# LKJCholesky

Source: https://www.tensorplay.cn/docs/generated/tensorplay.distributions.LKJCholesky.html

```python
class tensorplay.distributions.LKJCholesky(dim: int, concentration: Tensor | float = 1.0, validate_args: bool | None = None)
```

LKJ distribution for lower Cholesky factor of correlation matrices.
The distribution is controlled by concentration parameter $\eta$
to make the probability of the correlation matrix $M$ generated from
a Cholesky factor proportional to $\det(M)^{\eta - 1}$. Because of that,
when concentration == 1, we have a uniform distribution over Cholesky
factors of correlation matrices:

```
L ~ LKJCholesky(dim, concentration)
X = L @ L' ~ LKJCorr(dim, concentration)
```

Note that this distribution samples the
Cholesky factor of correlation matrices and not the correlation matrices
themselves and thereby differs slightly from the derivations in [1] for
the LKJCorr distribution. For sampling, this uses the Onion method from
[1] Section 3.

Example:

```
>>> # xdoctest: +IGNORE_WANT("non-deterministic")
>>> l = LKJCholesky(3, 0.5)
>>> l.sample()  # l @ l.T is a sample of a correlation 3x3 matrix
tensor([[ 1.0000,  0.0000,  0.0000],
        [ 0.3516,  0.9361,  0.0000],
        [-0.1899,  0.4748,  0.8593]])
```

Parameters:

- dimension (dim) – dimension of the matrices

- concentration ([float](https://docs.python.org/3/builtins/functions.html#float) or [Tensor](/docs/generated/tensorplay.Tensor.html#tensorplay.Tensor)) – concentration/shape parameter of the distribution (often referred to as eta)

References

[1] Generating random correlation matrices based on vines and extended onion method (2009),
Daniel Lewandowski, Dorota Kurowicka, Harry Joe.
Journal of Multivariate Analysis. 100. 10.1016/j.jmva.2009.04.008

```python
property batch_shape: Size
```

Returns the shape over which parameters are batched.

```python
cdf(value: Tensor) → Tensor
```

Returns the cumulative density/mass function evaluated at
value.

Parameters:

value ([Tensor](/docs/generated/tensorplay.Tensor.html#tensorplay.Tensor))

```python
entropy() → Tensor
```

Returns entropy of distribution, batched over batch_shape.

Returns:

Tensor of shape batch_shape.

```python
enumerate_support(expand: bool = True) → Tensor
```

Returns tensor containing all values supported by a discrete
distribution. The result will enumerate over dimension 0, so the shape
of the result will be (cardinality,) + batch_shape + event_shape
(where event_shape = () for univariate distributions).

Note that this enumerates over all batched tensors in lock-step
[[0, 0], [1, 1], …]. With expand=False, enumeration happens
along dim 0, but with the remaining batch dimensions being
singleton dimensions, [[0], [1], ...

To iterate over the full Cartesian product use
itertools.product(m.enumerate_support()).

Parameters:

expand ([bool](https://docs.python.org/3/builtins/functions.html#bool)) – whether to expand the support over the
batch dims to match the distribution’s batch_shape.

Returns:

Tensor iterating over dimension 0.

```python
property event_shape: Size
```

Returns the shape of a single sample (without batching).

```python
icdf(value: Tensor) → Tensor
```

Returns the inverse cumulative density/mass function evaluated at
value.

Parameters:

value ([Tensor](/docs/generated/tensorplay.Tensor.html#tensorplay.Tensor))

```python
property mean: Tensor
```

Returns the mean of the distribution.

```python
property mode: Tensor
```

Returns the mode of the distribution.

```python
perplexity() → Tensor
```

Returns perplexity of distribution, batched over batch_shape.

Returns:

Tensor of shape batch_shape.

```python
rsample(sample_shape: Size | Sequence[int] = ()) → Tensor
```

Generates a sample_shape shaped reparameterized sample or sample_shape
shaped batch of reparameterized samples if the distribution parameters
are batched.

```python
sample_n(n: int) → Tensor
```

Generates n samples or n batches of samples if the distribution
parameters are batched.

```python
static set_default_validate_args(value: bool) → None
```

Sets whether validation is enabled or disabled.

The default behavior mimics Python’s assert statement: validation
is on by default, but is disabled if Python is run in optimized mode
(via python -O). Validation may be expensive, so you may want to
disable it once a model is working.

Parameters:

value ([bool](https://docs.python.org/3/builtins/functions.html#bool)) – Whether to enable validation.

```python
property stddev: Tensor
```

Returns the standard deviation of the distribution.

```python
property variance: Tensor
```

Returns the variance of the distribution.