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Update of symbolic likelihoods.

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Neil Lawrence 2014-03-31 09:18:31 +01:00
parent 60a071f18f
commit 6f86b9b0fa
6 changed files with 319 additions and 134 deletions
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92
GPy/likelihoods/bernoulli.py
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@ -15,7 +15,7 @@ class Bernoulli(Likelihood):
p(y_{i}|\\lambda(f_{i})) = \\lambda(f_{i})^{y_{i}}(1-f_{i})^{1-y_{i}}
.. Note::
Y is expected to take values in {-1, 1} TODO: {0, 1}??
Y takes values in either {-1, 1} or {0, 1}.
link function should have the domain [0, 1], e.g. probit (default) or Heaviside
.. See also::
@ -54,10 +54,10 @@ class Bernoulli(Likelihood):
"""
if Y_i == 1:
sign = 1.
elif Y_i == 0:
elif Y_i == 0 or Y_i == -1:
sign = -1
else:
raise ValueError("bad value for Bernouilli observation (0, 1)")
raise ValueError("bad value for Bernoulli observation (0, 1)")
if isinstance(self.gp_link, link_functions.Probit):
z = sign*v_i/np.sqrt(tau_i**2 + tau_i)
Z_hat = std_norm_cdf(z)
@ -95,15 +95,15 @@ class Bernoulli(Likelihood):
else:
return np.nan
def pdf_link(self, link_f, y, Y_metadata=None):
def pdf_link(self, inv_link_f, y, Y_metadata=None):
"""
Likelihood function given link(f)
Likelihood function given inverse link of f.
.. math::
p(y_{i}|\\lambda(f_{i})) = \\lambda(f_{i})^{y_{i}}(1-f_{i})^{1-y_{i}}
:param link_f: latent variables link(f)
:type link_f: Nx1 array
:param inv_link_f: latent variables inverse link of f.
:type inv_link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param Y_metadata: Y_metadata not used in bernoulli
@ -113,102 +113,106 @@ class Bernoulli(Likelihood):
.. Note:
Each y_i must be in {0, 1}
"""
assert np.atleast_1d(link_f).shape == np.atleast_1d(y).shape
#objective = (link_f**y) * ((1.-link_f)**(1.-y))
objective = np.where(y, link_f, 1.-link_f)
assert np.atleast_1d(inv_link_f).shape == np.atleast_1d(y).shape
#objective = (inv_link_f**y) * ((1.-inv_link_f)**(1.-y))
objective = np.where(y, inv_link_f, 1.-inv_link_f)
return np.exp(np.sum(np.log(objective)))
def logpdf_link(self, link_f, y, Y_metadata=None):
def logpdf_link(self, inv_link_f, y, Y_metadata=None):
"""
Log Likelihood function given link(f)
Log Likelihood function given inverse link of f.
.. math::
\\ln p(y_{i}|\\lambda(f_{i})) = y_{i}\\log\\lambda(f_{i}) + (1-y_{i})\\log (1-f_{i})
:param link_f: latent variables link(f)
:type link_f: Nx1 array
:param inv_link_f: latent variables inverse link of f.
:type inv_link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param Y_metadata: Y_metadata not used in bernoulli
:returns: log likelihood evaluated at points link(f)
:returns: log likelihood evaluated at points inverse link of f.
:rtype: float
"""
assert np.atleast_1d(link_f).shape == np.atleast_1d(y).shape
#objective = y*np.log(link_f) + (1.-y)*np.log(link_f)
assert np.atleast_1d(inv_link_f).shape == np.atleast_1d(y).shape
#objective = y*np.log(inv_link_f) + (1.-y)*np.log(inv_link_f)
state = np.seterr(divide='ignore')
objective = np.where(y==1, np.log(link_f), np.log(1-link_f))
# TODO check y \in {0, 1} or {-1, 1}
objective = np.where(y==1, np.log(inv_link_f), np.log(1-inv_link_f))
np.seterr(**state)
return np.sum(objective)
def dlogpdf_dlink(self, link_f, y, Y_metadata=None):
def dlogpdf_dlink(self, inv_link_f, y, Y_metadata=None):
"""
Gradient of the pdf at y, given link(f) w.r.t link(f)
Gradient of the pdf at y, given inverse link of f w.r.t inverse link of f.
.. math::
\\frac{d\\ln p(y_{i}|\\lambda(f_{i}))}{d\\lambda(f)} = \\frac{y_{i}}{\\lambda(f_{i})} - \\frac{(1 - y_{i})}{(1 - \\lambda(f_{i}))}
:param link_f: latent variables link(f)
:type link_f: Nx1 array
:param inv_link_f: latent variables inverse link of f.
:type inv_link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param Y_metadata: Y_metadata not used in bernoulli
:returns: gradient of log likelihood evaluated at points link(f)
:returns: gradient of log likelihood evaluated at points inverse link of f.
:rtype: Nx1 array
"""
assert np.atleast_1d(link_f).shape == np.atleast_1d(y).shape
#grad = (y/link_f) - (1.-y)/(1-link_f)
assert np.atleast_1d(inv_link_f).shape == np.atleast_1d(y).shape
#grad = (y/inv_link_f) - (1.-y)/(1-inv_link_f)
state = np.seterr(divide='ignore')
grad = np.where(y, 1./link_f, -1./(1-link_f))
# TODO check y \in {0, 1} or {-1, 1}
grad = np.where(y, 1./inv_link_f, -1./(1-inv_link_f))
np.seterr(**state)
return grad
def d2logpdf_dlink2(self, link_f, y, Y_metadata=None):
def d2logpdf_dlink2(self, inv_link_f, y, Y_metadata=None):
"""
Hessian at y, given link_f, w.r.t link_f the hessian will be 0 unless i == j
i.e. second derivative logpdf at y given link(f_i) link(f_j) w.r.t link(f_i) and link(f_j)
Hessian at y, given inv_link_f, w.r.t inv_link_f the hessian will be 0 unless i == j
i.e. second derivative logpdf at y given inverse link of f_i and inverse link of f_j w.r.t inverse link of f_i and inverse link of f_j.
.. math::
\\frac{d^{2}\\ln p(y_{i}|\\lambda(f_{i}))}{d\\lambda(f)^{2}} = \\frac{-y_{i}}{\\lambda(f)^{2}} - \\frac{(1-y_{i})}{(1-\\lambda(f))^{2}}
:param link_f: latent variables link(f)
:type link_f: Nx1 array
:param inv_link_f: latent variables inverse link of f.
:type inv_link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param Y_metadata: Y_metadata not used in bernoulli
:returns: Diagonal of log hessian matrix (second derivative of log likelihood evaluated at points link(f))
:returns: Diagonal of log hessian matrix (second derivative of log likelihood evaluated at points inverse link of f.
:rtype: Nx1 array
.. Note::
Will return diagonal of hessian, since every where else it is 0, as the likelihood factorizes over cases
(the distribution for y_i depends only on link(f_i) not on link(f_(j!=i))
(the distribution for y_i depends only on inverse link of f_i not on inverse link of f_(j!=i)
"""
assert np.atleast_1d(link_f).shape == np.atleast_1d(y).shape
#d2logpdf_dlink2 = -y/(link_f**2) - (1-y)/((1-link_f)**2)
assert np.atleast_1d(inv_link_f).shape == np.atleast_1d(y).shape
#d2logpdf_dlink2 = -y/(inv_link_f**2) - (1-y)/((1-inv_link_f)**2)
state = np.seterr(divide='ignore')
d2logpdf_dlink2 = np.where(y, -1./np.square(link_f), -1./np.square(1.-link_f))
# TODO check y \in {0, 1} or {-1, 1}
d2logpdf_dlink2 = np.where(y, -1./np.square(inv_link_f), -1./np.square(1.-inv_link_f))
np.seterr(**state)
return d2logpdf_dlink2
def d3logpdf_dlink3(self, link_f, y, Y_metadata=None):
def d3logpdf_dlink3(self, inv_link_f, y, Y_metadata=None):
"""
Third order derivative log-likelihood function at y given link(f) w.r.t link(f)
Third order derivative log-likelihood function at y given inverse link of f w.r.t inverse link of f
.. math::
\\frac{d^{3} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{3}\\lambda(f)} = \\frac{2y_{i}}{\\lambda(f)^{3}} - \\frac{2(1-y_{i}}{(1-\\lambda(f))^{3}}
:param link_f: latent variables link(f)
:type link_f: Nx1 array
:param inv_link_f: latent variables passed through inverse link of f.
:type inv_link_f: Nx1 array
:param y: data
:type y: Nx1 array
:param Y_metadata: Y_metadata not used in bernoulli
:returns: third derivative of log likelihood evaluated at points link(f)
:returns: third derivative of log likelihood evaluated at points inverse_link(f)
:rtype: Nx1 array
"""
assert np.atleast_1d(link_f).shape == np.atleast_1d(y).shape
#d3logpdf_dlink3 = 2*(y/(link_f**3) - (1-y)/((1-link_f)**3))
assert np.atleast_1d(inv_link_f).shape == np.atleast_1d(y).shape
#d3logpdf_dlink3 = 2*(y/(inv_link_f**3) - (1-y)/((1-inv_link_f)**3))
state = np.seterr(divide='ignore')
d3logpdf_dlink3 = np.where(y, 2./(link_f**3), -2./((1.-link_f)**3))
# TODO check y \in {0, 1} or {-1, 1}
d3logpdf_dlink3 = np.where(y, 2./(inv_link_f**3), -2./((1.-inv_link_f)**3))
np.seterr(**state)
return d3logpdf_dlink3