def backsub_both_sides(L, X, transpose='left'):
""" Return L^-T * X * L^-1, assumuing X is symmetrical and L is lower cholesky"""
if transpose == 'left':
tmp, _ = lapack.dtrtrs(L, np.asfortranarray(X), lower=1, trans=1)
return lapack.dtrtrs(L, np.asfortranarray(tmp.T), lower=1, trans=1)[0].T
else:
tmp, _ = lapack.dtrtrs(L, np.asfortranarray(X), lower=1, trans=0)
return lapack.dtrtrs(L, np.asfortranarray(tmp.T), lower=1, trans=0)[0].T
def _qK_graminv(self):
"""
Inverse kernel mean multiplied with inverse kernel Gram matrix, all evaluated at training locations.
.. math::
\int k(x, X)\mathrm{d}x [k(X, X) + \sigma^2 I]^{-1}
:return: weights of shape (1, n_train_points)
"""
lower_chol = self.model.base_gp.gram_chol()
qK = self.model.base_gp.kern.qK(self.model.base_gp.X)
graminv_qK_trans = lapack.dtrtrs(lower_chol.T, (lapack.dtrtrs(lower_chol, qK.T, lower=1)[0]), lower=0)[0]
return np.transpose(graminv_qK_trans)
def _graminv_Kx(self, x):
"""
Inverse kernel Gram matrix multiplied with kernel function k(x, x') evaluated at existing training datapoints
and location x.
.. math::
[K(X, X) + \sigma^2 I]^{-1} K (X, x)
:param x: (n_points x input_dim) locations where to evaluate
:return: (n_train_points, n_points)
"""
lower_chol = self.model.base_gp.gram_chol()
KXx = self.model.base_gp.kern.K(self.model.base_gp.X, x)
return lapack.dtrtrs(lower_chol.T, (lapack.dtrtrs(lower_chol, KXx, lower=1)[0]), lower=0)[0]
def _qK_graminv(self):
"""
Inverse kernel mean multiplied with inverse kernel Gram matrix, all evaluated at training locations.
.. math::
\int k(x, X)\mathrm{d}x [k(X, X) + \sigma^2 I]^{-1}
:return: weights of shape (1, n_train_points)
"""
lower_chol = self.model.base_gp.gram_chol()
qK = self.model.base_gp.kern.qK(self.model.base_gp.X)
graminv_qK_trans = lapack.dtrtrs(
lower_chol.T, (lapack.dtrtrs(lower_chol, qK.T, lower=1)[0]),
lower=0)[0]
return np.transpose(graminv_qK_trans)
def _graminv_Kx(self, x):
"""
Inverse kernel Gram matrix multiplied with kernel function k(x, x') evaluated at existing training datapoints
and location x.
.. math::
[K(X, X) + \sigma^2 I]^{-1} K (X, x)
:param x: (n_points x input_dim) locations where to evaluate
:return: (n_train_points, n_points)
"""
lower_chol = self.model.base_gp.gram_chol()
KXx = self.model.base_gp.kern.K(self.model.base_gp.X, x)
return lapack.dtrtrs(lower_chol.T,
(lapack.dtrtrs(lower_chol, KXx, lower=1)[0]),
lower=0)[0]
def LAPACK_solve_ls_with_QR(A, b):
# Ref: https://stackoverflow.com/questions/21970510/solving-a-linear-system-with-lapacks-dgeqrf
# The corresponding procedure in LAPACK is https://www.netlib.org/lapack/lug/node40.html
qr, tau, work, info = dgeqrf(A)
cq, work, info = dormqr('L', 'T', qr, tau, b, qr.shape[0])
x_qr, info = dtrtrs(qr, cq)
return x_qr[0:A.shape[1]]
def _compute_integral_mean_and_variance(self):
integral_mean, kernel_mean_X = self._compute_integral_mean_and_kernel_mean(
)
integral_var = self.base_gp.kern.qKq() - np.square(
lapack.dtrtrs(self.base_gp.gram_chol(), kernel_mean_X.T,
lower=1)[0]).sum(axis=0, keepdims=True).T
return np.float(integral_mean), np.float(integral_var)
def residual_variable_projection(matrix: np.ndarray, data: np.ndarray) \
-> typing.Tuple[typing.List[str], np.ndarray]:
"""Calculates the conditionaly linear parameters and residual with the variable projection
method.
Parameters
----------
matrix :
The model matrix.
data : np.ndarray
The data to analyze.
"""
# TODO: Reference Kaufman paper
# Kaufman Q2 step 3
qr, tau, _, _ = lapack.dgeqrf(matrix)
# Kaufman Q2 step 4
temp, _, _ = lapack.dormqr("L", "T", qr, tau, data, max(1, matrix.shape[1]),
overwrite_c=0)
clp, _ = lapack.dtrtrs(qr, temp)
for i in range(matrix.shape[1]):
temp[i] = 0
# Kaufman Q2 step 5
residual, _, _ = lapack.dormqr("L", "N", qr, tau, temp, max(1, matrix.shape[1]),
overwrite_c=0)
return clp[:matrix.shape[1]], residual
def _solve_triangular(L, b, lower=True):
'''
Solve the triangular system of equations `Lx = b` using `dtrtrs`.
Parameters
----------
L : (n, n) float array
b : (n, *) float array
Returns
-------
(n, *) float array
'''
if any(i == 0 for i in b.shape):
return np.zeros(b.shape)
x, info = dtrtrs(L, b, lower=lower)
if info < 0:
raise ValueError('The %s-th argument had an illegal value' % (-info))
elif info > 0:
raise np.linalg.LinAlgError(
'The %s-th diagonal element of A is zero, indicating that the matrix is '
'singular and the solutions X have not been computed.' % info)
return x
def dtrtrs(A, B, lower=0, trans=0, unitdiag=0):
"""
Wrapper for lapack dtrtrs function
:param A: Matrix A
:param B: Matrix B
:param lower: is matrix lower (true) or upper (false)
:returns:
"""
return lapack.dtrtrs(A, B, lower=lower, trans=trans, unitdiag=unitdiag)
def solve_triangular(A, B, trans=False):
"""
Solve the system Ax=B where A is lower triangular. If `trans` is True then
solve the system A'x=B.
"""
X, info = lapack.dtrtrs(A, B, lower=1, trans=int(trans))
if info != 0:
raise LinAlgError('Matrix is singular')
return X
def solve_triangular(A, B, trans=False):
"""
Solve the system Ax=B where A is lower triangular. If `trans` is True then
solve the system A'x=B.
"""
X, info = lapack.dtrtrs(A, B, lower=1, trans=int(trans))
if info != 0:
raise LinAlgError('Matrix is singular')
return X
def dtrtrs(A, B, lower=0, trans=0, unitdiag=0):
"""Wrapper for lapack dtrtrs function
:param A: Matrix A
:param B: Matrix B
:param lower: is matrix lower (true) or upper (false)
:returns:
"""
return lapack.dtrtrs(A, B, lower=lower, trans=trans, unitdiag=unitdiag)
def integrate(self) -> Tuple[float, float]:
"""
Computes an estimator of the integral as well as its variance.
:returns: estimator of integral and its variance
"""
integral_mean, kernel_mean_X = self._compute_integral_mean_and_kernel_mean()
integral_var = self.base_gp.kern.qKq() - np.square(lapack.dtrtrs(self.base_gp.gram_chol(), kernel_mean_X.T,
lower=1)[0]).sum(axis=0, keepdims=True)[0][0]
return integral_mean, integral_var
def integrate(self) -> Tuple[float, float]:
"""
Computes an estimator of the integral as well as its variance.
:returns: estimator of integral and its variance
"""
kernel_mean_X = self.base_gp.kern.qK(self.X)
integral_mean = np.dot(kernel_mean_X, self.base_gp.graminv_residual())[0, 0]
integral_var = self.base_gp.kern.qKq() - np.square(lapack.dtrtrs(self.base_gp.gram_chol(), kernel_mean_X.T,
lower=1)[0]).sum(axis=0, keepdims=True)[0][0]
return integral_mean, integral_var
def integrate(self) -> Tuple[float, float]:
"""
Computes an estimator of the integral as well as its variance.
:returns: estimator of integral and its variance
"""
integral_mean, kernel_mean_X = self._compute_integral_mean_and_kernel_mean(
)
integral_var = self.base_gp.kern.qKq() - np.square(
lapack.dtrtrs(self.base_gp.gram_chol(), kernel_mean_X.T,
lower=1)[0]).sum(axis=0, keepdims=True)[0][0]
return integral_mean, integral_var
def _raw_predict(self, Xnew):
Kx = self.kernel.K(self._X, Xnew)
mu = np.dot(Kx.T, self._woodbury_vector)
if len(mu.shape)==1:
mu = mu.reshape(-1,1)
Kxx = self.kernel.Kdiag(Xnew)
tmp = lapack.dtrtrs(self._woodbury_chol, Kx, lower=1, trans=0, unitdiag=0)[0]
var = (Kxx - np.square(tmp).sum(0))[:,None]
return mu, var
def _solve_triangular(L, b, lower=True):
'''
Solves `Lx = b` for a triangular `L` using `dtrtrs`
'''
if any(i == 0 for i in b.shape):
return np.zeros(b.shape, dtype=float)
x, info = dtrtrs(L, b, lower=lower)
if info < 0:
raise ValueError('The %s-th argument had an illegal value' % -info)
elif info > 0:
raise np.linalg.LinAlgError('Singular matrix')
return x
def get_prediction_gradients(self, X: np.ndarray) -> Tuple:
"""
Computes and returns model gradients of mean and variance at given points
:param X: points to compute gradients at, shape (num_points, dim)
:returns: Tuple of gradients of mean and variance, shapes of both (num_points, dim)
"""
# gradient of mean
d_mean_dx = (self.base_gp.kern.dK_dx1(X, self.X)
@ self.base_gp.graminv_residual())[:, :, 0].T
# gradient of variance
dKdiag_dx = self.base_gp.kern.dKdiag_dx(X)
dKxX_dx1 = self.base_gp.kern.dK_dx1(X, self.X)
lower_chol = self.base_gp.gram_chol()
KXx = self.base_gp.kern.K(self.base_gp.X, X)
graminv_KXx = lapack.dtrtrs(
lower_chol.T, (lapack.dtrtrs(lower_chol, KXx, lower=1)[0]),
lower=0)[0]
d_var_dx = dKdiag_dx - 2. * (dKxX_dx1 * np.transpose(graminv_KXx)).sum(
axis=2, keepdims=False)
return d_mean_dx, d_var_dx.T
def _raw_predict_covar(self, Xnew, Xcond):
Kx = self.kernel.K(self._X, np.vstack((Xnew,Xcond)))
tmp = lapack.dtrtrs(self._woodbury_chol, Kx, lower=1, trans=0, unitdiag=0)[0]
n = Xnew.shape[0]
tmp1 = tmp[:,:n]
tmp2 = tmp[:,n:]
Kxx = self.kernel.K(Xnew, Xcond)
var = Kxx - (tmp1.T).dot(tmp2)
Kxx_new = self.kernel.Kdiag(Xnew)
var_Xnew = (Kxx_new - np.square(tmp1).sum(0))[:,None]
return var_Xnew, var
def _raw_predict(self, Xnew):
assert Xnew.shape[1] == self.active_d, ("Somehow, the input was not project")
Kx = self.kernel.K(self._X, Xnew)
mu = np.dot(Kx.T, self._woodbury_vector)
if len(mu.shape) == 1:
mu = mu.reshape(-1, 1)
Kxx = self.kernel.Kdiag(Xnew)
tmp = lapack.dtrtrs(self._woodbury_chol, Kx, lower=1, trans=0, unitdiag=0)[0]
var = (Kxx - np.square(tmp).sum(0))[:, None]
return mu, var
def residual_variable_projection(
matrix: np.ndarray,
data: np.ndarray) -> typing.Tuple[typing.List[str], np.ndarray]:
"""Calculates the conditionally linear parameters and residual with the variable projection
method.
Parameters
----------
matrix :
The model matrix.
data : np.ndarray
The data to analyze.
"""
# TODO: Reference Kaufman paper
# Kaufman Q2 step 3
qr, tau, _, _ = lapack.dgeqrf(matrix)
# Kaufman Q2 step 4
temp, _, _ = lapack.dormqr("L",
"T",
qr,
tau,
data,
max(1, matrix.shape[1]),
overwrite_c=0)
clp, _ = lapack.dtrtrs(qr, temp)
for i in range(matrix.shape[1]):
temp[i] = 0
# Kaufman Q2 step 5
residual, _, _ = lapack.dormqr("L",
"N",
qr,
tau,
temp,
max(1, matrix.shape[1]),
overwrite_c=0)
return clp[:matrix.shape[1]], residual
def dtrtrs(A, B, lower=1, trans=0, unitdiag=0):
"""
Wrapper for lapack dtrtrs function
DTRTRS solves a triangular system of the form
A * X = B or A**T * X = B,
where A is a triangular matrix of order N, and B is an N-by-NRHS
matrix. A check is made to verify that A is nonsingular.
:param A: Matrix A(triangular)
:param B: Matrix B
:param lower: is matrix lower (true) or upper (false)
:returns: Solution to A * X = B or A**T * X = B
"""
A = np.asfortranarray(A)
#Note: B does not seem to need to be F ordered!
return lapack.dtrtrs(A, B, lower=lower, trans=trans, unitdiag=unitdiag)
def dtrtrs(A, B, lower=1, trans=0, unitdiag=0):
"""
Wrapper for lapack dtrtrs function
DTRTRS solves a triangular system of the form
A * X = B or A**T * X = B,
where A is a triangular matrix of order N, and B is an N-by-NRHS
matrix. A check is made to verify that A is nonsingular.
:param A: Matrix A(triangular)
:param B: Matrix B
:param lower: is matrix lower (true) or upper (false)
:returns: Solution to A * X = B or A**T * X = B
"""
A = np.asfortranarray(A)
#Note: B does not seem to need to be F ordered!
return lapack.dtrtrs(A, B, lower=lower, trans=trans, unitdiag=unitdiag)
def solve_linear(self, z: np.ndarray) -> np.ndarray:
lower_chol = self.gpy_model.posterior.woodbury_chol
return lapack.dtrtrs(lower_chol.T, (lapack.dtrtrs(lower_chol, z, lower=1)[0]), lower=0)[0]
def _lbf(cov_term, stats, ldet_prior): ltri = np.linalg.cholesky(cov_term) ldet_term = np.log(np.prod(np.diagonal(ltri))) * 2 temp = np.asfortranarray(stats[:, np.newaxis]) lp.dtrtrs(ltri, temp, lower=1, overwrite_b=1) return (-ldet_term - ldet_prior * stats.size + np.sum(temp ** 2)) / 2
def test_draw_samples(self, dtype, X, X_isSamples, X_cond,
X_cond_isSamples, Y_cond, Y_cond_isSamples,
rbf_lengthscale, rbf_lengthscale_isSamples,
rbf_variance, rbf_variance_isSamples, rv_shape,
num_samples):
from scipy.linalg.lapack import dtrtrs
X_mx = prepare_mxnet_array(X, X_isSamples, dtype)
X_cond_mx = prepare_mxnet_array(X_cond, X_cond_isSamples, dtype)
Y_cond_mx = prepare_mxnet_array(Y_cond, Y_cond_isSamples, dtype)
rbf_lengthscale_mx = prepare_mxnet_array(rbf_lengthscale,
rbf_lengthscale_isSamples,
dtype)
rbf_variance_mx = prepare_mxnet_array(rbf_variance,
rbf_variance_isSamples, dtype)
rand = np.random.randn(num_samples, *rv_shape)
rand_gen = MockMXNetRandomGenerator(
mx.nd.array(rand.flatten(), dtype=dtype))
rbf = RBF(2, True, 1., 1., 'rbf', None, dtype)
X_var = Variable(shape=(5, 2))
X_cond_var = Variable(shape=(8, 2))
Y_cond_var = Variable(shape=(8, 1))
gp = ConditionalGaussianProcess.define_variable(
X=X_var,
X_cond=X_cond_var,
Y_cond=Y_cond_var,
kernel=rbf,
shape=rv_shape,
dtype=dtype,
rand_gen=rand_gen).factor
variables = {
gp.X.uuid: X_mx,
gp.X_cond.uuid: X_cond_mx,
gp.Y_cond.uuid: Y_cond_mx,
gp.rbf_lengthscale.uuid: rbf_lengthscale_mx,
gp.rbf_variance.uuid: rbf_variance_mx
}
samples_rt = gp.draw_samples(F=mx.nd,
variables=variables,
num_samples=num_samples).asnumpy()
samples_np = []
for i in range(num_samples):
X_i = X[i] if X_isSamples else X
X_cond_i = X_cond[i] if X_cond_isSamples else X_cond
Y_cond_i = Y_cond[i] if Y_cond_isSamples else Y_cond
lengthscale_i = rbf_lengthscale[
i] if rbf_lengthscale_isSamples else rbf_lengthscale
variance_i = rbf_variance[
i] if rbf_variance_isSamples else rbf_variance
rand_i = rand[i]
rbf_np = GPy.kern.RBF(input_dim=2, ARD=True)
rbf_np.lengthscale = lengthscale_i
rbf_np.variance = variance_i
K_np = rbf_np.K(X_i)
Kc_np = rbf_np.K(X_cond_i, X_i)
Kcc_np = rbf_np.K(X_cond_i)
L = np.linalg.cholesky(Kcc_np)
LInvY = dtrtrs(L, Y_cond_i, lower=1, trans=0)[0]
LinvKxt = dtrtrs(L, Kc_np, lower=1, trans=0)[0]
mu = LinvKxt.T.dot(LInvY)
cov = K_np - LinvKxt.T.dot(LinvKxt)
L_cov_np = np.linalg.cholesky(cov)
sample_np = mu + L_cov_np.dot(rand_i)
samples_np.append(sample_np)
samples_np = np.array(samples_np)
assert np.issubdtype(samples_rt.dtype, dtype)
assert get_num_samples(mx.nd, samples_rt) == num_samples
assert np.allclose(samples_np, samples_rt)
def test_draw_samples_w_mean(self, dtype, X, X_isSamples, X_cond,
X_cond_isSamples, Y_cond, Y_cond_isSamples,
rbf_lengthscale, rbf_lengthscale_isSamples,
rbf_variance, rbf_variance_isSamples,
rv_shape, num_samples):
net = nn.HybridSequential(prefix='nn_')
with net.name_scope():
net.add(
nn.Dense(rv_shape[-1],
flatten=False,
activation="tanh",
in_units=X.shape[-1],
dtype=dtype))
net.initialize(mx.init.Xavier(magnitude=3))
from scipy.linalg.lapack import dtrtrs
X_mx = prepare_mxnet_array(X, X_isSamples, dtype)
X_cond_mx = prepare_mxnet_array(X_cond, X_cond_isSamples, dtype)
Y_cond_mx = prepare_mxnet_array(Y_cond, Y_cond_isSamples, dtype)
rbf_lengthscale_mx = prepare_mxnet_array(rbf_lengthscale,
rbf_lengthscale_isSamples,
dtype)
rbf_variance_mx = prepare_mxnet_array(rbf_variance,
rbf_variance_isSamples, dtype)
mean_mx = net(X_mx)
mean_np = mean_mx.asnumpy()
mean_cond_mx = net(X_cond_mx)
mean_cond_np = mean_cond_mx.asnumpy()
rand = np.random.randn(num_samples, *rv_shape)
rand_gen = MockMXNetRandomGenerator(
mx.nd.array(rand.flatten(), dtype=dtype))
rbf = RBF(2, True, 1., 1., 'rbf', None, dtype)
X_var = Variable(shape=(5, 2))
X_cond_var = Variable(shape=(8, 2))
Y_cond_var = Variable(shape=(8, 1))
mean_func = MXFusionGluonFunction(net,
num_outputs=1,
broadcastable=True)
mean_var = mean_func(X_var)
mean_cond_var = mean_func(X_cond_var)
gp = ConditionalGaussianProcess.define_variable(
X=X_var,
X_cond=X_cond_var,
Y_cond=Y_cond_var,
mean=mean_var,
mean_cond=mean_cond_var,
kernel=rbf,
shape=rv_shape,
dtype=dtype,
rand_gen=rand_gen).factor
variables = {
gp.X.uuid: X_mx,
gp.X_cond.uuid: X_cond_mx,
gp.Y_cond.uuid: Y_cond_mx,
gp.rbf_lengthscale.uuid: rbf_lengthscale_mx,
gp.rbf_variance.uuid: rbf_variance_mx,
gp.mean.uuid: mean_mx,
gp.mean_cond.uuid: mean_cond_mx
}
samples_rt = gp.draw_samples(F=mx.nd,
variables=variables,
num_samples=num_samples).asnumpy()
samples_np = []
for i in range(num_samples):
X_i = X[i] if X_isSamples else X
X_cond_i = X_cond[i] if X_cond_isSamples else X_cond
Y_cond_i = Y_cond[i] if Y_cond_isSamples else Y_cond
Y_cond_i = Y_cond_i - mean_cond_np[
i] if X_cond_isSamples else Y_cond_i - mean_cond_np[0]
lengthscale_i = rbf_lengthscale[
i] if rbf_lengthscale_isSamples else rbf_lengthscale
variance_i = rbf_variance[
i] if rbf_variance_isSamples else rbf_variance
rand_i = rand[i]
rbf_np = GPy.kern.RBF(input_dim=2, ARD=True)
rbf_np.lengthscale = lengthscale_i
rbf_np.variance = variance_i
K_np = rbf_np.K(X_i)
Kc_np = rbf_np.K(X_cond_i, X_i)
Kcc_np = rbf_np.K(X_cond_i)
L = np.linalg.cholesky(Kcc_np)
LInvY = dtrtrs(L, Y_cond_i, lower=1, trans=0)[0]
LinvKxt = dtrtrs(L, Kc_np, lower=1, trans=0)[0]
mu = LinvKxt.T.dot(LInvY)
cov = K_np - LinvKxt.T.dot(LinvKxt)
L_cov_np = np.linalg.cholesky(cov)
sample_np = mu + L_cov_np.dot(rand_i)
samples_np.append(sample_np)
samples_np = np.array(samples_np) + mean_np
assert np.issubdtype(samples_rt.dtype, dtype)
assert get_num_samples(mx.nd, samples_rt) == num_samples
assert np.allclose(samples_np, samples_rt)
def dtrtrs(A, B, lower=1, trans=0, unitdiag=0):
A = np.asfortranarray(A)
#Note: B does not seem to need to be F ordered!
return lapack.dtrtrs(A, B, lower=lower, trans=trans, unitdiag=unitdiag)
def test_log_pdf(self, dtype, X, X_isSamples, X_cond, X_cond_isSamples,
Y_cond, Y_cond_isSamples, rbf_lengthscale,
rbf_lengthscale_isSamples, rbf_variance,
rbf_variance_isSamples, rv, rv_isSamples, num_samples):
from scipy.linalg.lapack import dtrtrs
X_mx = prepare_mxnet_array(X, X_isSamples, dtype)
X_cond_mx = prepare_mxnet_array(X_cond, X_cond_isSamples, dtype)
Y_cond_mx = prepare_mxnet_array(Y_cond, Y_cond_isSamples, dtype)
rbf_lengthscale_mx = prepare_mxnet_array(rbf_lengthscale,
rbf_lengthscale_isSamples,
dtype)
rbf_variance_mx = prepare_mxnet_array(rbf_variance,
rbf_variance_isSamples, dtype)
rv_mx = prepare_mxnet_array(rv, rv_isSamples, dtype)
rv_shape = rv.shape[1:] if rv_isSamples else rv.shape
rbf = RBF(2, True, 1., 1., 'rbf', None, dtype)
X_var = Variable(shape=(5, 2))
X_cond_var = Variable(shape=(8, 2))
Y_cond_var = Variable(shape=(8, 1))
gp = ConditionalGaussianProcess.define_variable(X=X_var,
X_cond=X_cond_var,
Y_cond=Y_cond_var,
kernel=rbf,
shape=rv_shape,
dtype=dtype).factor
variables = {
gp.X.uuid: X_mx,
gp.X_cond.uuid: X_cond_mx,
gp.Y_cond.uuid: Y_cond_mx,
gp.rbf_lengthscale.uuid: rbf_lengthscale_mx,
gp.rbf_variance.uuid: rbf_variance_mx,
gp.random_variable.uuid: rv_mx
}
log_pdf_rt = gp.log_pdf(F=mx.nd, variables=variables).asnumpy()
log_pdf_np = []
for i in range(num_samples):
X_i = X[i] if X_isSamples else X
X_cond_i = X_cond[i] if X_cond_isSamples else X_cond
Y_cond_i = Y_cond[i] if Y_cond_isSamples else Y_cond
lengthscale_i = rbf_lengthscale[
i] if rbf_lengthscale_isSamples else rbf_lengthscale
variance_i = rbf_variance[
i] if rbf_variance_isSamples else rbf_variance
rv_i = rv[i] if rv_isSamples else rv
rbf_np = GPy.kern.RBF(input_dim=2, ARD=True)
rbf_np.lengthscale = lengthscale_i
rbf_np.variance = variance_i
K_np = rbf_np.K(X_i)
Kc_np = rbf_np.K(X_cond_i, X_i)
Kcc_np = rbf_np.K(X_cond_i)
L = np.linalg.cholesky(Kcc_np)
LInvY = dtrtrs(L, Y_cond_i, lower=1, trans=0)[0]
LinvKxt = dtrtrs(L, Kc_np, lower=1, trans=0)[0]
mu = LinvKxt.T.dot(LInvY)
cov = K_np - LinvKxt.T.dot(LinvKxt)
log_pdf_np.append(
multivariate_normal.logpdf(rv_i[:, 0], mean=mu[:, 0], cov=cov))
log_pdf_np = np.array(log_pdf_np)
isSamples_any = any([
X_isSamples, rbf_lengthscale_isSamples, rbf_variance_isSamples,
rv_isSamples
])
assert np.issubdtype(log_pdf_rt.dtype, dtype)
assert array_has_samples(mx.nd, log_pdf_rt) == isSamples_any
if isSamples_any:
assert get_num_samples(mx.nd, log_pdf_rt) == num_samples
assert np.allclose(log_pdf_np, log_pdf_rt)
