def information_model(e, m, epsilon=1.0e-5):
I = pymc.Uniform(name='mutual information', lower=epsilon, upper=1.0-epsilon)
belb = ibeb.bayes_error_lower_bound(Y_entropy=1.0, num_classes=2)
@pymc.deterministic(name='Bayes error lower bound')
def bayes_error_lb(I=I):
return belb(I)
@pymc.deterministic(name='Bayes error upper bound')
def bayes_error_ub(I=I):
return ibeb.bayes_error_upper_bound(I)
epsilon_B = pymc.Uniform(name='Bayes error', lower=bayes_error_lb, upper=bayes_error_ub)
epsilon = pymc.Uniform(name='generalization error', lower=epsilon_B, upper=0.5)
error = pymc.Binomial(name='observed number of errors', n=m, p=epsilon, observed=True, value=e)
return locals()
def data_likelihood(m, lower=0.0, upper=1.0, N=100000, counts_min=5):
"""Compute p(e|0>I(X;Y)>=1,m) via Monte Carlo with N iterations.
upper, lower = bounds of mutual information.
"""
be_lb = ibeb.bayes_error_lower_bound(Y_entropy=1.0, num_classes=2, cache_size=100000)
be_ub = ibeb.bayes_error_upper_bound
# Monte Carlo:
MI = np.random.uniform(low=lower, high=upper, size=N)
epsilon_B_lower = be_lb(MI)
epsilon_B_upper = be_ub(MI)
epsilon_B = np.random.uniform(low=epsilon_B_lower, high=epsilon_B_upper)
epsilon = np.random.uniform(low=epsilon_B, high=0.5)
e = np.random.binomial(m,epsilon)
# Statistics:
e_counts, bins = np.histogram(e,bins=range(m+2))
e_frequency = e_counts / np.double(N)
e_frequency[e_counts<counts_min] = np.NaN # mark less reliable values with NaN
return e_frequency
