def dlogp(inputs, gradients):
(g_logp,) = gradients
cov, delta = inputs
g_logp.tag.test_value = floatX(1.0)
n, k = delta.shape
chol_cov = cholesky(cov)
diag = at.diag(chol_cov)
ok = at.all(diag > 0)
chol_cov = at.switch(ok, chol_cov, at.fill(chol_cov, 1))
delta_trans = solve_lower(chol_cov, delta.T).T
inner = n * at.eye(k) - at.dot(delta_trans.T, delta_trans)
g_cov = solve_upper(chol_cov.T, inner)
g_cov = solve_upper(chol_cov.T, g_cov.T)
tau_delta = solve_upper(chol_cov.T, delta_trans.T)
g_delta = tau_delta.T
g_cov = at.switch(ok, g_cov, -np.nan)
g_delta = at.switch(ok, g_delta, -np.nan)
return [-0.5 * g_cov * g_logp, -g_delta * g_logp]
def _build_marginal_likelihood_logp(self, y, X, Xu, sigma):
sigma2 = at.square(sigma)
Kuu = self.cov_func(Xu)
Kuf = self.cov_func(Xu, X)
Luu = cholesky(stabilize(Kuu))
A = solve_lower(Luu, Kuf)
Qffd = at.sum(A * A, 0)
if self.approx == "FITC":
Kffd = self.cov_func(X, diag=True)
Lamd = at.clip(Kffd - Qffd, 0.0, np.inf) + sigma2
trace = 0.0
elif self.approx == "VFE":
Lamd = at.ones_like(Qffd) * sigma2
trace = (1.0 /
(2.0 * sigma2)) * (at.sum(self.cov_func(X, diag=True)) -
at.sum(at.sum(A * A, 0)))
else: # DTC
Lamd = at.ones_like(Qffd) * sigma2
trace = 0.0
A_l = A / Lamd
L_B = cholesky(at.eye(Xu.shape[0]) + at.dot(A_l, at.transpose(A)))
r = y - self.mean_func(X)
r_l = r / Lamd
c = solve_lower(L_B, at.dot(A, r_l))
constant = 0.5 * X.shape[0] * at.log(2.0 * np.pi)
logdet = 0.5 * at.sum(at.log(Lamd)) + at.sum(at.log(at.diag(L_B)))
quadratic = 0.5 * (at.dot(r, r_l) - at.dot(c, c))
return -1.0 * (constant + logdet + quadratic + trace)
def merge_factors(self, X, Xs=None, diag=False):
factor_list = []
for factor in self.factor_list:
# make sure diag=True is handled properly
if isinstance(factor, Covariance):
factor_list.append(factor(X, Xs, diag))
elif isinstance(factor, np.ndarray):
if np.ndim(factor) == 2 and diag:
factor_list.append(np.diag(factor))
else:
factor_list.append(factor)
elif isinstance(
factor,
(
TensorConstant,
TensorVariable,
TensorSharedVariable,
),
):
if factor.ndim == 2 and diag:
factor_list.append(at.diag(factor))
else:
factor_list.append(factor)
else:
factor_list.append(factor)
return factor_list
def check_jacobian_det(
transform,
domain,
constructor=at.dscalar,
test=0,
make_comparable=None,
elemwise=False,
rv_var=None,
):
y = constructor("y")
y.tag.test_value = test
if rv_var is None:
rv_var = y
rv_inputs = rv_var.owner.inputs if rv_var.owner else []
x = transform.backward(y, *rv_inputs)
if make_comparable:
x = make_comparable(x)
if not elemwise:
jac = at.log(at.nlinalg.det(jacobian(x, [y])))
else:
jac = at.log(at.abs_(at.diag(jacobian(x, [y]))))
# ljd = log jacobian det
actual_ljd = aesara.function([y], jac)
computed_ljd = aesara.function(
[y], at.as_tensor_variable(transform.log_jac_det(y, *rv_inputs)), on_unused_input="ignore"
)
for yval in domain.vals:
close_to(actual_ljd(yval), computed_ljd(yval), tol)
def make_model(cls):
with pm.Model() as model:
sd_mu = np.array([1, 2, 3, 4, 5])
sd_dist = pm.LogNormal.dist(mu=sd_mu, sigma=sd_mu / 10.0, size=5)
chol_packed = pm.LKJCholeskyCov("chol_packed", eta=3, n=5, sd_dist=sd_dist)
chol = pm.expand_packed_triangular(5, chol_packed, lower=True)
cov = at.dot(chol, chol.T)
stds = at.sqrt(at.diag(cov))
pm.Deterministic("log_stds", at.log(stds))
corr = cov / stds[None, :] / stds[:, None]
corr_entries_unit = (corr[np.tril_indices(5, -1)] + 1) / 2
pm.Deterministic("corr_entries_unit", corr_entries_unit)
return model
def predict_mean_i(i, x_star, s_star, X, beta, h):
n, D = shape(X)
# rescale every dimension by the corresponding inverse lengthscale
iL = at.diag(h[i, :D])
inp = (X - x_star).dot(iL)
# compute the mean
B = iL.dot(s_star).dot(iL)
t = inp.dot(B)
lb = (inp * t).sum() + beta.sum()
Mi = at_sum(lb) * h[i, D]
return Mi
def L_op(self, inputs, outputs, gradients):
# Modified from aesara/tensor/slinalg.py
# No handling for on_error = 'nan'
dz = gradients[0]
chol_x = outputs[0]
# this is for nan mode
#
# ok = ~tensor.any(tensor.isnan(chol_x))
# chol_x = tensor.switch(ok, chol_x, 1)
# dz = tensor.switch(ok, dz, 1)
# deal with upper triangular by converting to lower triangular
if not self.lower:
chol_x = chol_x.T
dz = dz.T
def tril_and_halve_diagonal(mtx):
"""Extracts lower triangle of square matrix and halves diagonal."""
return tensor.tril(mtx) - tensor.diag(tensor.diagonal(mtx) / 2.0)
def conjugate_solve_triangular(outer, inner):
"""Computes L^{-T} P L^{-1} for lower-triangular L."""
return gpu_solve_upper_triangular(
outer.T, gpu_solve_upper_triangular(outer.T, inner.T).T
)
s = conjugate_solve_triangular(
chol_x, tril_and_halve_diagonal(chol_x.T.dot(dz))
)
if self.lower:
grad = tensor.tril(s + s.T) - tensor.diag(tensor.diagonal(s))
else:
grad = tensor.triu(s + s.T) - tensor.diag(tensor.diagonal(s))
return [grad]
def __init__(self, input_dim, W=None, kappa=None, B=None, active_dims=None):
super().__init__(input_dim, active_dims)
if len(self.active_dims) != 1:
raise ValueError("Coregion requires exactly one dimension to be active")
make_B = W is not None or kappa is not None
if make_B and B is not None:
raise ValueError("Exactly one of (W, kappa) and B must be provided to Coregion")
if make_B:
self.W = at.as_tensor_variable(W)
self.kappa = at.as_tensor_variable(kappa)
self.B = at.dot(self.W, self.W.T) + at.diag(self.kappa)
elif B is not None:
self.B = at.as_tensor_variable(B)
else:
raise ValueError("Exactly one of (W, kappa) and B must be provided to Coregion")
def cov(self):
var = rho2sigma(self.rho)**2
if self.batched:
return batched_diag(var)
else:
return at.diag(var)
def std(self):
return at.sqrt(at.diag(self.cov))
def std(self):
if self.batched:
return at.sqrt(batched_diag(self.cov))
else:
return at.sqrt(at.diag(self.cov))
def MvNormalLogp():
"""Compute the log pdf of a multivariate normal distribution.
This should be used in MvNormal.logp once Theano#5908 is released.
Parameters
----------
cov: at.matrix
The covariance matrix.
delta: at.matrix
Array of deviations from the mean.
"""
cov = at.matrix("cov")
cov.tag.test_value = floatX(np.eye(3))
delta = at.matrix("delta")
delta.tag.test_value = floatX(np.zeros((2, 3)))
solve_lower = Solve(A_structure="lower_triangular")
solve_upper = Solve(A_structure="upper_triangular")
cholesky = Cholesky(lower=True, on_error="nan")
n, k = delta.shape
n, k = f(n), f(k)
chol_cov = cholesky(cov)
diag = at.diag(chol_cov)
ok = at.all(diag > 0)
chol_cov = at.switch(ok, chol_cov, at.fill(chol_cov, 1))
delta_trans = solve_lower(chol_cov, delta.T).T
result = n * k * at.log(f(2) * np.pi)
result += f(2) * n * at.sum(at.log(diag))
result += (delta_trans ** f(2)).sum()
result = f(-0.5) * result
logp = at.switch(ok, result, -np.inf)
def dlogp(inputs, gradients):
(g_logp,) = gradients
cov, delta = inputs
g_logp.tag.test_value = floatX(1.0)
n, k = delta.shape
chol_cov = cholesky(cov)
diag = at.diag(chol_cov)
ok = at.all(diag > 0)
chol_cov = at.switch(ok, chol_cov, at.fill(chol_cov, 1))
delta_trans = solve_lower(chol_cov, delta.T).T
inner = n * at.eye(k) - at.dot(delta_trans.T, delta_trans)
g_cov = solve_upper(chol_cov.T, inner)
g_cov = solve_upper(chol_cov.T, g_cov.T)
tau_delta = solve_upper(chol_cov.T, delta_trans.T)
g_delta = tau_delta.T
g_cov = at.switch(ok, g_cov, -np.nan)
g_delta = at.switch(ok, g_delta, -np.nan)
return [-0.5 * g_cov * g_logp, -g_delta * g_logp]
return OpFromGraph([cov, delta], [logp], grad_overrides=dlogp, inline=True)
def diag(self, X):
X, _ = self._slice(X, None)
index = at.cast(X, "int32")
return at.diag(self.B)[index.ravel()]
def full(self, X, Xs=None):
if Xs is None:
return at.diag(self.diag(X))
else:
return at.alloc(0.0, X.shape[0], Xs.shape[0])
def tril_and_halve_diagonal(mtx):
"""Extracts lower triangle of square matrix and halves diagonal."""
return tensor.tril(mtx) - tensor.diag(tensor.diagonal(mtx) / 2.0)
def cov(self):
var = rho2sigma(self.rho) ** 2
return at.diag(var)