def test_UniformPrior():
lower = 2
upper = 4
P1 = UniformPrior(lower=lower, upper=upper, variable_indices=[0])
# Check some basic properties
assert P1.lower == lower
assert P1.upper == upper
assert P1.bounds == [(lower, upper)]
# Check sampling produces expected distribution
sample = array([P1.sample() for _ in range(100)])
assert all(sample >= lower)
assert all(sample <= upper)
# Check that probability is constant within the range
theta_in_range = linspace(lower, upper, endpoint=True)
log_probability = [P1(array([i])) for i in theta_in_range]
assert all(log_probability == log_probability[0])
assert exp(log_probability[0]) == 1.0 / (upper - lower)
# Check that the probability is very small outside the range
theta_left_range = linspace(-15, lower, endpoint=False)
log_probability_left = array([P1(array([i])) for i in theta_left_range])
assert all(log_probability_left < -1e32)
theta_right_range = linspace(upper + 1e-5, 15)
log_probability_right = array([P1(array([i])) for i in theta_right_range])
assert all(log_probability_right < -1e32)
def test_UniformPrior_gradient():
# test combining multiple uniforms
P1 = UniformPrior(lower=2, upper=4, variable_indices=[0])
P2 = UniformPrior(lower=4, upper=8, variable_indices=[1])
P3 = UniformPrior(lower=[8, 16], upper=[16, 32], variable_indices=[2, 3])
combo = UniformPrior.combine([P1, P2, P3])
# evaluate the prior at a test point
test_point = array([3.0, 5.0, 15.0, 19.0])
# check the analytic gradient calculation against finite difference
analytic_gradient = combo.gradient(test_point)
numeric_gradient = finite_difference(
func=combo, x0=test_point, vectorised_arguments=True
)
assert allclose(analytic_gradient, numeric_gradient)
def test_UniformPrior_combine():
# test combining multiple uniforms
P1 = UniformPrior(lower=2, upper=4, variable_indices=[0])
P2 = UniformPrior(lower=4, upper=8, variable_indices=[1])
P3 = UniformPrior(lower=[8, 16], upper=[16, 32], variable_indices=[2, 3])
combo = UniformPrior.combine([P1, P2, P3])
# check the combined distribution has the right values
assert (combo.lower == array([2.0, 4.0, 8.0, 16.0])).all()
assert (combo.upper == array([4.0, 8.0, 16.0, 32.0])).all()
assert combo.bounds == [(2, 4), (4, 8), (8, 16), (16, 32)]
assert combo.variables == [0, 1, 2, 3]
# Check sampling produces expected distribution
sample = array([combo.sample() for _ in range(100)])
for index in combo.variables:
assert all(sample[:, index] >= combo.lower[index])
assert all(sample[:, index] <= combo.upper[index])
def test_JointPrior_gradient():
P1 = ExponentialPrior(beta=10, variable_indices=[1])
P2 = GaussianPrior(mean=20, sigma=2, variable_indices=[0])
P3 = UniformPrior(lower=8, upper=16, variable_indices=[2])
JP = JointPrior(components=[P1, P2, P3], n_variables=3)
test_point = array([4.0, 23.0, 12.0])
analytic_gradient = JP.gradient(test_point)
numeric_gradient = finite_difference(
func=JP, x0=test_point, vectorised_arguments=True
)
assert allclose(analytic_gradient, numeric_gradient)
def test_JointPrior():
P1 = ExponentialPrior(beta=10, variable_indices=[1])
P2 = GaussianPrior(mean=20, sigma=2, variable_indices=[0])
P3 = UniformPrior(lower=8, upper=16, variable_indices=[2])
components = [P1, P2, P3]
JP = JointPrior(components=components, n_variables=3)
sample = array([JP.sample() for _ in range(1000)])
assert sample.shape[1] == len(components)
assert isclose(sample[:, 0].mean(), P2.mean, rtol=0.2)
assert isclose(sample[:, 0].std(), P2.sigma, rtol=0.2)
for index, (lower, upper) in enumerate(JP.bounds):
if lower is not None:
assert all(sample[:, index] >= lower)
if upper is not None:
assert all(sample[:, index] <= upper)
# We could at this stage pair the likelihood object with an optimiser in order to obtain
# the maximum-likelihood estimate of the parameters. In this example however, we want to
# construct the posterior distribution for the model parameters, and that means we need
# a prior.
# The inference.priors module contains classes which allow for easy construction of
# prior distributions across all model parameters.
from inference.priors import ExponentialPrior, UniformPrior, JointPrior
# If we want different model parameters to have different prior distributions, as in this
# case where we give three variables an exponential prior and one a uniform prior, we first
# construct each type of prior separately:
prior_components = [
ExponentialPrior(beta=[50., 20., 20.], variable_indices=[0, 1, 3]),
UniformPrior(lower=0., upper=12., variable_indices=[2])
]
# Now we use the JointPrior class to combine the various components into a single prior
# distribution which covers all the model parameters.
prior = JointPrior(components=prior_components, n_variables=4)
# As with the likelihood, prior objects can also be called as function to return a
# log-probability value when passed a vector of model parameters. We can also draw
# samples from the prior directly using the sample() method:
prior_sample = prior.sample()
print(prior_sample)
# The likelihood and prior can be easily combined into a posterior distribution
# using the Posterior class:
from inference.posterior import Posterior
posterior = Posterior(likelihood=likelihood, prior=prior)
def test_JointPrior_repeat_variable():
P1 = ExponentialPrior(beta=10, variable_indices=[0])
P2 = GaussianPrior(mean=20, sigma=2, variable_indices=[0])
P3 = UniformPrior(lower=8, upper=16, variable_indices=[2])
with pytest.raises(ValueError):
JointPrior(components=[P1, P2, P3], n_variables=3)
def test_UniformPrior_bad_upper():
with pytest.raises(ValueError):
UniformPrior(lower=zeros(2), upper=[1.0, 0.0], variable_indices=[0])
def test_UniformPrior_inconsistent_dimensions():
with pytest.raises(ValueError):
UniformPrior(lower=zeros((2, 2)), upper=ones(4), variable_indices=[0])
with pytest.raises(ValueError):
UniformPrior(lower=zeros(4), upper=ones((2, 2)), variable_indices=[0])
def test_UniformPrior_inconsistent_size():
with pytest.raises(ValueError):
UniformPrior(lower=zeros(1), upper=ones(2), variable_indices=[0])