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The Multivariate Normal distribution on `R^k`.
The distribution has mean and covariance parameters mu (1-D), sigma (2-D),
or alternatively mean `mu` and factored covariance (cholesky decomposed
`sigma`) called `sigma_chol`.
#### Mathematical details
The PDF of this distribution is:
```
f(x) = (2*pi)^(-k/2) |det(sigma)|^(-1/2) exp(-1/2*(x-mu)^*.sigma^{-1}.(x-mu))
```
where `.` denotes the inner product on `R^k` and `^*` denotes transpose.
Alternatively, if `sigma` is positive definite, it can be represented in terms
of its lower triangular cholesky factorization
```sigma = sigma_chol . sigma_chol^*```
and the pdf above allows simpler computation:
```
|det(sigma)| = reduce_prod(diag(sigma_chol))^2
x_whitened = sigma^{-1/2} . (x - mu) = tri_solve(sigma_chol, x - mu)
(x-mu)^* .sigma^{-1} . (x-mu) = x_whitened^* . x_whitened
```
where `tri_solve()` solves a triangular system of equations.
#### Examples
A single multi-variate Gaussian distribution is defined by a vector of means
of length `k`, and a covariance matrix of shape `k x k`.
Extra leading dimensions, if provided, allow for batches.
```python
# Initialize a single 3-variate Gaussian with diagonal covariance.
mu = [1, 2, 3]
sigma = [[1, 0, 0], [0, 3, 0], [0, 0, 2]]
dist = tf.contrib.distributions.MultivariateNormal(mu=mu, sigma=sigma)
# Evaluate this on an observation in R^3, returning a scalar.
dist.pdf([-1, 0, 1])
# Initialize a batch of two 3-variate Gaussians.
mu = [[1, 2, 3], [11, 22, 33]]
sigma = ... # shape 2 x 3 x 3
dist = tf.contrib.distributions.MultivariateNormal(mu=mu, sigma=sigma)
# Evaluate this on a two observations, each in R^3, returning a length two
# tensor.
x = [[-1, 0, 1], [-11, 0, 11]] # Shape 2 x 3.
dist.pdf(x)
```
- - -
#### `tf.contrib.distributions.MultivariateNormal.__init__(mu, sigma=None, sigma_chol=None, name=None)` {#MultivariateNormal.__init__}
Multivariate Normal distributions on `R^k`.
User must provide means `mu`, which are tensors of rank `N+1` (`N >= 0`)
with the last dimension having length `k`.
User must provide exactly one of `sigma` (the covariance matrices) or
`sigma_chol` (the cholesky decompositions of the covariance matrices).
`sigma` or `sigma_chol` must be of rank `N+2`. The last two dimensions
must both have length `k`. The first `N` dimensions correspond to batch
indices.
If `sigma_chol` is not provided, the batch cholesky factorization of `sigma`
is calculated for you.
The shapes of `mu` and `sigma` must match for the first `N` dimensions.
Regardless of which parameter is provided, the covariance matrices must all
be **positive definite** (an error is raised if one of them is not).
##### Args:
* <b>`mu`</b>: (N+1)-D. `float` or `double` tensor, the means of the distributions.
* <b>`sigma`</b>: (N+2)-D. (optional) `float` or `double` tensor, the covariances
of the distribution(s). The first `N+1` dimensions must match
those of `mu`. Must be batch-positive-definite.
* <b>`sigma_chol`</b>: (N+2)-D. (optional) `float` or `double` tensor, a
lower-triangular factorization of `sigma`
(`sigma = sigma_chol . sigma_chol^*`). The first `N+1` dimensions
must match those of `mu`. The tensor itself need not be batch
lower triangular: we ignore the upper triangular part. However,
the batch diagonals must be positive (i.e., sigma_chol must be
batch-positive-definite).
* <b>`name`</b>: The name to give Ops created by the initializer.
##### Raises:
* <b>`ValueError`</b>: if neither sigma nor sigma_chol is provided.
* <b>`TypeError`</b>: if mu and sigma (resp. sigma_chol) are different dtypes.
- - -
#### `tf.contrib.distributions.MultivariateNormal.dtype` {#MultivariateNormal.dtype}
- - -
#### `tf.contrib.distributions.MultivariateNormal.entropy(name=None)` {#MultivariateNormal.entropy}
The entropies of these Multivariate Normals.
##### Args:
* <b>`name`</b>: The name to give this op.
##### Returns:
* <b>`entropy`</b>: tensor of dtype `dtype`, the entropies.
- - -
#### `tf.contrib.distributions.MultivariateNormal.is_reparameterized` {#MultivariateNormal.is_reparameterized}
- - -
#### `tf.contrib.distributions.MultivariateNormal.log_pdf(x, name=None)` {#MultivariateNormal.log_pdf}
Log pdf of observations `x` given these Multivariate Normals.
##### Args:
* <b>`x`</b>: tensor of dtype `dtype`, must be broadcastable with `mu`.
* <b>`name`</b>: The name to give this op.
##### Returns:
* <b>`log_pdf`</b>: tensor of dtype `dtype`, the log-PDFs of `x`.
- - -
#### `tf.contrib.distributions.MultivariateNormal.mean` {#MultivariateNormal.mean}
- - -
#### `tf.contrib.distributions.MultivariateNormal.mu` {#MultivariateNormal.mu}
- - -
#### `tf.contrib.distributions.MultivariateNormal.pdf(x, name=None)` {#MultivariateNormal.pdf}
The PDF of observations `x` under these Multivariate Normals.
##### Args:
* <b>`x`</b>: tensor of dtype `dtype`, must be broadcastable with `mu` and `sigma`.
* <b>`name`</b>: The name to give this op.
##### Returns:
* <b>`pdf`</b>: tensor of dtype `dtype`, the pdf values of `x`.
- - -
#### `tf.contrib.distributions.MultivariateNormal.sample(n, seed=None, name=None)` {#MultivariateNormal.sample}
Sample `n` observations from the Multivariate Normal Distributions.
##### Args:
* <b>`n`</b>: `Scalar`, type int32, the number of observations to sample.
* <b>`seed`</b>: Python integer, the random seed.
* <b>`name`</b>: The name to give this op.
##### Returns:
* <b>`samples`</b>: `[n, ...]`, a `Tensor` of `n` samples for each
of the distributions determined by broadcasting the hyperparameters.
- - -
#### `tf.contrib.distributions.MultivariateNormal.sigma` {#MultivariateNormal.sigma}
- - -
#### `tf.contrib.distributions.MultivariateNormal.sigma_det` {#MultivariateNormal.sigma_det}
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