def plot_wikipedia_cdfs():
"""
https://en.wikipedia.org/wiki/Beta_distribution#/media/File:Beta_distribution_cdf.svg
"""
ax = new_axes(width=10, height=10)
Beta(0.5, 0.5).cdf().plot(x=x, color='red', ax=ax)
Beta(5, 1).cdf().plot(x=x, color='blue', ax=ax)
Beta(1, 3).cdf().plot(x=x, color='green', ax=ax)
Beta(2, 2).cdf().plot(x=x, color='purple', ax=ax)
Beta(2, 5).cdf().plot(x=x, color='orange', ax=ax)
ax.set_title('Cumulative distribution function')
ax.legend(loc='upper left')
plt.show()
def plot_wikipedia_pdfs():
"""
https://en.wikipedia.org/wiki/Beta_distribution#/media/File:Beta_distribution_pdf.svg
"""
ax = new_axes(width=10, height=10)
Beta(0.5, 0.5).pdf().plot(x=x, color='red', ax=ax)
Beta(5, 1).pdf().plot(x=x, color='blue', ax=ax)
Beta(1, 3).pdf().plot(x=x, color='green', ax=ax)
Beta(2, 2).pdf().plot(x=x, color='purple', ax=ax)
Beta(2, 5).pdf().plot(x=x, color='orange', ax=ax)
ax.set_ylim(0, 2.5)
ax.set_title('Probability density function')
ax.legend(loc='upper center')
plt.show()
def test_infer_posteriors(self):
g__1_1 = Beta(1 + 1, 1 + 1)
g__1_2 = Beta(1 + 1, 1 + 2)
g__1_3 = Beta(1 + 1, 1 + 3)
g__1_4 = Beta(1 + 1, 1 + 4)
g__2_0 = Beta(1 + 2, 1 + 0)
g__2_1 = Beta(1 + 2, 1 + 1)
g__2_2 = Beta(1 + 2, 1 + 2)
g__2_3 = Beta(1 + 2, 1 + 3)
expected = DataFrame(
data=[(1, 1, 'c', 1, g__1_1), (2, 1, 'c', 1, g__1_2),
(1, 2, 'c', 1, g__1_3), (2, 2, 'c', 1, g__1_4),
(1, 1, 'd', 1, g__2_0), (2, 1, 'd', 1, g__2_1),
(1, 2, 'd', 1, g__2_2), (2, 2, 'd', 1, g__2_3)],
columns=['a', 'b', 'prob_var', 'prob_val', 'Beta'])
actual = BetaGeometricConjugate.infer_posteriors(
data=self.geometric_data,
prob_vars=['c', 'd'],
cond_vars=['a', 'b'])
for _, row in expected.iterrows():
actual_beta = actual.loc[(actual['a'] == row['a']) &
(actual['b'] == row['b']) &
(actual['prob_var'] == row['prob_var']) &
(actual['prob_val'] == row['prob_val']),
'Beta'].iloc[0]
self.assertTrue(row['Beta'] == actual_beta)
def test_fit(self):
for alpha, beta in zip((0.5, 5, 1, 2, 2), (0.5, 1, 3, 2, 5)):
beta_orig = Beta(alpha, beta)
beta_fit = Beta.fit(beta_orig.rvs(100_000))
self.assertAlmostEqual(beta_fit.alpha, beta_orig.alpha, 1)
self.assertAlmostEqual(beta_fit.beta, beta_orig.beta, 1)
def plot_ml_app():
"""
Machine Learning: A Probabilistic Perspective. Figure 3.6
"""
_, ax = plt.subplots(nrows=1, ncols=2, figsize=(16, 9))
bb_1 = BetaBinomial(n=20, alpha=2, beta=2)
bb_1.prior().plot(x=x, color='red', ax=ax[0])
Beta(alpha=3, beta=17).plot(x=x, color='black', ax=ax[0]) # using a Beta to plot likelihood on the same scale
bb_1.posterior(m=3).plot(x=x, color='blue', ax=ax[0])
ax[0].legend()
bb_2 = BetaBinomial(n=20, alpha=5, beta=2)
bb_2.prior().plot(x=x, color='red', ax=ax[1])
Beta(alpha=11, beta=13).plot(x=x, color='black', ax=ax[1]) # using a Beta to plot likelihood on the same scale
bb_2.posterior(m=11).plot(x=x, color='blue', ax=ax[1])
ax[1].legend()
plt.show()
def from_proportions(data: DataFrame):
"""
Fit to a DataFrame of proportions. Returns a Series with one item for
each column in data.
"""
return BetaSeries(
Series({
column: Beta.fit(data[column].dropna())
for column in data.columns
}))
def setUp(self) -> None:
self.b1 = Beta(700, 300)
self.b2 = Beta(600, 400)
self.b3 = Beta(500, 500)
self.d1 = Dirichlet([500, 300, 200])
self.d2 = Dirichlet({'x': 100, 'y': 200, 'z': 300})
self.b1__mul__b2 = self.b1 * self.b2
self.b3__mul__b1__mul__b2 = self.b3 * self.b1__mul__b2
self.b1__mul__comp__b1 = self.b1 * (1 - self.b1)
self.b_series = Series({'b1': self.b1, 'b2': self.b2, 'b3': self.b3})
self.b_frame = DataFrame({
'c1': {
'r1': self.b1,
'r2': self.b2
},
'c2': {
'r1': self.b2,
'r2': self.b3
}
})
self.float_series = Series({'$100': 0.8, '$200': 0.6})
def conditional_beta_table(
self, condition: 'DataProbabilityTableMixin') -> BetaFrame:
"""
Return the conditional probability of each category given different
values of condition.
"""
counts = self.joint_count_table(condition)
row_sums = counts.sum(axis=1).to_list()
beta_dicts = []
for r in range(len(row_sums)):
beta_dicts.append(counts.iloc[r].map(
lambda count: Beta(count, row_sums[r] - count)))
return BetaFrame(DataFrame(beta_dicts))
def test_distribution_table__significance(self):
a = array([2, 4, 6, 0, 3])
n = a.sum()
b = n - a
a_others = array([(a.sum() - a[i]) / (len(a) - 1)
for i in range(len(a))])
b_others = n - a_others
expected = DataFrame(data=[
('1 - strongly disagree', a[0], Beta(1 + a[0], 1 + b[0]) > Beta(
1 + a_others[0], 1 + b_others[0])),
('2 - disagree', a[1], Beta(1 + a[1], 1 + b[1]) > Beta(
1 + a_others[1], 1 + b_others[1])),
('3 - neither agree nor disagree', a[2], Beta(1 + a[2], 1 + b[2]) >
Beta(1 + a_others[2], 1 + b_others[2])),
('4 - agree', a[3], Beta(1 + a[3], 1 + b[3]) > Beta(
1 + a_others[3], 1 + b_others[3])),
('5 - strongly agree', a[4],
Beta(1 + a[4], 1 + b[4]) > Beta(1 + a_others[4], 1 + b_others[4]))
],
columns=['Value', 'Count', 'Significance'])
actual = self.question.distribution_table(significance=True)
self.assertTrue(expected.equals(actual))
def test_infer_posteriors(self):
b__0_3 = Beta(1 + 0, 1 + 3)
b__1_2 = Beta(1 + 1, 1 + 2)
b__2_1 = Beta(1 + 2, 1 + 1)
b__3_0 = Beta(1 + 3, 1 + 0)
expected = DataFrame(
data=[(1, 1, 'c', 1, b__0_3), (2, 1, 'c', 1, b__1_2),
(1, 2, 'c', 1, b__2_1), (2, 2, 'c', 1, b__3_0),
(1, 1, 'd', 1, b__3_0), (2, 1, 'd', 1, b__2_1),
(1, 2, 'd', 1, b__1_2), (2, 2, 'd', 1, b__0_3)],
columns=['a', 'b', 'prob_var', 'prob_val', 'Beta'])
actual = BetaBinomialConjugate.infer_posteriors(
data=self.binomial_data,
prob_vars=['c', 'd'],
cond_vars=['a', 'b'])
for _, row in expected.iterrows():
actual_beta = actual.loc[(actual['a'] == row['a']) &
(actual['b'] == row['b']) &
(actual['prob_var'] == row['prob_var']) &
(actual['prob_val'] == row['prob_val']),
'Beta'].iloc[0]
self.assertTrue(row['Beta'] == actual_beta)
def test_distribution_table__significance(self):
a = array([3, 2, 1, 0])
n = a.sum()
b = n - a
a_others = array([(a.sum() - a[i]) / (len(a) - 1)
for i in range(len(a))])
b_others = n - a_others
expected = DataFrame(data=[
('apples', a[0], Beta(1 + a[0], 1 + b[0]) > Beta(
1 + a_others[0], 1 + b_others[0])),
('bananas', a[1], Beta(1 + a[1], 1 + b[1]) > Beta(
1 + a_others[1], 1 + b_others[1])),
('cherries', a[2], Beta(1 + a[2], 1 + b[2]) > Beta(
1 + a_others[2], 1 + b_others[2])),
('dates', a[3], Beta(1 + a[3], 1 + b[3]) > Beta(
1 + a_others[3], 1 + b_others[3])),
],
columns=['Value', 'Count', 'Significance'])
actual = self.question.distribution_table(significance=True)
self.assertTrue(expected.equals(actual))
def from_bool_frame(data: DataFrame,
prior_alpha: float = 0,
prior_beta: float = 0,
name: str = ''):
"""
Create a new BetaSeries using the counts of True and False or 1 and 0
in a DataFrame.
:param data: Data with True / False counts.
:param prior_alpha: Value for alpha assuming these represent posterior
distributions.
:param prior_beta: Value for alpha assuming these represent posterior
distributions.
:param name: Name for the Series.
"""
betas = {}
for col in data.columns:
betas[col] = Beta(alpha=prior_alpha + (data[col] == 1).sum(),
beta=prior_beta + (data[col] == 0).sum())
betas = Series(data=betas, name=name)
return BetaSeries(betas)
def setUp(self) -> None:
self.prior_float = 0.3
self.prior_beta = Beta(1 + 3, 1 + 7)
self.prior_float_map = Series({'$100': 0.3, '$200': 0.2})
self.prior_beta_map = Series({
'$100': Beta(1 + 3, 1 + 7),
'$200': Beta(1 + 2, 1 + 8)
})
self.likelihood_float = 0.8
self.likelihood_float_map = Series({'$100': 0.8, '$200': 0.6})
self.likelihood_beta = Beta(1 + 8, 1 + 2)
self.likelihood_beta_map = Series({
'$100': Beta(1 + 8, 1 + 2),
'$200': Beta(1 + 6, 1 + 4)
})
def test_distribution_table__significance(self):
n_one = 3
n_rest = 6
a_one = array([3, 2, 1])
b_one = n_one - a_one
a_rest = array([sum(a_one) - a_one[i]
for i in range(len(a_one))])
b_rest = n_rest - a_rest
expected = DataFrame(data=[
('apples', 3,
Beta(1 + a_one[0], 1 + b_one[0]) >
Beta(1 + a_rest[0], 1 + b_rest[0])),
('bananas', 2,
Beta(1 + a_one[1], 1 + b_one[1]) >
Beta(1 + a_rest[1], 1 + b_rest[1])),
('cherries', 1,
Beta(1 + a_one[2], 1 + b_one[2]) >
Beta(1 + a_rest[2], 1 + b_rest[2])),
], columns=['Value', 'Count', 'Significance'])
actual = self.question.distribution_table(significance=True)
self.assertTrue(expected.equals(actual))
def test_get_item(self):
for k, v in self.d_series.alpha.items():
expected = Beta(alpha=v, beta=1 - v)
actual = self.d_series[k]
self.assertTrue(expected == actual)
def test_infer_posteriors_with_stats(self):
b__0_3 = Beta(1 + 0, 1 + 3)
b__1_2 = Beta(1 + 1, 1 + 2)
b__2_1 = Beta(1 + 2, 1 + 1)
b__3_0 = Beta(1 + 3, 1 + 0)
expected = DataFrame(data=[
(1, 1, 'c', 1, b__0_3, b__0_3.mean(), b__0_3.interval(.95)),
(2, 1, 'c', 1, b__1_2, b__1_2.mean(), b__1_2.interval(.95)),
(1, 2, 'c', 1, b__2_1, b__2_1.mean(), b__2_1.interval(.95)),
(2, 2, 'c', 1, b__3_0, b__3_0.mean(), b__3_0.interval(.95)),
(1, 1, 'd', 1, b__3_0, b__3_0.mean(), b__3_0.interval(.95)),
(2, 1, 'd', 1, b__2_1, b__2_1.mean(), b__2_1.interval(.95)),
(1, 2, 'd', 1, b__1_2, b__1_2.mean(), b__1_2.interval(.95)),
(2, 2, 'd', 1, b__0_3, b__0_3.mean(), b__0_3.interval(.95))
],
columns=[
'a', 'b', 'prob_var', 'prob_val', 'Beta',
'mean', 'interval__0.95'
])
actual = BetaBinomialConjugate.infer_posteriors(
data=self.binomial_data,
prob_vars=['c', 'd'],
cond_vars=['a', 'b'],
stats=['mean', {
'interval': 0.95
}])
for _, row in expected.iterrows():
actual_beta = actual.loc[(actual['a'] == row['a']) &
(actual['b'] == row['b']) &
(actual['prob_var'] == row['prob_var']) &
(actual['prob_val'] == row['prob_val']),
'Beta'].iloc[0]
self.assertTrue(row['Beta'] == actual_beta)
def test_infer_posterior(self):
expected = Beta(alpha=1 + 4, beta=1 + 6)
actual = BetaBinomialConjugate.infer_posterior(self.series)
self.assertEqual(expected, actual)
def test_infer_posterior(self):
expected = Beta(alpha=1 + 2, beta=1 + 10 - 2)
actual = BetaGeometricConjugate.infer_posterior(self.series)
self.assertEqual(expected, actual)