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hib_stats.py
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# -*- coding: utf-8 -*-
import numpy as np
from mpmath import mp, fp
from .horn_numeric import horn_phi1, mp_ctx
def m_hib_single(y, sigma=1., tau=1., a=0.5, b=0.5, s=0.,
horn_phi1_fn=horn_phi1):
r""" Computation of the marginal posterior for the hypergeometric
inverted-beta model.
In its most general form, we have for the hypergeometric
inverted-beta model given by
.. math:
p(y_i, \kappa_i) \propto \kappa_i^{a^\prime - 1} (1-\kappa_i)^{b-1}
\left(1/\tau^2 + (1 - 1/\tau^2) \kappa_i\right)^{-1}
e^{-\kappa_i s^\prime}
\;.
where ::math::`s^\prime = s + y_i^2 / (2\sigma^2)` and
::math::`a^\prime = a + 1/2`.
The marginal posterior is
.. math:
m(y_i; \sigma, \tau) = \frac{1}{\sqrt(2 \pi \sigma^2}
\exp\left(-\frac{y_i^2}{2 \sigma^2}\right)
\frac{\operatorname{B}(a^\prime, b)}{\operatorname{B}(a,b)}
\frac{\Phi_1(b, 1, a^\prime + b, s^\prime, 1 - 1/\tau^2)}{
\Phi_1(b, 1, a + b, s, 1 - 1/\tau^2)}
The Horseshoe prior has ::math::`a = b = 1/2, s = 0` and
::math::`\tau = 1`.
Parameters
==========
y: float
A single observation.
sigma: float
Observation variance.
tau: float
Prior variance scale factor.
a: float
Hypergeometric inverted-beta model parameter
b: float
Hypergeometric inverted-beta model parameter
s: float
Hypergeometric inverted-beta model parameter
Returns
=======
ndarray of mpmath.mpf
Numeric value of `m(y; sigma, tau)`.
"""
y_2_sig = 0.5 * np.square(y / sigma)
s_p = s + y_2_sig
a_p = a + 0.5
C = 1. / np.sqrt(2. * np.pi) / sigma
res = C * np.exp(-y_2_sig)
res *= mp_ctx.beta(a_p, b) / mp_ctx.beta(a, b)
if tau > 0:
tau_term = 1. - tau**(-2)
else:
tau_term = mp.ninf
# TODO, FIXME: Replace with direct computation of this ratio.
res *= horn_phi1_fn(b, 1., a_p + b, s_p, tau_term, keep_exp_const=False)
res /= horn_phi1_fn(b, 1., a + b, s, tau_term, keep_exp_const=False)
return res
m_hib = np.vectorize(m_hib_single)
def m_hs(y, sigma=1., tau=1.):
r""" Exact evaluation of the marginal posterior of the HS prior via
special functions.
Parameters
==========
y: float
A single observation.
sigma: float
Observation variance.
tau: float
Prior variance scale factor.
Returns
=======
ndarray of mpmath.mpf
Numeric value of `m(y; sigma, tau)`.
"""
return m_hib(y, sigma, tau, 0.5, 0.5, 0)
def E_kappa(y, sigma=1., tau=1., a=0.5, b=0.5, s=0., n=1.,
horn_phi1_fn=horn_phi1):
r""" Moments of the hypergeometric inverted-beta model
in the ::math::`\kappa` parameterization.
In its most general form, we have
.. math:
E(\kappa^n \mid y, \sigma, \tau) &=
\frac{(a^\prime)_n}{(a^\prime + b)_n}
\frac{\Phi_1(b, 1, a^\prime + b + n, s^\prime, 1 - 1/\tau^2)}{
\Phi_1(b, 1, a^\prime + b, s^\prime, 1 - 1/\tau^2)}
\;,
where ::math::`s^\prime = s + y_i^2 / (2\sigma^2)` and
::math::`a^\prime = a + 1/2` for the hypergeometric inverted-beta
given by
.. math:
p(y_i, \kappa_i) \propto \kappa_i^{a^\prime - 1} (1-\kappa_i)^{b-1}
\left(1/\tau^2 + (1 - 1/\tau^2) \kappa_i\right)^{-1}
e^{-\kappa_i s^\prime}
\;.
Parameters
==========
y: float
A single observation.
sigma: float
Observation variance.
tau: float
Prior variance scale factor.
a: float
Hypergeometric inverted-beta model parameter
b: float
Hypergeometric inverted-beta model parameter
s: float
Hypergeometric inverted-beta model parameter
n: int
Order of the moment.
Results
=======
ndarray of mpmath.mpf
"""
s_p = s + 0.5 * np.square(y / sigma)
a_p = a + 0.5
res = mp_ctx.rf(a_p, n) / mp_ctx.rf(a_p + b, n)
if np.iterable(tau):
# TODO: mp_ctx.mpf(t_) ?
tau_term = np.fromiter((1. - t_**(-2) if t_ > 0 else mp.ninf
for t_ in tau), dtype=np.float)
else:
if tau > 0:
# TODO: mp_ctx.mpf(t_) ?
tau_term = 1. - tau**(-2)
else:
tau_term = mp.ninf
# TODO: Replace with direct computation of this ratio.
res_num = horn_phi1_fn(b, 1., a_p + b + n, s_p, tau_term,
keep_exp_const=False)
res_denom = horn_phi1_fn(b, 1., a_p + b, s_p, tau_term,
keep_exp_const=False)
# XXX FIXME: A finite precision hack! We're just assuming that the
# numerator diverges faster. Is this *always* the case? What if the ratio
# converges to 1?
res_ratio = np.array([0 if mp_ctx.isinf(n_) else n_/d_
for n_, d_ in zip(np.atleast_1d(res_num),
np.atleast_1d(res_denom))],
dtype=np.object)
res *= res_ratio
#assert all(0 <= res) and all(res <= 1)
return res
def E_beta(y, sigma=1., tau=1., a=0.5, b=0.5, s=0.,
horn_phi1_fn=horn_phi1):
r""" Moments of the hypergeometric inverted-beta model
in the ::math::`\beta` parameterization.
In its most general form, we have
.. math:
E(\beta^n \mid y, \sigma, \tau) &=
\left( 1 - \frac{a^\prime}{a^\prime + b}
\frac{\Phi_1(b, 1, a^\prime + b + n, s^\prime, 1 - 1/\tau^2)}{
\Phi_1(b, 1, a^\prime + b, s^\prime, 1 - 1/\tau^2)}
\right) y
\;,
where ::math::`s^\prime = s + y_i^2 / (2\sigma^2)` and
::math::`a^\prime = a + 1/2` for the hypergeometric inverted-beta
given by
.. math:
p(y_i, \kappa_i) \propto \kappa_i^{a^\prime - 1} (1-\kappa_i)^{b-1}
\left(1/\tau^2 + (1 - 1/\tau^2) \kappa_i\right)^{-1}
e^{-\kappa_i s^\prime}
\;.
Parameters
==========
y: float
A single observation.
sigma: float
Observation variance.
tau: float
Prior variance scale factor.
a: float
Hypergeometric inverted-beta model parameter
b: float
Hypergeometric inverted-beta model parameter
s: float
Hypergeometric inverted-beta model parameter
Results
=======
ndarray of mpmath.mpf
"""
res = E_kappa(y, sigma, tau, a, b, s, n=1,
horn_phi1_fn=horn_phi1)
res *= y
res = y - res
return res
def SURE_hib(y, sigma=1., tau=1., a=0.5, b=0.5, s=0., d=1.):
r""" Compute the SURE value for the HIB prior model.
Parameters
==========
y: float
A single observation.
sigma: float
Observation variance.
tau: float
Prior variance scale factor.
a: float
Hypergeometric inverted-beta model parameter
b: float
Hypergeometric inverted-beta model parameter
s: float
Hypergeometric inverted-beta model parameter
d: float
Optional observation scaling parameter.
Returns
=======
TODO
"""
res = 2 * sigma**2
E_1 = E_kappa(y, sigma, tau, a, b, s, n=1)
y_d_2 = (y * d)**2
res = res - y_d_2 * E_1**2
E_2 = E_kappa(y, sigma, tau, a, b, s, n=2)
res2 = -sigma**2 * E_1 + y_d_2 * E_2
res = res + 2 * res2
return res
def DIC_hib(y, sigma=1., tau=1., a=0.5, b=0.5, s=0., d=1.):
r""" Computes DIC estimate for the hypergeometric inverted-beta model.
.. math::
\text{DIC} = \sum_{i=1}^N \left\{
2 \left(1 - E\left[\kappa_i \mid y_i\right] \right) +
\frac{y_i^2}{\sigma^2} \left( 2 E\left[\kappa_i^2 \mid y_i\right] -
{E\left[\kappa_i \mid y_i\right]}^2 \right)
\right\}
Parameters
==========
y: float
A single observation.
sigma: float
Observation variance.
tau: float
Prior variance scale factor.
a: float
Hypergeometric inverted-beta model parameter
b: float
Hypergeometric inverted-beta model parameter
s: float
Hypergeometric inverted-beta model parameter
d: float
Optional observation scaling parameter.
Returns
=======
ndarray (float)
"""
E_1 = E_kappa(y, sigma, tau, a, b, s, n=1)
E_2 = E_kappa(y, sigma, tau, a, b, s, n=2)
res = 2.*(1. - E_1) + (y * d / sigma)**2 * (2. * E_2 - E_1**2)
return res