dmnorm: Density of the (conditional) multivariate normal distribution
In mnorm: Multivariate Normal Distribution
dmnorm R Documentation
Density of the (conditional) multivariate normal distribution
Description
This function calculates and differentiates the density of
the (conditional) multivariate normal distribution.
Usage
dmnorm(
x,
mean,
sigma,
given_ind = numeric(),
log = FALSE,
grad_x = FALSE,
grad_sigma = FALSE,
is_validation = TRUE,
control = NULL,
n_cores = 1L
)
Arguments
x
numeric vector representing the point at which the density
should be calculated. If x is a matrix, then each row determines
a new point.
mean
numeric vector representing an expectation of the multivariate
normal vector (distribution).
sigma
positively defined numeric matrix representing the covariance
matrix of the multivariate normal vector (distribution).
given_ind
numeric vector representing indexes of a multivariate
normal vector which are conditioned on the values of x with the
corresponding indexes, i.e., x[given_ind] or x[, given_ind] if
x is a matrix.
log
logical; if TRUE, then the probabilities (or densities) p
are given as log(p), and the derivatives will be given
with respect to log(p).
grad_x
logical; if TRUE, then the vector of partial derivatives
of the density function will be calculated with respect to each
element of x. If x is a matrix, then the gradients will be
estimated for each row of x.
grad_sigma
logical; if TRUE, then the vector of partial
derivatives (gradient) of the density function will be calculated with
respect to each element of sigma. If x is a matrix, then
the gradients will be estimated for each row of x.
is_validation
logical value indicating whether input
arguments should be validated. Set it to FALSE to get a
performance boost (default value is TRUE).
control
a list of control parameters. See Details.
n_cores
positive integer representing the number of CPU cores
used for parallel computing. Currently it is not recommended to set
n_cores > 1 if vectorized arguments include less than 100000 elements.
Details
Consider the notations from the Details section of
cmnorm. The function calculates the density
f(x^{(d)}|x^{(g)}) of conditioned multivariate normal vector
X_{I_{d}} | X_{I_{g}} = x^{(g)}, where x^{(d)} is a subvector of
x associated with X_{I_{d}}, i.e., the unconditioned components.
Therefore x[given_ind] represents x^{(g)}, while
x[-given_ind] is x^{(d)}.
If grad_x is TRUE, then the function additionally estimates the
gradient with respect to both the unconditioned and conditioned components:
\nabla f(x^{(d)}|x^{(g)})=
\left(\frac{\partial f(x^{(d)}|x^{(g)})}{\partial x_{1}}
,...,
\frac{\partial f(x^{(d)}|x^{(g)})}{\partial x_{m}}\right),
where each x_{i} belongs either to x^{(d)} or x^{(g)}
depending on whether i\in I_{d} or i\in I_{g}.
In particular, the subgradients of the density function with respect to
x^{(d)} and x^{(g)} are of the form:
\nabla_{x^{(d)}}\ln f(x^{(d)}|x^{(g)}) =
-\left(x^{(d)}-\mu_{c}\right)\Sigma_{c}^{-1}
\nabla_{x^{(g)}}\ln f(x^{(d)}|x^{(g)}) =
-\nabla_{x^{(d)}}f(x^{(d)}|x^{(g)})\Sigma_{d,g}\Sigma_{g,g}^{-1}
If grad_sigma is TRUE, then the function additionally estimates
the gradient with respect to the elements of covariance matrix \Sigma.
For i\in I_{d} and j\in I_{d}, the function calculates:
\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial \Sigma_{i, j}} =
\left(\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial x_{i}} \times
\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial x_{j}} -
\Sigma_{c,(i, j)}^{-1}\right) /
\left(1 + I(i=j)\right),
where I(i=j) is an indicator function which equals 1 when
the condition i=j is satisfied and 0 otherwise.
For i\in I_{d} and j\in I_{g}, the following formula is used:
\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial \Sigma_{i, j}} =
-\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial x_{i}} \times
\left(\left(x^{(g)}-\mu_{g}\right)\Sigma_{g,g}^{-1}\right)_{q_{g}(j)}-
-\sum\limits_{k=1}^{n_{d}}(1+I(q_{d}(i)=k))\times
(\Sigma_{d,g}\Sigma_{g,g}^{-1})_{k,q_{g}(j)}\times
\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial \Sigma_{i, q^{-1}_{d}(k)}},
where q_{g}(j)=\sum\limits_{k=1}^{m} I\left(I_{g,k} \leq j\right)
and q_{d}(i)=\sum\limits_{k=1}^{m} I\left(I_{d,k} \leq i\right)
represent the order of the i-th and j-th elements
in I_{g} and I_{d}, respectively, i.e.,
x_{i}=x^{(d)}_{q_{d}(i)}=x_{I_{d, q_{d}(i)}} and
x_{j}=x^{(g)}_{q_{g}(j)}=x_{I_{g, q_{g}(j)}}.
Note that q_{g}(j)^{-1} and q_{d}(i)^{-1} are inverse functions.
The number of conditioned and unconditioned components is denoted by
n_{g}=\sum\limits_{k=1}^{m}I(k\in I_{g}) and
n_{d}=\sum\limits_{k=1}^{m}I(k\in I_{d}) respectively.
For the case i\in I_{g} and j\in I_{d}, the formula is similar.
For i\in I_{g} and j\in I_{g}, the following formula is used:
\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial \Sigma_{i, j}} =
-\nabla_{x^{(d)}}\ln f(x^{(d)}|x^{(g)})\times
\left(x^{(g)}\times(\Sigma_{d,g} \times \Sigma_{g,g}^{-1} \times I_{g}^{*} \times
\Sigma_{g,g}^{-1})^{T}\right)^T -
-\sum\limits_{k_{1}=1}^{n_{d}}\sum\limits_{k_{2}=k_{1}}^{n_{d}}
\frac{\partial \ln f(x^{(d)}|x^{(g)})}
{\partial \Sigma_{q_{d}(k_{1})^{-1}, q_{d}(k_{2})^{-1}}}
\left(\Sigma_{d,g} \times \Sigma_{g,g}^{-1} \times I_{g}^{*} \times
\Sigma_{g,g}^{-1}\times\Sigma_{d,g}^T\right)_{q_{d}(k_{1})^{-1},
q_{d}(k_{2})^{-1}},
where I_{g}^{*} is a square n_{g}-dimensional matrix of
zeros except I_{g,(i,j)}^{*}=I_{g,(j,i)}^{*}=1.
Argument given_ind represents I_{g} and it should not
contain any duplicates. The order of given_ind elements
does not matter, so it has no impact on the output.
More details on aforementioned differentiation formulas could
be found in the appendix of E. Kossova and B. Potanin (2018).
Currently control has no input arguments intended for
the users. This argument is used for some internal purposes
of the package.
Value
This function returns an object of class "mnorm_dmnorm".
An object of class "mnorm_dmnorm" is a list containing the
following components:
-
den - the density function value at x.
-
grad_x - a gradient of the density with respect to x, if
the grad_x or grad_sigma input argument is set to TRUE.
-
grad_sigma - a gradient with respect to the elements of
sigma, if the grad_sigma input argument is set to TRUE.
If log is TRUE, then den is a log-density,
so the outputs grad_x and grad_sigma are calculated with
respect to the log-density.
Output grad_x is a Jacobian matrix whose rows are the gradients of
the density function calculated for each row of x. Therefore,
grad_x[i, j] is a derivative of the density function with respect to
the j-th argument at the point x[i, ].
Output grad_sigma is a 3D array such that grad_sigma[i, j, k]
is a partial derivative of the density function with respect to the
sigma[i, j] estimated for the observation x[k, ].
References
E. Kossova, B. Potanin (2018).
Heckman method and switching regression model multivariate generalization.
Applied Econometrics, vol. 50, pages 114-143.
Examples
# Consider a multivariate normal vector:
# X = (X1, X2, X3, X4, X5) ~ N(mean, sigma)
# Prepare the multivariate normal vector parameters
# the expected value
mean <- c(-2, -1, 0, 1, 2)
n_dim <- length(mean)
# the correlation matrix
cor <- c( 1, 0.1, 0.2, 0.3, 0.4,
0.1, 1, -0.1, -0.2, -0.3,
0.2, -0.1, 1, 0.3, 0.2,
0.3, -0.2, 0.3, 1, -0.05,
0.4, -0.3, 0.2, -0.05, 1)
cor <- matrix(cor, ncol = n_dim, nrow = n_dim, byrow = TRUE)
# the covariance matrix
sd_mat <- diag(c(1, 1.5, 2, 2.5, 3))
sigma <- sd_mat %*% cor %*% t(sd_mat)
# Estimate the density of X at the point (-1, 0, 1, 2, 3)
x <- c(-1, 0, 1, 2, 3)
d.list <- dmnorm(x = x, mean = mean, sigma = sigma)
d <- d.list$den
print(d)
# Estimate the density of X at points
# x=(-1, 0, 1, 2, 3) and y=(-1.5, -1.2, 0, 1.2, 1.5)
y <- c(-1.5, -1.2, 0, 1.2, 1.5)
xy <- rbind(x, y)
d.list.1 <- dmnorm(x = xy, mean = mean, sigma = sigma)
d.1 <- d.list.1$den
print(d.1)
# Estimate the density of Xc=(X2, X4, X5 | X1 = -1, X3 = 1) at
# the point xd=(0, 2, 3) given the conditioning values xg = (-1, 1)
given_ind <- c(1, 3)
d.list.2 <- dmnorm(x = x, mean = mean, sigma = sigma,
given_ind = given_ind)
d.2 <- d.list.2$den
print(d.2)
# Estimate the gradient of the density with respect to the argument and
# covariance matrix at points 'x' and 'y'
d.list.3 <- dmnorm(x = xy, mean = mean, sigma = sigma,
grad_x = TRUE, grad_sigma = TRUE)
# Gradient with respect to the argument
grad_x.3 <- d.list.3$grad_x
# at the point 'x'
print(grad_x.3[1, ])
# at the point 'y'
print(grad_x.3[2, ])
# Partial derivative at the point 'y' with respect
# to the 3-rd argument
print(grad_x.3[2, 3])
# Gradient respect to the covariance matrix
grad_sigma.3 <- d.list.3$grad_sigma
# Partial derivative with respect to sigma(3, 5) at
# point 'y'
print(grad_sigma.3[3, 5, 2])
# Estimate the gradient of the log-density function of
# Xc=(X2, X4, X5 | X1 = -1, X3 = 1) and Yc=(X2, X4, X5 | X1 = -1.5, X3 = 0)
# with respect to the argument and covariance matrix at
# points xd=(0, 2, 3) and yd=(-1.2, 0, 1.5), respectively, given
# the conditioning values xg=(-1, 1) and yg=(-1.5, 0) correspondingly
d.list.4 <- dmnorm(x = xy, mean = mean, sigma = sigma,
grad_x = TRUE, grad_sigma = TRUE,
given_ind = given_ind, log = TRUE)
# Gradient respect to the argument
grad_x.4 <- d.list.4$grad_x
# at point 'xd'
print(grad_x.4[1, ])
# at point 'yd'
print(grad_x.4[2, ])
# Partial derivative at the point 'xd' with respect to 'xg[2]'
print(grad_x.4[1, 3])
# Partial derivative at the point 'yd' with respect to 'yd[5]'
print(grad_x.4[2, 5])
# Gradient with respect to the covariance matrix
grad_sigma.4 <- d.list.4$grad_sigma
# Partial derivative with respect to sigma(3, 5) at
# point 'yd'
print(grad_sigma.4[3, 5, 2])
# Compare analytical gradients from the previous example with
# their numeric (forward difference) analogues at point 'xd'
# given the conditioning 'xg'
delta <- 1e-6
grad_x.num <- rep(NA, 5)
grad_sigma.num <- matrix(NA, nrow = 5, ncol = 5)
for (i in 1:5)
{
x.delta <- x
x.delta[i] <- x[i] + delta
d.list.delta <- dmnorm(x = x.delta, mean = mean, sigma = sigma,
grad_x = TRUE, grad_sigma = TRUE,
given_ind = given_ind, log = TRUE)
grad_x.num[i] <- (d.list.delta$den - d.list.4$den[1]) / delta
for (j in 1:5)
{
sigma.delta <- sigma
sigma.delta[i, j] <- sigma[i, j] + delta
sigma.delta[j, i] <- sigma[j, i] + delta
d.list.delta <- dmnorm(x = x, mean = mean, sigma = sigma.delta,
grad_x = TRUE, grad_sigma = TRUE,
given_ind = given_ind, log = TRUE)
grad_sigma.num[i, j] <- (d.list.delta$den - d.list.4$den[1]) / delta
}
}
# Comparison of gradients with respect to the argument
h.x <- cbind(analytical = grad_x.4[1, ], numeric = grad_x.num)
rownames(h.x) <- c("xg[1]", "xd[1]", "xg[2]", "xd[3]", "xd[4]")
print(h.x)
# Comparison of gradients with respect to the covariance matrix
h.sigma <- list(analytical = grad_sigma.4[, , 1], numeric = grad_sigma.num)
print(h.sigma)
mnorm documentation built on April 14, 2026, 5:07 p.m.
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| dmnorm | R Documentation |
Density of the (conditional) multivariate normal distribution
Description
This function calculates and differentiates the density of the (conditional) multivariate normal distribution.
Usage
dmnorm(
x,
mean,
sigma,
given_ind = numeric(),
log = FALSE,
grad_x = FALSE,
grad_sigma = FALSE,
is_validation = TRUE,
control = NULL,
n_cores = 1L
)
Arguments
x |
numeric vector representing the point at which the density
should be calculated. If |
mean |
numeric vector representing an expectation of the multivariate normal vector (distribution). |
sigma |
positively defined numeric matrix representing the covariance matrix of the multivariate normal vector (distribution). |
given_ind |
numeric vector representing indexes of a multivariate
normal vector which are conditioned on the values of |
log |
logical; if |
grad_x |
logical; if |
grad_sigma |
logical; if |
is_validation |
logical value indicating whether input
arguments should be validated. Set it to |
control |
a list of control parameters. See Details. |
n_cores |
positive integer representing the number of CPU cores
used for parallel computing. Currently it is not recommended to set
|
Details
Consider the notations from the Details section of
cmnorm. The function calculates the density
f(x^{(d)}|x^{(g)}) of conditioned multivariate normal vector
X_{I_{d}} | X_{I_{g}} = x^{(g)}, where x^{(d)} is a subvector of
x associated with X_{I_{d}}, i.e., the unconditioned components.
Therefore x[given_ind] represents x^{(g)}, while
x[-given_ind] is x^{(d)}.
If grad_x is TRUE, then the function additionally estimates the
gradient with respect to both the unconditioned and conditioned components:
\nabla f(x^{(d)}|x^{(g)})=
\left(\frac{\partial f(x^{(d)}|x^{(g)})}{\partial x_{1}}
,...,
\frac{\partial f(x^{(d)}|x^{(g)})}{\partial x_{m}}\right),
where each x_{i} belongs either to x^{(d)} or x^{(g)}
depending on whether i\in I_{d} or i\in I_{g}.
In particular, the subgradients of the density function with respect to
x^{(d)} and x^{(g)} are of the form:
\nabla_{x^{(d)}}\ln f(x^{(d)}|x^{(g)}) =
-\left(x^{(d)}-\mu_{c}\right)\Sigma_{c}^{-1}
\nabla_{x^{(g)}}\ln f(x^{(d)}|x^{(g)}) =
-\nabla_{x^{(d)}}f(x^{(d)}|x^{(g)})\Sigma_{d,g}\Sigma_{g,g}^{-1}
If grad_sigma is TRUE, then the function additionally estimates
the gradient with respect to the elements of covariance matrix \Sigma.
For i\in I_{d} and j\in I_{d}, the function calculates:
\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial \Sigma_{i, j}} =
\left(\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial x_{i}} \times
\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial x_{j}} -
\Sigma_{c,(i, j)}^{-1}\right) /
\left(1 + I(i=j)\right),
where I(i=j) is an indicator function which equals 1 when
the condition i=j is satisfied and 0 otherwise.
For i\in I_{d} and j\in I_{g}, the following formula is used:
\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial \Sigma_{i, j}} =
-\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial x_{i}} \times
\left(\left(x^{(g)}-\mu_{g}\right)\Sigma_{g,g}^{-1}\right)_{q_{g}(j)}-
-\sum\limits_{k=1}^{n_{d}}(1+I(q_{d}(i)=k))\times
(\Sigma_{d,g}\Sigma_{g,g}^{-1})_{k,q_{g}(j)}\times
\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial \Sigma_{i, q^{-1}_{d}(k)}},
where q_{g}(j)=\sum\limits_{k=1}^{m} I\left(I_{g,k} \leq j\right)
and q_{d}(i)=\sum\limits_{k=1}^{m} I\left(I_{d,k} \leq i\right)
represent the order of the i-th and j-th elements
in I_{g} and I_{d}, respectively, i.e.,
x_{i}=x^{(d)}_{q_{d}(i)}=x_{I_{d, q_{d}(i)}} and
x_{j}=x^{(g)}_{q_{g}(j)}=x_{I_{g, q_{g}(j)}}.
Note that q_{g}(j)^{-1} and q_{d}(i)^{-1} are inverse functions.
The number of conditioned and unconditioned components is denoted by
n_{g}=\sum\limits_{k=1}^{m}I(k\in I_{g}) and
n_{d}=\sum\limits_{k=1}^{m}I(k\in I_{d}) respectively.
For the case i\in I_{g} and j\in I_{d}, the formula is similar.
For i\in I_{g} and j\in I_{g}, the following formula is used:
\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial \Sigma_{i, j}} =
-\nabla_{x^{(d)}}\ln f(x^{(d)}|x^{(g)})\times
\left(x^{(g)}\times(\Sigma_{d,g} \times \Sigma_{g,g}^{-1} \times I_{g}^{*} \times
\Sigma_{g,g}^{-1})^{T}\right)^T -
-\sum\limits_{k_{1}=1}^{n_{d}}\sum\limits_{k_{2}=k_{1}}^{n_{d}}
\frac{\partial \ln f(x^{(d)}|x^{(g)})}
{\partial \Sigma_{q_{d}(k_{1})^{-1}, q_{d}(k_{2})^{-1}}}
\left(\Sigma_{d,g} \times \Sigma_{g,g}^{-1} \times I_{g}^{*} \times
\Sigma_{g,g}^{-1}\times\Sigma_{d,g}^T\right)_{q_{d}(k_{1})^{-1},
q_{d}(k_{2})^{-1}},
where I_{g}^{*} is a square n_{g}-dimensional matrix of
zeros except I_{g,(i,j)}^{*}=I_{g,(j,i)}^{*}=1.
Argument given_ind represents I_{g} and it should not
contain any duplicates. The order of given_ind elements
does not matter, so it has no impact on the output.
More details on aforementioned differentiation formulas could be found in the appendix of E. Kossova and B. Potanin (2018).
Currently control has no input arguments intended for
the users. This argument is used for some internal purposes
of the package.
Value
This function returns an object of class "mnorm_dmnorm".
An object of class "mnorm_dmnorm" is a list containing the
following components:
-
den- the density function value atx. -
grad_x- a gradient of the density with respect tox, if thegrad_xorgrad_sigmainput argument is set toTRUE. -
grad_sigma- a gradient with respect to the elements ofsigma, if thegrad_sigmainput argument is set toTRUE.
If log is TRUE, then den is a log-density,
so the outputs grad_x and grad_sigma are calculated with
respect to the log-density.
Output grad_x is a Jacobian matrix whose rows are the gradients of
the density function calculated for each row of x. Therefore,
grad_x[i, j] is a derivative of the density function with respect to
the j-th argument at the point x[i, ].
Output grad_sigma is a 3D array such that grad_sigma[i, j, k]
is a partial derivative of the density function with respect to the
sigma[i, j] estimated for the observation x[k, ].
References
E. Kossova, B. Potanin (2018). Heckman method and switching regression model multivariate generalization. Applied Econometrics, vol. 50, pages 114-143.
Examples
# Consider a multivariate normal vector:
# X = (X1, X2, X3, X4, X5) ~ N(mean, sigma)
# Prepare the multivariate normal vector parameters
# the expected value
mean <- c(-2, -1, 0, 1, 2)
n_dim <- length(mean)
# the correlation matrix
cor <- c( 1, 0.1, 0.2, 0.3, 0.4,
0.1, 1, -0.1, -0.2, -0.3,
0.2, -0.1, 1, 0.3, 0.2,
0.3, -0.2, 0.3, 1, -0.05,
0.4, -0.3, 0.2, -0.05, 1)
cor <- matrix(cor, ncol = n_dim, nrow = n_dim, byrow = TRUE)
# the covariance matrix
sd_mat <- diag(c(1, 1.5, 2, 2.5, 3))
sigma <- sd_mat %*% cor %*% t(sd_mat)
# Estimate the density of X at the point (-1, 0, 1, 2, 3)
x <- c(-1, 0, 1, 2, 3)
d.list <- dmnorm(x = x, mean = mean, sigma = sigma)
d <- d.list$den
print(d)
# Estimate the density of X at points
# x=(-1, 0, 1, 2, 3) and y=(-1.5, -1.2, 0, 1.2, 1.5)
y <- c(-1.5, -1.2, 0, 1.2, 1.5)
xy <- rbind(x, y)
d.list.1 <- dmnorm(x = xy, mean = mean, sigma = sigma)
d.1 <- d.list.1$den
print(d.1)
# Estimate the density of Xc=(X2, X4, X5 | X1 = -1, X3 = 1) at
# the point xd=(0, 2, 3) given the conditioning values xg = (-1, 1)
given_ind <- c(1, 3)
d.list.2 <- dmnorm(x = x, mean = mean, sigma = sigma,
given_ind = given_ind)
d.2 <- d.list.2$den
print(d.2)
# Estimate the gradient of the density with respect to the argument and
# covariance matrix at points 'x' and 'y'
d.list.3 <- dmnorm(x = xy, mean = mean, sigma = sigma,
grad_x = TRUE, grad_sigma = TRUE)
# Gradient with respect to the argument
grad_x.3 <- d.list.3$grad_x
# at the point 'x'
print(grad_x.3[1, ])
# at the point 'y'
print(grad_x.3[2, ])
# Partial derivative at the point 'y' with respect
# to the 3-rd argument
print(grad_x.3[2, 3])
# Gradient respect to the covariance matrix
grad_sigma.3 <- d.list.3$grad_sigma
# Partial derivative with respect to sigma(3, 5) at
# point 'y'
print(grad_sigma.3[3, 5, 2])
# Estimate the gradient of the log-density function of
# Xc=(X2, X4, X5 | X1 = -1, X3 = 1) and Yc=(X2, X4, X5 | X1 = -1.5, X3 = 0)
# with respect to the argument and covariance matrix at
# points xd=(0, 2, 3) and yd=(-1.2, 0, 1.5), respectively, given
# the conditioning values xg=(-1, 1) and yg=(-1.5, 0) correspondingly
d.list.4 <- dmnorm(x = xy, mean = mean, sigma = sigma,
grad_x = TRUE, grad_sigma = TRUE,
given_ind = given_ind, log = TRUE)
# Gradient respect to the argument
grad_x.4 <- d.list.4$grad_x
# at point 'xd'
print(grad_x.4[1, ])
# at point 'yd'
print(grad_x.4[2, ])
# Partial derivative at the point 'xd' with respect to 'xg[2]'
print(grad_x.4[1, 3])
# Partial derivative at the point 'yd' with respect to 'yd[5]'
print(grad_x.4[2, 5])
# Gradient with respect to the covariance matrix
grad_sigma.4 <- d.list.4$grad_sigma
# Partial derivative with respect to sigma(3, 5) at
# point 'yd'
print(grad_sigma.4[3, 5, 2])
# Compare analytical gradients from the previous example with
# their numeric (forward difference) analogues at point 'xd'
# given the conditioning 'xg'
delta <- 1e-6
grad_x.num <- rep(NA, 5)
grad_sigma.num <- matrix(NA, nrow = 5, ncol = 5)
for (i in 1:5)
{
x.delta <- x
x.delta[i] <- x[i] + delta
d.list.delta <- dmnorm(x = x.delta, mean = mean, sigma = sigma,
grad_x = TRUE, grad_sigma = TRUE,
given_ind = given_ind, log = TRUE)
grad_x.num[i] <- (d.list.delta$den - d.list.4$den[1]) / delta
for (j in 1:5)
{
sigma.delta <- sigma
sigma.delta[i, j] <- sigma[i, j] + delta
sigma.delta[j, i] <- sigma[j, i] + delta
d.list.delta <- dmnorm(x = x, mean = mean, sigma = sigma.delta,
grad_x = TRUE, grad_sigma = TRUE,
given_ind = given_ind, log = TRUE)
grad_sigma.num[i, j] <- (d.list.delta$den - d.list.4$den[1]) / delta
}
}
# Comparison of gradients with respect to the argument
h.x <- cbind(analytical = grad_x.4[1, ], numeric = grad_x.num)
rownames(h.x) <- c("xg[1]", "xd[1]", "xg[2]", "xd[3]", "xd[4]")
print(h.x)
# Comparison of gradients with respect to the covariance matrix
h.sigma <- list(analytical = grad_sigma.4[, , 1], numeric = grad_sigma.num)
print(h.sigma)
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