# Multivariate¶

 MvNormal(mu[, cov, tau, chol, lower]) Multivariate normal log-likelihood. MatrixNormal([mu, rowcov, rowchol, rowtau, …]) Matrix-valued normal log-likelihood. KroneckerNormal(mu[, covs, chols, evds, sigma]) Multivariate normal log-likelihood with Kronecker-structured covariance. MvStudentT(nu[, Sigma, mu, cov, tau, chol, …]) Multivariate Student-T log-likelihood. Wishart(nu, V, *args, **kwargs) Wishart log-likelihood. LKJCholeskyCov(eta, n, sd_dist, *args, **kwargs) Covariance matrix with LKJ distributed correlations. LKJCorr([eta, n, p, transform]) The LKJ (Lewandowski, Kurowicka and Joe) log-likelihood. Multinomial(n, p, *args, **kwargs) Multinomial log-likelihood. Dirichlet(a[, transform]) Dirichlet log-likelihood.
class pymc3.distributions.multivariate.MvNormal(mu, cov=None, tau=None, chol=None, lower=True, *args, **kwargs)

Multivariate normal log-likelihood.

$f(x \mid \pi, T) = \frac{|T|^{1/2}}{(2\pi)^{k/2}} \exp\left\{ -\frac{1}{2} (x-\mu)^{\prime} T (x-\mu) \right\}$
 Support $$x \in \mathbb{R}^k$$ Mean $$\mu$$ Variance $$T^{-1}$$
Parameters: mu : array Vector of means. cov : array Covariance matrix. Exactly one of cov, tau, or chol is needed. tau : array Precision matrix. Exactly one of cov, tau, or chol is needed. chol : array Cholesky decomposition of covariance matrix. Exactly one of cov, tau, or chol is needed. lower : bool, default=True Whether chol is the lower tridiagonal cholesky factor.

Examples

Define a multivariate normal variable for a given covariance matrix:

cov = np.array([[1., 0.5], [0.5, 2]])
mu = np.zeros(2)
vals = pm.MvNormal('vals', mu=mu, cov=cov, shape=(5, 2))


Most of the time it is preferable to specify the cholesky factor of the covariance instead. For example, we could fit a multivariate outcome like this (see the docstring of LKJCholeskyCov for more information about this):

mu = np.zeros(3)
true_cov = np.array([[1.0, 0.5, 0.1],
[0.5, 2.0, 0.2],
[0.1, 0.2, 1.0]])
data = np.random.multivariate_normal(mu, true_cov, 10)

sd_dist = pm.HalfCauchy.dist(beta=2.5, shape=3)
chol_packed = pm.LKJCholeskyCov('chol_packed',
n=3, eta=2, sd_dist=sd_dist)
chol = pm.expand_packed_triangular(3, chol_packed)
vals = pm.MvNormal('vals', mu=mu, chol=chol, observed=data)


For unobserved values it can be better to use a non-centered parametrization:

sd_dist = pm.HalfCauchy.dist(beta=2.5, shape=3)
chol_packed = pm.LKJCholeskyCov('chol_packed',
n=3, eta=2, sd_dist=sd_dist)
chol = pm.expand_packed_triangular(3, chol_packed)
vals_raw = pm.Normal('vals_raw', mu=0, sd=1, shape=(5, 3))
vals = pm.Deterministic('vals', tt.dot(chol, vals_raw.T).T)

class pymc3.distributions.multivariate.MvStudentT(nu, Sigma=None, mu=None, cov=None, tau=None, chol=None, lower=True, *args, **kwargs)

Multivariate Student-T log-likelihood.

$f(\mathbf{x}| \nu,\mu,\Sigma) = \frac {\Gamma\left[(\nu+p)/2\right]} {\Gamma(\nu/2)\nu^{p/2}\pi^{p/2} \left|{\Sigma}\right|^{1/2} \left[ 1+\frac{1}{\nu} ({\mathbf x}-{\mu})^T {\Sigma}^{-1}({\mathbf x}-{\mu}) \right]^{(\nu+p)/2}}$
 Support $$x \in \mathbb{R}^k$$ Mean $$\mu$$ if $$\nu > 1$$ else undefined Variance $$\frac{\nu}{\mu-2}\Sigma$$ if $$\nu>2$$ else undefined
Parameters: nu : int Degrees of freedom. Sigma : matrix Covariance matrix. Use cov in new code. mu : array Vector of means. cov : matrix The covariance matrix. tau : matrix The precision matrix. chol : matrix The cholesky factor of the covariance matrix. lower : bool, default=True Whether the cholesky fatcor is given as a lower triangular matrix.
class pymc3.distributions.multivariate.Dirichlet(a, transform=<pymc3.distributions.transforms.StickBreaking object>, *args, **kwargs)

Dirichlet log-likelihood.

$f(\mathbf{x}|\mathbf{a}) = \frac{\Gamma(\sum_{i=1}^k a_i)}{\prod_{i=1}^k \Gamma(a_i)} \prod_{i=1}^k x_i^{a_i - 1}$
 Support $$x_i \in (0, 1)$$ for $$i \in \{1, \ldots, K\}$$ such that $$\sum x_i = 1$$ Mean $$\dfrac{a_i}{\sum a_i}$$ Variance $$\dfrac{a_i - \sum a_0}{a_0^2 (a_0 + 1)}$$ where $$a_0 = \sum a_i$$
Parameters: a : array Concentration parameters (a > 0).
class pymc3.distributions.multivariate.Multinomial(n, p, *args, **kwargs)

Multinomial log-likelihood.

Generalizes binomial distribution, but instead of each trial resulting in “success” or “failure”, each one results in exactly one of some fixed finite number k of possible outcomes over n independent trials. ‘x[i]’ indicates the number of times outcome number i was observed over the n trials.

$f(x \mid n, p) = \frac{n!}{\prod_{i=1}^k x_i!} \prod_{i=1}^k p_i^{x_i}$
 Support $$x \in \{0, 1, \ldots, n\}$$ such that $$\sum x_i = n$$ Mean $$n p_i$$ Variance $$n p_i (1 - p_i)$$ Covariance $$-n p_i p_j$$ for $$i \ne j$$
Parameters: n : int or array Number of trials (n > 0). If n is an array its shape must be (N,) with N = p.shape p : one- or two-dimensional array Probability of each one of the different outcomes. Elements must be non-negative and sum to 1 along the last axis. They will be automatically rescaled otherwise.
class pymc3.distributions.multivariate.Wishart(nu, V, *args, **kwargs)

Wishart log-likelihood.

The Wishart distribution is the probability distribution of the maximum-likelihood estimator (MLE) of the precision matrix of a multivariate normal distribution. If V=1, the distribution is identical to the chi-square distribution with nu degrees of freedom.

$f(X \mid nu, T) = \frac{{\mid T \mid}^{nu/2}{\mid X \mid}^{(nu-k-1)/2}}{2^{nu k/2} \Gamma_p(nu/2)} \exp\left\{ -\frac{1}{2} Tr(TX) \right\}$

where $$k$$ is the rank of $$X$$.

 Support $$X(p x p)$$ positive definite matrix Mean $$nu V$$ Variance $$nu (v_{ij}^2 + v_{ii} v_{jj})$$
Parameters: nu : int Degrees of freedom, > 0. V : array p x p positive definite matrix.

Notes

This distribution is unusable in a PyMC3 model. You should instead use LKJCholeskyCov or LKJCorr.

pymc3.distributions.multivariate.WishartBartlett(name, S, nu, is_cholesky=False, return_cholesky=False, testval=None)

Bartlett decomposition of the Wishart distribution. As the Wishart distribution requires the matrix to be symmetric positive semi-definite it is impossible for MCMC to ever propose acceptable matrices.

Instead, we can use the Barlett decomposition which samples a lower diagonal matrix. Specifically:

\begin{align}\begin{aligned}\begin{split}\text{If} L \sim \begin{pmatrix} \sqrt{c_1} & 0 & 0 \\ z_{21} & \sqrt{c_2} & 0 \\ z_{31} & z_{32} & \sqrt{c_3} \end{pmatrix}\end{split}\\\begin{split}\text{with} c_i \sim \chi^2(n-i+1) \text{ and } n_{ij} \sim \mathcal{N}(0, 1), \text{then} \\ L \times A \times A.T \times L.T \sim \text{Wishart}(L \times L.T, \nu)\end{split}\end{aligned}\end{align}

Parameters: S : ndarray p x p positive definite matrix Or: p x p lower-triangular matrix that is the Cholesky factor of the covariance matrix. nu : int Degrees of freedom, > dim(S). is_cholesky : bool (default=False) Input matrix S is already Cholesky decomposed as S.T * S return_cholesky : bool (default=False) Only return the Cholesky decomposed matrix. testval : ndarray p x p positive definite matrix used to initialize

Notes

This is not a standard Distribution class but follows a similar interface. Besides the Wishart distribution, it will add RVs name_c and name_z to your model which make up the matrix.

This distribution is usually a bad idea to use as a prior for multivariate normal. You should instead use LKJCholeskyCov or LKJCorr.

class pymc3.distributions.multivariate.LKJCorr(eta=None, n=None, p=None, transform='interval', *args, **kwargs)

The LKJ (Lewandowski, Kurowicka and Joe) log-likelihood.

The LKJ distribution is a prior distribution for correlation matrices. If eta = 1 this corresponds to the uniform distribution over correlation matrices. For eta -> oo the LKJ prior approaches the identity matrix.

 Support Upper triangular matrix with values in [-1, 1]
Parameters: n : int Dimension of the covariance matrix (n > 1). eta : float The shape parameter (eta > 0) of the LKJ distribution. eta = 1 implies a uniform distribution of the correlation matrices; larger values put more weight on matrices with few correlations.

Notes

This implementation only returns the values of the upper triangular matrix excluding the diagonal. Here is a schematic for n = 5, showing the indexes of the elements:

[[- 0 1 2 3]
[- - 4 5 6]
[- - - 7 8]
[- - - - 9]
[- - - - -]]


References

 [LKJ200918] Lewandowski, D., Kurowicka, D. and Joe, H. (2009). “Generating random correlation matrices based on vines and extended onion method.” Journal of multivariate analysis, 100(9), pp.1989-2001.
class pymc3.distributions.multivariate.LKJCholeskyCov(eta, n, sd_dist, *args, **kwargs)

Covariance matrix with LKJ distributed correlations.

This defines a distribution over cholesky decomposed covariance matrices, such that the underlying correlation matrices follow an LKJ distribution  and the standard deviations follow an arbitray distribution specified by the user.

Parameters: n : int Dimension of the covariance matrix (n > 1). eta : float The shape parameter (eta > 0) of the LKJ distribution. eta = 1 implies a uniform distribution of the correlation matrices; larger values put more weight on matrices with few correlations. sd_dist : pm.Distribution A distribution for the standard deviations.

Notes

Since the cholesky factor is a lower triangular matrix, we use packed storge for the matrix: We store and return the values of the lower triangular matrix in a one-dimensional array, numbered by row:

[[0 - - -]
[1 2 - -]
[3 4 5 -]
[6 7 8 9]]


You can use pm.expand_packed_triangular(packed_cov, lower=True) to convert this to a regular two-dimensional array.

References

 [R20] Lewandowski, D., Kurowicka, D. and Joe, H. (2009). “Generating random correlation matrices based on vines and extended onion method.” Journal of multivariate analysis, 100(9), pp.1989-2001.
 [R21] J. M. isn’t a mathematician (http://math.stackexchange.com/users/498/ j-m-isnt-a-mathematician), Different approaches to evaluate this determinant, URL (version: 2012-04-14): http://math.stackexchange.com/q/130026

Examples

with pm.Model() as model:
# Note that we access the distribution for the standard
# deviations, and do not create a new random variable.
sd_dist = pm.HalfCauchy.dist(beta=2.5)
packed_chol = pm.LKJCholeskyCov('chol_cov', eta=2, n=10, sd_dist=sd_dist)
chol = pm.expand_packed_triangular(10, packed_chol, lower=True)

# Define a new MvNormal with the given covariance
vals = pm.MvNormal('vals', mu=np.zeros(10), chol=chol, shape=10)

# Or transform an uncorrelated normal:
vals_raw = pm.Normal('vals_raw', mu=0, sd=1, shape=10)
vals = tt.dot(chol, vals_raw)

# Or compute the covariance matrix
cov = tt.dot(chol, chol.T)

# Extract the standard deviations
stds = tt.sqrt(tt.diag(cov))


Implementation In the unconstrained space all values of the cholesky factor are stored untransformed, except for the diagonal entries, where we use a log-transform to restrict them to positive values.

To correctly compute log-likelihoods for the standard deviations and the correlation matrix seperatly, we need to consider a second transformation: Given a cholesky factorization $$LL^T = \Sigma$$ of a covariance matrix we can recover the standard deviations $$\sigma$$ as the euclidean lengths of the rows of $$L$$, and the cholesky factor of the correlation matrix as $$U = \text{diag}(\sigma)^{-1}L$$. Since each row of $$U$$ has length 1, we do not need to store the diagonal. We define a transformation $$\phi$$ such that $$\phi(L)$$ is the lower triangular matrix containing the standard deviations $$\sigma$$ on the diagonal and the correlation matrix $$U$$ below. In this form we can easily compute the different likelihoods seperatly, as the likelihood of the correlation matrix only depends on the values below the diagonal, and the likelihood of the standard deviation depends only on the diagonal values.

We still need the determinant of the jacobian of $$\phi^{-1}$$. If we think of $$\phi$$ as an automorphism on $$\mathbb{R}^{\tfrac{n(n+1)}{2}}$$, where we order the dimensions as described in the notes above, the jacobian is a block-diagonal matrix, where each block corresponds to one row of $$U$$. Each block has arrowhead shape, and we can compute the determinant of that as described in . Since the determinant of a block-diagonal matrix is the product of the determinants of the blocks, we get

$\text{det}(J_{\phi^{-1}}(U)) = \left[ \prod_{i=2}^N u_{ii}^{i - 1} L_{ii} \right]^{-1}$
class pymc3.distributions.multivariate.MatrixNormal(mu=0, rowcov=None, rowchol=None, rowtau=None, colcov=None, colchol=None, coltau=None, shape=None, *args, **kwargs)

Matrix-valued normal log-likelihood.

$f(x \mid \mu, U, V) = \frac{1}{(2\pi |U|^n |V|^m)^{1/2}} \exp\left\{ -\frac{1}{2} \mathrm{Tr}[ V^{-1} (x-\mu)^{\prime} U^{-1} (x-\mu)] \right\}$
 Support $$x \in \mathbb{R}^{m \times n}$$ Mean $$\mu$$ Row Variance $$U$$ Column Variance $$V$$
Parameters: mu : array Array of means. Must be broadcastable with the random variable X such that the shape of mu + X is (m,n). rowcov : mxm array Among-row covariance matrix. Defines variance within columns. Exactly one of rowcov or rowchol is needed. rowchol : mxm array Cholesky decomposition of among-row covariance matrix. Exactly one of rowcov or rowchol is needed. colcov : nxn array Among-column covariance matrix. If rowcov is the identity matrix, this functions as cov in MvNormal. Exactly one of colcov or colchol is needed. colchol : nxn array Cholesky decomposition of among-column covariance matrix. Exactly one of colcov or colchol is needed.

Examples

Define a matrixvariate normal variable for given row and column covariance matrices:

colcov = np.array([[1., 0.5], [0.5, 2]])
rowcov = np.array([[1, 0, 0], [0, 4, 0], [0, 0, 16]])
m = rowcov.shape
n = colcov.shape
mu = np.zeros((m, n))
vals = pm.MatrixNormal('vals', mu=mu, colcov=colcov,
rowcov=rowcov, shape=(m, n))


Above, the ith row in vals has a variance that is scaled by 4^i. Alternatively, row or column cholesky matrices could be substituted for either covariance matrix. The MatrixNormal is quicker way compute MvNormal(mu, np.kron(rowcov, colcov)) that takes advantage of kronecker product properties for inversion. For example, if draws from MvNormal had the same covariance structure, but were scaled by different powers of an unknown constant, both the covariance and scaling could be learned as follows (see the docstring of LKJCholeskyCov for more information about this)

# Setup data
true_colcov = np.array([[1.0, 0.5, 0.1],
[0.5, 1.0, 0.2],
[0.1, 0.2, 1.0]])
m = 3
n = true_colcov.shape
true_scale = 3
true_rowcov = np.diag([true_scale**(2*i) for i in range(m)])
mu = np.zeros((m, n))
true_kron = np.kron(true_rowcov, true_colcov)
data = np.random.multivariate_normal(mu.flatten(), true_kron)
data = data.reshape(m, n)

with pm.Model() as model:
# Setup right cholesky matrix
sd_dist = pm.HalfCauchy.dist(beta=2.5, shape=3)
colchol_packed = pm.LKJCholeskyCov('colcholpacked', n=3, eta=2,
sd_dist=sd_dist)
colchol = pm.expand_packed_triangular(3, colchol_packed)

# Setup left covariance matrix
scale = pm.Lognormal('scale', mu=np.log(true_scale), sd=0.5)
rowcov = tt.nlinalg.diag([scale**(2*i) for i in range(m)])

vals = pm.MatrixNormal('vals', mu=mu, colchol=colchol, rowcov=rowcov,
observed=data, shape=(m, n))

class pymc3.distributions.multivariate.KroneckerNormal(mu, covs=None, chols=None, evds=None, sigma=None, *args, **kwargs)

Multivariate normal log-likelihood with Kronecker-structured covariance.

$f(x \mid \mu, K) = \frac{1}{(2\pi |K|)^{1/2}} \exp\left\{ -\frac{1}{2} (x-\mu)^{\prime} K^{-1} (x-\mu) \right\}$
 Support $$x \in \mathbb{R}^N$$ Mean $$\mu$$ Variance $$K = \bigotimes K_i$$ + sigma^2 I_N
Parameters: mu : array Vector of means, just as in MvNormal. covs : list of arrays The set of covariance matrices $$[K_1, K_2, ...]$$ to be Kroneckered in the order provided $$\bigotimes K_i$$. chols : list of arrays The set of lower cholesky matrices $$[L_1, L_2, ...]$$ such that $$K_i = L_i L_i'$$. evds : list of tuples The set of eigenvalue-vector, eigenvector-matrix pairs $$[(v_1, Q_1), (v_2, Q_2), ...]$$ such that $$K_i = Q_i \text{diag}(v_i) Q_i'$$. For example: v_i, Q_i = tt.nlinalg.eigh(K_i)  sigma : scalar, variable Standard deviation of the Gaussian white noise.

References

 [R22] Saatchi, Y. (2011). “Scalable inference for structured Gaussian process models”

Examples

Define a multivariate normal variable with a covariance $$K = K_1 \otimes K_2$$

K1 = np.array([[1., 0.5], [0.5, 2]])
K2 = np.array([[1., 0.4, 0.2], [0.4, 2, 0.3], [0.2, 0.3, 1]])
covs = [K1, K2]
N = 6
mu = np.zeros(N)
with pm.Model() as model:
vals = pm.KroneckerNormal('vals', mu=mu, covs=covs, shape=N)


Effeciency gains are made by cholesky decomposing $$K_1$$ and $$K_2$$ individually rather than the larger $$K$$ matrix. Although only two matrices $$K_1$$ and $$K_2$$ are shown here, an arbitrary number of submatrices can be combined in this way. Choleskys and eigendecompositions can be provided instead

chols = [np.linalg.cholesky(Ki) for Ki in covs]
evds = [np.linalg.eigh(Ki) for Ki in covs]
with pm.Model() as model:
vals2 = pm.KroneckerNormal('vals2', mu=mu, chols=chols, shape=N)
# or
vals3 = pm.KroneckerNormal('vals3', mu=mu, evds=evds, shape=N)


neither of which will be converted. Diagonal noise can also be added to the covariance matrix, $$K = K_1 \otimes K_2 + \sigma^2 I_N$$. Despite the noise removing the overall Kronecker structure of the matrix, KroneckerNormal can continue to make efficient calculations by utilizing eigendecompositons of the submatrices behind the scenes . Thus,

sigma = 0.1
with pm.Model() as noise_model:
vals = pm.KroneckerNormal('vals', mu=mu, covs=covs, sigma=sigma, shape=N)
vals2 = pm.KroneckerNormal('vals2', mu=mu, chols=chols, sigma=sigma, shape=N)
vals3 = pm.KroneckerNormal('vals3', mu=mu, evds=evds, sigma=sigma, shape=N)


are identical, with covs and chols each converted to eigendecompositions.