# ------------------------------------------------------------------------------ #
# Helper functions that are used by other parts of the modeling
# ------------------------------------------------------------------------------ #
import platform
import logging
import theano
import theano.tensor as tt
import pymc3 as pm
from .model import *
log = logging.getLogger(__name__)
# workaround for macos, sufficient to do this once
if platform.system() == "Darwin":
theano.config.gcc.cxxflags = "-Wno-c++11-narrowing"
[docs]def hierarchical_normal(
pr_mean,
pr_sigma,
name_L1="delay_hc_L1",
name_L2="delay_hc_L2",
name_sigma="delay_hc_sigma",
model=None,
error_fact=2.0,
error_cauchy=True,
):
r"""
Implements an hierarchical normal model:
.. math::
x_\text{L1} &= Normal(\text{pr\_mean}, \text{pr\_sigma})\\
y_{i, \text{L2}} &= Normal(x_\text{L1}, \sigma_\text{L2})\\
\sigma_\text{L2} &= HalfCauchy(\text{error\_fact} \cdot \text{pr\_sigma})
It is however implemented in a non-centered way, that the second line is changed to:
.. math::
y_{i, \text{L2}} &= x_\text{L1} + Normal(0,1) \cdot \sigma_\text{L2}
See for example https://arxiv.org/pdf/1312.0906.pdf
Parameters
----------
name_L1 : str
Name under which :math:`x_\text{L1}` is saved in the trace.
name_L2 : str
Name under which :math:`x_\text{L2}` is saved in the trace. The non-centered distribution in addition
saved with a suffix _raw added.
name_sigma : str
Name under which :math:`\sigma_\text{L2}` is saved in the trace.
pr_mean : float
Prior mean of :math:`x_\text{L1}`
pr_sigma : float
Prior sigma for :math:`x_\text{L1}` and (muliplied by ``error_fact``) for :math:`\sigma_\text{L2}`
len_L2 : int
length of :math:`y_\text{L2}`
error_fact : float
Factor by which ``pr_sigma`` is multiplied as prior for `\sigma_\text{L2}`
error_cauchy : bool
if False, a :math:`HalfNormal` distribution is used for :math:`\sigma_\text{L2}` instead of :math:`HalfCauchy`
Returns
-------
y : :class:`~theano.tensor.TensorVariable`
the random variable :math:`y_\text{L2}`
x : :class:`~theano.tensor.TensorVariable`
the random variable :math:`x_\text{L1}`
"""
model = modelcontext(model)
if not model.is_hierarchical:
raise RuntimeError("Model is not hierarchical.")
if error_cauchy:
sigma_Y = pm.HalfCauchy(name_sigma, beta=error_fact * pr_sigma)
else:
sigma_Y = pm.HalfNormal(name_sigma, sigma=error_fact * pr_sigma)
X = pm.Normal(name_L1, mu=pr_mean, sigma=pr_sigma)
phi = pm.Normal(
name_L2 + "_raw_", mu=0, sigma=1, shape=model.shape_of_regions
) # (1-w**2)*sigma_X+1*w**2, shape=len_Y)
Y = X + phi * sigma_Y
pm.Deterministic(name_L2, Y)
return Y, X
# utility.py
# names
# still do decide
def hierarchical_beta(name, name_sigma, pr_mean, pr_sigma, len_L2, model=None):
model = modelcontext(model)
if not model.is_hierarchical: # not hierarchical
Y = pm.Beta(name, alpha=pr_mean / pr_sigma, beta=1 / pr_sigma * (1 - pr_mean))
X = None
else:
sigma_Y = pm.HalfCauchy(name_sigma + "_hc_L2", beta=pr_sigma)
X = pm.Beta(
name + "_hc_L1", alpha=pr_mean / pr_sigma, beta=1 / pr_sigma * (1 - pr_mean)
)
Y = pm.Beta(
name + "_hc_L2", alpha=X / sigma_Y, beta=1 / sigma_Y * (1 - X), shape=len_L2
)
return Y, X
# utility.py
[docs]def tt_lognormal(x, mu, sigma):
"""
Calculates a lognormal pdf for integer spaced x input.
"""
x = tt.nnet.relu(x - 1e-12) + 1e-12 # clip values at 1e-12
distr = 1 / x * tt.exp(-((tt.log(x) - mu) ** 2) / (2 * sigma ** 2))
# normalize, add a small offset in case the sum is zero
return distr / tt.sum(distr, axis=0) + 1e-8
[docs]def tt_gamma(x, mu=None, sigma=None, alpha=None, beta=None):
"""
Calculates a gamma distribution pdf for integer spaced x input. Parametrized similarly to
:class:`~pymc3.distributions.continuous.Gamma`
"""
assert (alpha is None and beta is None) != (mu is None and sigma is None)
if alpha is None and beta is None:
alpha = mu ** 2 / sigma ** 2
beta = mu / sigma ** 2
x = tt.nnet.relu(x - 1e-12) + 1e-12 # clip values at 1e-12
distr = beta ** alpha * x ** (alpha - 1) * tt.exp(-beta * x)
# normalize, add a small offset in case the sum is zero
return distr / tt.sum(distr, axis=0) + 1e-8