Cython module: bq_c
The functions in this module are called by the
BQ object, and in most cases
shouldn’t need to be called directly by user code. However, they are
provided here for reference.
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bayesian_quadrature.bq_c.Z_mean(__Pyx_memviewslice x_sc, __Pyx_memviewslice alpha_l, float64_t h_l, __Pyx_memviewslice w_l, __Pyx_memviewslice mu, __Pyx_memviewslice cov)
Compute the mean of the integral:
\[Z = \int \ell(x)\mathcal{N}(x \big\vert \mu, \Sigma)\ \mathrm{d}x\]
where the mean is defined as:
\[m(Z) = E[Z \big\vert \bar{\ell}(x_sc)]\]
| Parameters : | x_sc : float64_t[::1, :]
\(d\times n\) vector of observed and candidate locations
alpha_l : float64_t[::1]
\(K_\ell(x_{sc}, x_{sc})^{-1}\bar{\ell}(x_{sc})\)
h_l : float64_t
output scale parameter for kernel \(K_\ell\)
w_l : float64_t[::1]
\(d\) vector of lengthscales for kernel \(K_\ell\)
mu : float64_t[::1]
cov : float64_t[::1, :]
\(d\times d\) prior covariance
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| Returns : | out : mean of \(Z\)
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bayesian_quadrature.bq_c.Z_var(__Pyx_memviewslice x_s, __Pyx_memviewslice x_sc, __Pyx_memviewslice alpha_l, __Pyx_memviewslice L_tl, float64_t h_l, __Pyx_memviewslice w_l, float64_t h_tl, __Pyx_memviewslice w_tl, __Pyx_memviewslice mu, __Pyx_memviewslice cov)
Compute the variance of the integral:
\[Z = \int \ell(x)\mathcal{N}(x \big\vert \mu, \Sigma)\ \mathrm{d}x\]
where the variance is defined as:
\[V(Z) = \int\int m_\ell(x) C_{\log\ell}(x, x^\prime) m_\ell(x^\prime) p(x)p(x^\prime)\ \mathrm{d}x\ \mathrm{d}x^\prime\]
| Parameters : | x_s : float64_t[::1, :]
\(d\times n_s\) vector of observed locations
x_sc : float64_t[::1, :]
\(d\times n_{sc}\) vector of observed and candidate locations
alpha_l : float64_t[::1]
\(K_\ell(x_{sc}, x_{sc})^{-1}\bar{\ell}(x_{sc})\)
L_tl : float64_t[::1, :]
lower-triangular Cholesky factor of \(K_{\log\ell}(x_{s}, x_{s})\)
h_l : float64_t
output scale parameter for kernel \(K_\ell\)
w_l : float64_t[::1]
\(d\) vector of lengthscales for kernel \(K_\ell\)
h_tl : float64_t
output scale parameter for kernel \(K_{\log\ell}\)
w_tl : float64_t[::1]
\(d\) vector of lengthscales for kernel \(K_{\log\ell}\)
mu : float64_t[::1]
cov : float64_t[::1, :]
\(d\times d\) prior covariance
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| Returns : | out : variance of \(Z\)
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bayesian_quadrature.bq_c.approx_Z_mean(__Pyx_memviewslice xo, __Pyx_memviewslice p_xo, __Pyx_memviewslice l)
Approximate the mean of the integral:
\[Z = \int \ell(x)p(x)\ \mathrm{d}x\]
| Parameters : | xo : float64_t[::1, :]
\(d\times n\) vector of approximation locations
p_xo : float64_t[::1]
\(n\) vector prior probabilities at approximation locations
l : float64_t[::1]
\(n\) vector of likelihood evaluated at approximation locations
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| Returns : | out : approximate mean of \(Z\)
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bayesian_quadrature.bq_c.approx_Z_var(__Pyx_memviewslice xo, __Pyx_memviewslice p_xo, __Pyx_memviewslice m_l, __Pyx_memviewslice C_tl)
Approximate the variance of the integral:
\[Z = \int \ell(x)p(x)\ \mathrm{d}x\]
where the variance is defined as:
\[V(Z) = \int\int m_\ell(x) C_{\log\ell}(x, x^\prime) m_\ell(x^\prime) p(x)p(x^\prime)\ \mathrm{d}x\ \mathrm{d}x^\prime\]
| Parameters : | xo : float64_t[::1, :]
\(d\times n\) vector of approximation locations
p_xo : float64_t[::1]
\(n\) vector prior probabilities at approximation locations
m_l : float64_t[::1]
\(n\) vector of likelihoods evaluated at approximation locations
C_tl : float64_t[::1, :]
\(n\times n\) covariance matrix for \(\log\ell\) evaluated at approximation locations
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| Returns : | out : approximate variance of \(Z\)
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bayesian_quadrature.bq_c.approx_expected_squared_mean_and_mean(__Pyx_memviewslice l_sc, __Pyx_memviewslice K_l, float64_t tm_a, float64_t tC_a, __Pyx_memviewslice xo, __Pyx_memviewslice p_xo, __Pyx_memviewslice Kxxo)
Approximates the expected squared mean and expected mean of
\(Z\) given a new observation at \(x_a\).
\[E[m(Z)^2 \big\vert x_a] = \int m(Z | \ell_s, \ell_a)^2 \mathcal{N}(\log\ell_a | \hat{m}_a, \hat{C}_a)\ \mathrm{d}\log\ell_a\]
| Parameters : | l_sc : float64_t[::1]
\(n_{sc}\) vector of observed and candidate locations
K_l : float64_t[::1, :]
\(n_{sca}\times n_{sca}\) kernel matrix \(K_{\ell}(x_{sca}, x_{sca})\)
tm_a : float64_t
prior mean of \(\log\ell_a\)
tC_a : float64_t
prior variance of \(\log\ell_a\)
xo : float64_t[::1, :]
\(d\times m\) vector of approximation locations
Kxxo : float64_t[::1, :]
\(n_{sca}\times m\) kernel matrix
mu : float64_t[::1]
cov : float64_t[::1, :]
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| Returns : | out : approximate expected squared mean and expected mean of \(Z\)
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bayesian_quadrature.bq_c.expected_squared_mean_and_mean(__Pyx_memviewslice l_sc, __Pyx_memviewslice K_l, float64_t tm_a, float64_t tC_a, __Pyx_memviewslice x_sca, float64_t h_l, __Pyx_memviewslice w_l, __Pyx_memviewslice mu, __Pyx_memviewslice cov)
Computes the expected squared mean and expected mean of \(Z\)
given a new observation at \(x_a\).
\[E[m(Z)^2 \big\vert x_a] = \int m(Z | \ell_s, \ell_a)^2 \mathcal{N}(\log\ell_a | \hat{m}_a, \hat{C}_a)\ \mathrm{d}\log\ell_a\]
| Parameters : | l_sc : float64_t[::1]
\(n_{sc}\) vector of observed and candidate locations
K_l : float64_t[::1, :]
\(n_{sca}\times n_{sca}\) kernel matrix \(K_{\ell}(x_{sca}, x_{sca})\)
tm_a : float64_t
prior mean of \(\log\ell_a\)
tC_a : float64_t
prior variance of \(\log\ell_a\)
x_sca : float64_t[::1, :]
\(d\times n_{sca}\) input vector
h_l : float64_t
output scale kernel parameter for \(K_\ell\)
w_l : float64_t[::1]
\(d\) vector of lengthscales for \(K_\ell\)
mu : float64_t[::1]
cov : float64_t[::1, :]
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| Returns : | out : expected squared mean and expected mean of \(Z\)
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bayesian_quadrature.bq_c.filter_candidates(__Pyx_memviewslice x_c, __Pyx_memviewslice x_s, float64_t thresh)
Given a vector of possible candidate locations, \(x_c\),
filter out locations which are close to one or more observations
\(x_s\), or to other candidate locations.
| Parameters : | x_c : float64_t[::1]
potential candidate locations
x_s : float64_t[::1]
thresh : float64_t
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bayesian_quadrature.bq_c.improve_covariance_conditioning(__Pyx_memviewslice M, __Pyx_memviewslice jitters, __Pyx_memviewslice idx)
Add noise to the indices idx along the diagonal of M. Update
the corresponding locations in the jitters vector to include
this new noise.
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bayesian_quadrature.bq_c.p_x_gaussian(__Pyx_memviewslice p_x, __Pyx_memviewslice x, __Pyx_memviewslice mu, __Pyx_memviewslice cov)
Computes the Gaussian PDF of input locations \(x\).
| Parameters : | p_x : float64_t[::1]
\(n\) output vector of probabilities
x : float64_t[::1, :]
\(d\times n\) input locations
mu : float64_t[::1]
cov : float64_t[::1, :]
\(d\times d\) covariance matrix
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bayesian_quadrature.bq_c.p_x_vonmises(__Pyx_memviewslice p_x, __Pyx_memviewslice x, float64_t mu, float64_t kappa)
Computes the Von Mises PDF of input locations \(x\).
| Parameters : | p_x : float64_t[::1]
\(n\) output vector of probabilities
x : float64_t[::1, :]
\(n\) input vector of locations
mu : float64_t[::1]
kappa : float64_t[::1, :]
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bayesian_quadrature.bq_c.remove_jitter(__Pyx_memviewslice M, __Pyx_memviewslice jitters, __Pyx_memviewslice idx)
Remove noise from the indices idx along the diagonal of M. Set
the corresponding locations in the jitters vector to zero.
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bayesian_quadrature.bq_c.vonmises_logpdf(float64_t x, float64_t mu, float64_t kappa) → float64_t
Computes the log-PDF for the Von Mises distribution at location
\(x\):
\[p(x | \mu, \kappa) = \frac{\exp(\kappa\cos(x-\mu))}{2\pi I_0(\kappa)}\]
Where \(I_0\) is the Bessel function of order 0.
| Parameters : | x : float64_t
mu : float64_t
kappa : float64_t
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| Returns : | out : value of the log-PDF
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