kd.random.PRNGKey#

class kauldron.random.PRNGKey( seed_or_rng: int | jax.Array | kauldron.random.random.PRNGKey = 0, *, impl: str | None = None, _allow_seed_array: bool = True, )[source]

Bases: object

Small wrapper around jax.random key arrays to reduce boilerplate.

Benefits:

Object oriented API (jax.random.uniform(key) -> key.uniform())
fold_in supports str (key.fold_in(‘dropout’))
Additional as_seed() method to get a seed int from the rng (to pass to third party APIs, like grain, np.random,…)

Usage:

key = kd.random.PRNGKey(0)
key0, key1 = key.split()
x = key0.uniform()

x = jax.random.uniform(key)  # Jax API still works

rng: jax.Array

split(n: int = 2) → kauldron.random.random.PRNGKey[source]: Returns the next rng key.

fold_in( data: int | str, ) → kauldron.random.random.PRNGKey[source]: Folds in delta into the random state.

next() → kauldron.random.random.PRNGKey[source]: Returns the next rng key (alias for key.split(1)[0]).

tree_flatten() → tuple[list[jax.Array], dict[str, Any]][source]: jax.tree_utils support.

classmethod tree_unflatten( metadata: dict[str, Any], array_field_values: list[jax.Array], ) → kauldron.random.random.PRNGKey[source]: jax.tree_utils support.

as_seed() → int[source]

Returns a seed integer (alias of int(rng.bits())).

Note this is non-reversible (the returned seed is not the one passed to construct the rng).

Returns:: An integer seed.

ball( d: int, p: float = 2, shape: collections.abc.Sequence[int] = (), dtype: str | type[Any] | numpy.dtype | jax._src.typing.SupportsDType | None = None, )

Sample uniformly from the unit Lp ball.

Reference: https://arxiv.org/abs/math/0503650.

Parameters:

key – a PRNG key used as the random key.
d – a nonnegative int representing the dimensionality of the ball.
p – a float representing the p parameter of the Lp norm.
shape – optional, the batch dimensions of the result. Default ().
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array of shape (*shape, d) and specified dtype.

Sample Bernoulli random values with given shape and mean.

The values are distributed according to the probability mass function:

\[f(k; p) = p^k(1 - p)^{1 - k}\]

where \(k \in \{0, 1\}\) and \(0 \le p \le 1\).

Parameters:

key – a PRNG key used as the random key.
p – optional, a float or array of floats for the mean of the random variables. Must be broadcast-compatible with shape. Default 0.5.
shape – optional, a tuple of nonnegative integers representing the result shape. Must be broadcast-compatible with p.shape. The default (None) produces a result shape equal to p.shape.
mode – optional, “high” or “low” for how many bits to use when sampling. default=’low’. Set to “high” for correct sampling at small values of p. When sampling in float32, bernoulli samples with mode=’low’ produce incorrect results for p < ~1E-7. mode=”high” approximately doubles the cost of sampling.

Returns:

A random array with boolean dtype and shape given by shape if shape is not None, or else p.shape.

Sample Beta random values with given shape and float dtype.

The values are distributed according to the probability density function:

\[f(x;a,b) \propto x^{a - 1}(1 - x)^{b - 1}\]

on the domain \(0 \le x \le 1\).

Parameters:

key – a PRNG key used as the random key.
a – a float or array of floats broadcast-compatible with shape representing the first parameter “alpha”.
b – a float or array of floats broadcast-compatible with shape representing the second parameter “beta”.
shape – optional, a tuple of nonnegative integers specifying the result shape. Must be broadcast-compatible with a and b. The default (None) produces a result shape by broadcasting a and b.
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified dtype and shape given by shape if shape is not None, or else by broadcasting a and b.

bits( shape: collections.abc.Sequence[int] = (), dtype: str | type[Any] | numpy.dtype | jax._src.typing.SupportsDType | None = None, *, out_sharding=None, ) → jax.Array

Sample uniform bits in the form of unsigned integers.

Parameters:

key – a PRNG key used as the random key.
shape – optional, a tuple of nonnegative integers representing the result shape. Default ().
dtype – optional, an unsigned integer dtype for the returned values (default uint64 if jax_enable_x64 is true, otherwise uint32).

Returns:

A random array with the specified shape and dtype.

Sample random values from categorical distributions.

Sampling with replacement uses the Gumbel max trick. Sampling without replacement uses the Gumbel top-k trick. See [1] for reference.

Parameters:

key – a PRNG key used as the random key.
logits – Unnormalized log probabilities of the categorical distribution(s) to sample from, so that softmax(logits, axis) gives the corresponding probabilities.
axis – Axis along which logits belong to the same categorical distribution.
shape – Optional, a tuple of nonnegative integers representing the result shape. Must be broadcast-compatible with np.delete(logits.shape, axis). The default (None) produces a result shape equal to np.delete(logits.shape, axis).
replace – If True (default), perform sampling with replacement. If False, perform sampling without replacement.
mode – optional, “high” or “low” for how many bits to use in the gumbel sampler. The default is determined by the use_high_dynamic_range_gumbel config, which defaults to “low”. With mode=”low”, in float32 sampling will be biased for events with probability less than about 1E-7; with mode=”high” this limit is pushed down to about 1E-14. mode=”high” approximately doubles the cost of sampling.

Returns:

A random array with int dtype and shape given by shape if shape is not None, or else np.delete(logits.shape, axis).

References

cauchy( shape: collections.abc.Sequence[int] = (), dtype: str | type[Any] | numpy.dtype | jax._src.typing.SupportsDType | None = None, ) → jax.Array

Sample Cauchy random values with given shape and float dtype.

The values are distributed according to the probability density function:

\[f(x) \propto \frac{1}{x^2 + 1}\]

on the domain \(-\infty < x < \infty\)

Parameters:

key – a PRNG key used as the random key.
shape – optional, a tuple of nonnegative integers representing the result shape. Default ().
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified shape and dtype.

Sample Chisquare random values with given shape and float dtype.

The values are distributed according to the probability density function:

\[f(x; \nu) \propto x^{\nu/2 - 1}e^{-x/2}\]

on the domain \(0 < x < \infty\), where \(\nu > 0\) represents the degrees of freedom, given by the parameter df.

Parameters:

key – a PRNG key used as the random key.
df – a float or array of floats broadcast-compatible with shape representing the parameter of the distribution.
shape – optional, a tuple of nonnegative integers specifying the result shape. Must be broadcast-compatible with df. The default (None) produces a result shape equal to df.shape.
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified dtype and with shape given by shape if shape is not None, or else by df.shape.

Generates a random sample from a given array.

Warning

If p has fewer non-zero elements than the requested number of samples, as specified in shape, and replace=False, the output of this function is ill-defined. Please make sure to use appropriate inputs.

Parameters:

key – a PRNG key used as the random key.
a – array or int. If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if a were arange(a).
shape – tuple of ints, optional. Output shape. If the given shape is, e.g., (m, n), then m * n samples are drawn. Default is (), in which case a single value is returned.
replace – boolean. Whether the sample is with or without replacement. Default is True.
p – 1-D array-like, The probabilities associated with each entry in a. If not given the sample assumes a uniform distribution over all entries in a.
axis – int, optional. The axis along which the selection is performed. The default, 0, selects by row.
mode – optional, “high” or “low” for how many bits to use in the gumbel sampler when p is None and replace = False. The default is determined by the use_high_dynamic_range_gumbel config, which defaults to “low”. With mode=”low”, in float32 sampling will be biased for choices with probability less than about 1E-7; with mode=”high” this limit is pushed down to about 1E-14. mode=”high” approximately doubles the cost of sampling.

Returns:

An array of shape shape containing samples from a.

Sample Dirichlet random values with given shape and float dtype.

The values are distributed according to the probability density function:

\[f(\{x_i\}; \{\alpha_i\}) \propto \prod_{i=1}^k x_i^{\alpha_i - 1}\]

Where \(k\) is the dimension, and \(\{x_i\}\) satisfies

\[\sum_{i=1}^k x_i = 1\]

and \(0 \le x_i \le 1\) for all \(x_i\).

Parameters:

key – a PRNG key used as the random key.
alpha – an array of shape (..., n) used as the concentration parameter of the random variables.
shape – optional, a tuple of nonnegative integers specifying the result batch shape; that is, the prefix of the result shape excluding the last element of value n. Must be broadcast-compatible with alpha.shape[:-1]. The default (None) produces a result shape equal to alpha.shape.
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified dtype and shape given by shape + (alpha.shape[-1],) if shape is not None, or else alpha.shape.

Sample from a double sided Maxwell distribution.

The values are distributed according to the probability density function:

\[f(x;\mu,\sigma) \propto z^2 e^{-z^2 / 2}\]

where \(z = (x - \mu) / \sigma\), with the center \(\mu\) specified by loc and the scale \(\sigma\) specified by scale.

Parameters:

key – a PRNG key.
loc – The location parameter of the distribution.
scale – The scale parameter of the distribution.
shape – The shape added to the parameters loc and scale broadcastable shape.
dtype – The type used for samples.

Returns:

A jnp.array of samples.

exponential( shape: collections.abc.Sequence[int] = (), dtype: str | type[Any] | numpy.dtype | jax._src.typing.SupportsDType | None = None, ) → jax.Array

Sample Exponential random values with given shape and float dtype.

The values are distributed according to the probability density function:

\[f(x) = e^{-x}\]

on the domain \(0 \le x < \infty\).

Parameters:

key – a PRNG key used as the random key.
shape – optional, a tuple of nonnegative integers representing the result shape. Default ().
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified shape and dtype.

Sample F-distribution random values with given shape and float dtype.

The values are distributed according to the probability density function:

\[f(x; \nu_1, \nu_2) \propto x^{\nu_1/2 - 1}\left(1 + \frac{\nu_1}{\nu_2}x\right)^{ -(\nu_1 + \nu_2) / 2}\]

on the domain \(0 < x < \infty\). Here \(\nu_1\) is the degrees of freedom of the numerator (dfnum), and \(\nu_2\) is the degrees of freedom of the denominator (dfden).

Parameters:

key – a PRNG key used as the random key.
dfnum – a float or array of floats broadcast-compatible with shape representing the numerator’s df of the distribution.
dfden – a float or array of floats broadcast-compatible with shape representing the denominator’s df of the distribution.
shape – optional, a tuple of nonnegative integers specifying the result shape. Must be broadcast-compatible with dfnum and dfden. The default (None) produces a result shape equal to dfnum.shape, and dfden.shape.
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified dtype and with shape given by shape if shape is not None, or else by df.shape.

Sample Gamma random values with given shape and float dtype.

The values are distributed according to the probability density function:

\[f(x;a) \propto x^{a - 1} e^{-x}\]

on the domain \(0 \le x < \infty\), with \(a > 0\).

This is the standard gamma density, with a unit scale/rate parameter. Dividing the sample output by the rate is equivalent to sampling from gamma(a, rate), and multiplying the sample output by the scale is equivalent to sampling from gamma(a, scale).

Parameters:

key – a PRNG key used as the random key.
a – a float or array of floats broadcast-compatible with shape representing the parameter of the distribution.
shape – optional, a tuple of nonnegative integers specifying the result shape. Must be broadcast-compatible with a. The default (None) produces a result shape equal to a.shape.
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified dtype and with shape given by shape if shape is not None, or else by a.shape.

See also

gamma : standard gamma sampler.

laplace( shape: collections.abc.Sequence[int] = (), dtype: str | type[Any] | numpy.dtype | jax._src.typing.SupportsDType | None = None, ) → jax.Array

Sample Laplace random values with given shape and float dtype.

The values are distributed according to the probability density function:

\[f(x) = \frac{1}{2}e^{-|x|}\]

Parameters:

key – a PRNG key used as the random key.
shape – optional, a tuple of nonnegative integers representing the result shape. Default ().
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified shape and dtype.

logistic( shape: collections.abc.Sequence[int] = (), dtype: str | type[Any] | numpy.dtype | jax._src.typing.SupportsDType | None = None, ) → jax.Array

Sample logistic random values with given shape and float dtype.

The values are distributed according to the probability density function:

\[f(x) = \frac{e^{-x}}{(1 + e^{-x})^2}\]

Parameters:

key – a PRNG key used as the random key.
shape – optional, a tuple of nonnegative integers representing the result shape. Default ().
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified shape and dtype.

maxwell( shape: collections.abc.Sequence[int] = (), dtype: str | type[Any] | numpy.dtype | jax._src.typing.SupportsDType | None = None, ) → jax.Array

Sample from a one sided Maxwell distribution.

The values are distributed according to the probability density function:

\[f(x) \propto x^2 e^{-x^2 / 2}\]

on the domain \(0 \le x < \infty\).

Parameters:

key – a PRNG key.
shape – The shape of the returned samples.
dtype – The type used for samples.

Returns:

A jnp.array of samples, of shape shape.

Sample multivariate normal random values with given mean and covariance.

The values are returned according to the probability density function:

\[f(x;\mu, \Sigma) = (2\pi)^{-k/2} \det(\Sigma)^{-1}e^{-\frac{1}{2}(x - \mu)^T \Sigma^{-1} (x - \mu)}\]

where \(k\) is the dimension, \(\mu\) is the mean (given by mean) and \(\Sigma\) is the covariance matrix (given by cov).

Parameters:

key – a PRNG key used as the random key.
mean – a mean vector of shape (..., n).
cov – a positive definite covariance matrix of shape (..., n, n). The batch shape ... must be broadcast-compatible with that of mean.
shape – optional, a tuple of nonnegative integers specifying the result batch shape; that is, the prefix of the result shape excluding the last axis. Must be broadcast-compatible with mean.shape[:-1] and cov.shape[:-2]. The default (None) produces a result batch shape by broadcasting together the batch shapes of mean and cov.
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).
method – optional, a method to compute the factor of cov. Must be one of ‘svd’, ‘eigh’, and ‘cholesky’. Default ‘cholesky’. For singular covariance matrices, use ‘svd’ or ‘eigh’.

Returns:

A random array with the specified dtype and shape given by shape + mean.shape[-1:] if shape is not None, or else broadcast_shapes(mean.shape[:-1], cov.shape[:-2]) + mean.shape[-1:].

normal( shape: collections.abc.Sequence[int] = (), dtype: str | type[Any] | numpy.dtype | jax._src.typing.SupportsDType | None = None, *, out_sharding=None, ) → jax.Array

Sample standard normal random values with given shape and float dtype.

The values are returned according to the probability density function:

\[f(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}\]

on the domain \(-\infty < x < \infty\)

Parameters:

key – a PRNG key used as the random key.
shape – optional, a tuple of nonnegative integers representing the result shape. Default ().
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified shape and dtype.

Sample uniformly from the orthogonal group O(n).

If the dtype is complex, sample uniformly from the unitary group U(n).

For unequal rows and columns, this samples a semi-orthogonal matrix instead. That is, if \(A\) is the resulting matrix and \(A^*\) is its conjugate transpose, then:

If \(n \leq m\), the rows are mutually orthonormal: \(A A^* = I_n\).
If \(m \leq n\), the columns are mutually orthonormal: \(A^* A = I_m\).

Parameters:

key – a PRNG key used as the random key.
n – an integer indicating the number of rows.
shape – optional, the batch dimensions of the result. Default ().
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).
m – an integer indicating the number of columns. Defaults to n.

Returns:

A random array of shape (*shape, n, m) and specified dtype.

References

Sample Pareto random values with given shape and float dtype.

The values are distributed according to the probability density function:

\[f(x; b) = b / x^{b + 1}\]

on the domain \(1 \le x < \infty\) with \(b > 0\)

Parameters:

key – a PRNG key used as the random key.
b – a float or array of floats broadcast-compatible with shape representing the parameter of the distribution.
shape – optional, a tuple of nonnegative integers specifying the result shape. Must be broadcast-compatible with b. The default (None) produces a result shape equal to b.shape.
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified dtype and with shape given by shape if shape is not None, or else by b.shape.

Returns a randomly permuted array or range.

Parameters:

key – a PRNG key used as the random key.
x – int or array. If x is an integer, randomly shuffle np.arange(x). If x is an array, randomly shuffle its elements.
axis – int, optional. The axis which x is shuffled along. Default is 0.
independent – bool, optional. If set to True, each individual vector along the given axis is shuffled independently. Default is False.

Returns:

A shuffled version of x or array range

Sample Poisson random values with given shape and integer dtype.

The values are distributed according to the probability mass function:

\[f(k; \lambda) = \frac{\lambda^k e^{-\lambda}}{k!}\]

Where k is a non-negative integer and \(\lambda > 0\).

Parameters:

key – a PRNG key used as the random key.
lam – rate parameter (mean of the distribution), must be >= 0. Must be broadcast-compatible with shape
shape – optional, a tuple of nonnegative integers representing the result shape. Default (None) produces a result shape equal to lam.shape.
dtype – optional, a integer dtype for the returned values (default int64 if jax_enable_x64 is true, otherwise int32).

Returns:

A random array with the specified dtype and with shape given by shape if shape is not None, or else by ``lam.shape.

rademacher( shape: collections.abc.Sequence[int] = (), dtype: str | type[Any] | numpy.dtype | jax._src.typing.SupportsDType | None = None, ) → jax.Array

Sample from a Rademacher distribution.

The values are distributed according to the probability mass function:

\[f(k) = \frac{1}{2}(\delta(k - 1) + \delta(k + 1))\]

on the domain \(k \in \{-1, 1\}\), where \(\delta(x)\) is the dirac delta function.

Parameters:

key – a PRNG key.
shape – The shape of the returned samples. Default ().
dtype – The type used for samples.

Returns:

A jnp.array of samples, of shape shape. Each element in the output has a 50% change of being 1 or -1.

Sample uniform random values in [minval, maxval) with given shape/dtype.

Parameters:

key – a PRNG key used as the random key.
shape – a tuple of nonnegative integers representing the shape.
minval – int or array of ints broadcast-compatible with shape, a minimum (inclusive) value for the range.
maxval – int or array of ints broadcast-compatible with shape, a maximum (exclusive) value for the range.
dtype – optional, an int dtype for the returned values (default int64 if jax_enable_x64 is true, otherwise int32).

Returns:

A random array with the specified shape and dtype.

Note

randint() uses a modulus-based computation that is known to produce slightly biased values in some cases. The magnitude of the bias scales as (maxval - minval) * ((2 ** nbits ) % (maxval - minval)) / 2 ** nbits: in words, the bias goes to zero when (maxval - minval) is a power of 2, and otherwise the bias will be small whenever (maxval - minval) is small compared to the range of the sampled type.

To reduce this bias, 8-bit and 16-bit values will always be sampled at 32-bit and then cast to the requested type. If you find yourself sampling values for which this bias may be problematic, a possible alternative is to sample via uniform:

def randint_via_uniform(key, shape, minval, maxval, dtype):
  u = jax.random.uniform(key, shape, minval=minval - 0.5, maxval=maxval - 0.5)
  return u.round().astype(dtype)

But keep in mind this method has its own biases due to floating point rounding errors, and in particular there may be some integers in the range [minval, maxval) that are impossible to produce with this approach.

Sample Rayleigh random values with given shape and float dtype.

The values are returned according to the probability density function:

\[f(x;\sigma) \propto xe^{-x^2/(2\sigma^2)}\]

on the domain \(-\infty < x < \infty\), and where \(\sigma > 0\) is the scale parameter of the distribution.

Parameters:

key – a PRNG key used as the random key.
scale – a float or array of floats broadcast-compatible with shape representing the parameter of the distribution.
shape – optional, a tuple of nonnegative integers specifying the result shape. Must be broadcast-compatible with scale. The default (None) produces a result shape equal to scale.shape.
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified dtype and with shape given by shape if shape is not None, or else by scale.shape.

Sample Student’s t random values with given shape and float dtype.

The values are distributed according to the probability density function:

\[f(t; \nu) \propto \left(1 + \frac{t^2}{\nu}\right)^{-(\nu + 1)/2}\]

Where \(\nu > 0\) is the degrees of freedom, given by the parameter df.

Parameters:

key – a PRNG key used as the random key.
df – a float or array of floats broadcast-compatible with shape representing the degrees of freedom parameter of the distribution.
shape – optional, a tuple of nonnegative integers specifying the result shape. Must be broadcast-compatible with df. The default (None) produces a result shape equal to df.shape.
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified dtype and with shape given by shape if shape is not None, or else by df.shape.

Sample truncated standard normal random values with given shape and dtype.

The values are returned according to the probability density function:

\[f(x) \propto e^{-x^2/2}\]

on the domain \(\rm{lower} < x < \rm{upper}\).

Parameters:

key – a PRNG key used as the random key.
lower – a float or array of floats representing the lower bound for truncation. Must be broadcast-compatible with upper.
upper – a float or array of floats representing the upper bound for truncation. Must be broadcast-compatible with lower.
shape – optional, a tuple of nonnegative integers specifying the result shape. Must be broadcast-compatible with lower and upper. The default (None) produces a result shape by broadcasting lower and upper.
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified dtype and shape given by shape if shape is not None, or else by broadcasting lower and upper. Returns values in the open interval (lower, upper).

Sample uniform random values in [minval, maxval) with given shape/dtype.

Parameters:

key – a PRNG key used as the random key.
shape – optional, a tuple of nonnegative integers representing the result shape. Default ().
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).
minval – optional, a minimum (inclusive) value broadcast-compatible with shape for the range (default 0).
maxval – optional, a maximum (exclusive) value broadcast-compatible with shape for the range (default 1).

Returns:

A random array with the specified shape and dtype.

Sample Wald random values with given shape and float dtype.

The values are returned according to the probability density function:

\[f(x;\mu) = \frac{1}{\sqrt{2\pi x^3}} \exp\left(-\frac{(x - \mu)^2}{2\mu^2 x}\right)\]

on the domain \(-\infty < x < \infty\), and where \(\mu > 0\) is the location parameter of the distribution.

Parameters:

key – a PRNG key used as the random key.
mean – a float or array of floats broadcast-compatible with shape representing the mean parameter of the distribution.
shape – optional, a tuple of nonnegative integers specifying the result shape. Must be broadcast-compatible with mean. The default (None) produces a result shape equal to np.shape(mean).
dtype – optional, a float dtype for the returned values (default float64 if jax_enable_x64 is true, otherwise float32).

Returns:

A random array with the specified dtype and with shape given by shape if shape is not None, or else by mean.shape.

Sample from a Weibull distribution.

The values are distributed according to the probability density function:

\[f(x;\sigma,c) \propto x^{c - 1} \exp(-(x / \sigma)^c)\]

on the domain \(0 < x < \infty\), where \(c > 0\) is the concentration parameter, and \(\sigma > 0\) is the scale parameter.

Parameters:

key – a PRNG key.
scale – The scale parameter of the distribution.
concentration – The concentration parameter of the distribution.
shape – The shape added to the parameters loc and scale broadcastable shape.
dtype – The type used for samples.

Returns:

A jnp.array of samples.