.grid(True, color='0.7', linestyle='-', which='major', axis='both')