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# tfp.distributions.CholeskyLKJ

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The CholeskyLKJ distribution on cholesky factors of correlation matrices.

Inherits From: `Distribution`, `AutoCompositeTensor`

This is a one-parameter family of distributions on cholesky factors of correlation matrices.

In other words, if If `X ~ CholeskyLKJ(c)`, then `X @ X^T ~ LKJ(c)`.

The distribution is supported on lower triangular N x N matrices which are Cholesky factors of correlation matrices; equivalently, matrices whose rows have Euclidean norm 1 and diagonal entries are positive. The probability density function is given by:

pdf(X; c) = (1/Z(c)) prod_i X_ii^{n-i+2c-3} (0 <= i < n)

where there are n(n-1)/2 independent variables X_ij (0 <= i < j < n) and Z(c) is the normalizing constant; the same one as that of LKJ(c).

For more details on the LKJ distribution, see `tfp.distributions.LKJ`.

#### Examples

``````# Initialize a single 3x3 LKJ with concentration parameter 1.5
dist = tfp.distributions.CholeskyLKJ(dimension=3, concentration=1.5)

# Evaluate this at a batch of two observations, each in R^{3x3}.
x = ...  # Shape is [2, 3, 3].
dist.prob(x)  # Shape is [2].

# Draw 6 Cholesky LKJ-distributed 3x3 lower triangular matrices
ans = dist.sample(sample_shape=[2, 3], seed=42)
# shape of ans is [2, 3, 3, 3]
``````

The sampler follows the 'onion' method from

[1] Daniel Lewandowski, Dorota Kurowicka, and Harry Joe, 'Generating random correlation matrices based on vines and extended onion method,' Journal of Multivariate Analysis 100 (2009), pp 1989-2001.

`dimension` Python `int`. The dimension of the correlation matrices to sample.
`concentration` `float` or `double` `Tensor`. The positive concentration parameter of the CholeskyLKJ distributions.
`validate_args` Python `bool`, default `False`. When `True`, distribution parameters are checked for validity despite possibly degrading runtime performance. When `False`, invalid inputs may silently render incorrect outputs.
`allow_nan_stats` Python `bool`, default `True`. When `True`, statistics (e.g., mean, mode, variance) use the value `NaN` to indicate the result is undefined. When `False`, an exception is raised if one or more of the statistic's batch members are undefined.
`name` Python `str` name prefixed to Ops created by this class.

`ValueError` If `dimension` is negative.

`allow_nan_stats` Python `bool` describing behavior when a stat is undefined.

Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.

`batch_shape` Shape of a single sample from a single event index as a `TensorShape`.

May be partially defined or unknown.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

`concentration` Concentration parameter.
`dimension` Dimension of returned cholesky factors of correlation matrices.
`dtype` The `DType` of `Tensor`s handled by this `Distribution`.
`event_shape` Shape of a single sample from a single batch as a `TensorShape`.

May be partially defined or unknown.

`experimental_shard_axis_names` The list or structure of lists of active shard axis names.
`name` Name prepended to all ops created by this `Distribution`.
`name_scope` Returns a `tf.name_scope` instance for this class.
`non_trainable_variables` Sequence of non-trainable variables owned by this module and its submodules.

`parameters` Dictionary of parameters used to instantiate this `Distribution`.
`reparameterization_type` Describes how samples from the distribution are reparameterized.

Currently this is one of the static instances `tfd.FULLY_REPARAMETERIZED` or `tfd.NOT_REPARAMETERIZED`.

`submodules` Sequence of all sub-modules.

Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on).

````a = tf.Module()`
`b = tf.Module()`
`c = tf.Module()`
`a.b = b`
`b.c = c`
`list(a.submodules) == [b, c]`
`True`
`list(b.submodules) == [c]`
`True`
`list(c.submodules) == []`
`True`
```

`trainable_variables` Sequence of trainable variables owned by this module and its submodules.

`validate_args` Python `bool` indicating possibly expensive checks are enabled.
`variables` Sequence of variables owned by this module and its submodules.

## Methods

### `batch_shape_tensor`

View source

Shape of a single sample from a single event index as a 1-D `Tensor`.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

Args
`name` name to give to the op

Returns
`batch_shape` `Tensor`.

### `cdf`

View source

Cumulative distribution function.

Given random variable `X`, the cumulative distribution function `cdf` is:

``````cdf(x) := P[X <= x]
``````

Args
`value` `float` or `double` `Tensor`.
`name` Python `str` prepended to names of ops created by this function.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`cdf` a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `copy`

View source

Creates a deep copy of the distribution.

Args
`**override_parameters_kwargs` String/value dictionary of initialization arguments to override with new values.

Returns
`distribution` A new instance of `type(self)` initialized from the union of self.parameters and override_parameters_kwargs, i.e., `dict(self.parameters, **override_parameters_kwargs)`.

### `covariance`

View source

Covariance.

Covariance is (possibly) defined only for non-scalar-event distributions.

For example, for a length-`k`, vector-valued distribution, it is calculated as,

``````Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]
``````

where `Cov` is a (batch of) `k x k` matrix, `0 <= (i, j) < k`, and `E` denotes expectation.

Alternatively, for non-vector, multivariate distributions (e.g., matrix-valued, Wishart), `Covariance` shall return a (batch of) matrices under some vectorization of the events, i.e.,

``````Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]
``````

where `Cov` is a (batch of) `k' x k'` matrices, `0 <= (i, j) < k' = reduce_prod(event_shape)`, and `Vec` is some function mapping indices of this distribution's event dimensions to indices of a length-`k'` vector.

Args
`name` Python `str` prepended to names of ops created by this function.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`covariance` Floating-point `Tensor` with shape `[B1, ..., Bn, k', k']` where the first `n` dimensions are batch coordinates and `k' = reduce_prod(self.event_shape)`.

### `cross_entropy`

View source

Computes the (Shannon) cross entropy.

Denote this distribution (`self`) by `P` and the `other` distribution by `Q`. Assuming `P, Q` are absolutely continuous with respect to one another and permit densities `p(x) dr(x)` and `q(x) dr(x)`, (Shannon) cross entropy is defined as:

``````H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)
``````

where `F` denotes the support of the random variable `X ~ P`.

Args
`other` `tfp.distributions.Distribution` instance.
`name` Pyth