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Ordered logistic distribution.

Inherits From: `Distribution`

```
tfp.experimental.substrates.jax.distributions.OrderedLogistic(
cutpoints, loc, dtype=tf.int32, validate_args=False, allow_nan_stats=True,
name='OrderedLogistic'
)
```

The OrderedLogistic distribution is parameterized by a location and a set of
cutpoints. It is defined over the integers `{0, 1, ..., K}`

for `K`

non-decreasing cutpoints.

One often useful way to interpret this distribution is by imagining a draw
from a latent/unobserved logistic distribution with location `loc`

and
scale `1`

and then only considering the index of the bin defined by the `K`

cutpoints this draw falls between. An example implementation of this idea is
as follows:

```
cutpoints = [0.0, 1.0]
loc = 0.5
def probs_from_latent(latent):
augmented_cutpoints = tf.concat([[-np.inf], cutpoints, [np.inf]], axis=0)
below = latent[..., tf.newaxis] < augmented_cutpoints[1:]
above = latent[..., tf.newaxis] > augmented_cutpoints[:-1]
return tf.cast(below & above, tf.float32)
latent_and_ordered_logistic = tfd.JointDistributionSequential([
tfd.Logistic(loc=loc, scale=1.),
lambda l: tfd.Categorical(probs=probs_from_latent(l), dtype=tf.float32)
])
tf.stack(latent_and_ordered_logistic.sample(5), axis=1)
# ==> array([[ 0.6434291, 1. ],
[ 3.0963311, 2. ],
[-1.2692463, 0. ],
[-3.3595495, 0. ],
[ 0.8468886, 1. ]], dtype=float32)
```

which displays that latent draws < `cutpoints[0] = 0.`

are category 0, latent
draws between `cutpoints[0] = 0.`

and `cutpoints[1] = 1.`

are category 1, and
finally latent draws > `cutpoints[1] = 1.`

(the final cutpoint in this
example) are the top category of 2.

This distribution can be useful for modelling outcomes which have inherent
ordering but no real numerical values, for example modelling the outcome of a
survey question where the responses are `[bad, mediocre, good]`

, which would
be coded as `[0, 1, 2]`

and the model would contain two cutpoints (`K = 2`

).

#### Mathematical Details

The survival function (s) is:

```
s(x; c, eta) = P(X > x)
= sigmoid(eta - concat([-inf, c, inf])[x+1])
```

where `loc = eta`

is the location of a latent logistic distribution and
`cutpoints = c`

define points to split up this latent distribution. The
concatenation of the cutpoints, `concat([-inf, c, inf])`

, ensures that ```
s(K) =
P(X > K) = 0
```

and `s(-1) = P(X > -1) = 1`

which aids in the definition of the
probability mass function (pmf):

```
pmf(x; c, eta) = P(X > x-1) - P(x > x)
= s(x-1; c, eta) - s(x; c, eta)
```

#### Examples

Create a symmetric 4-class distribution:

```
import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.jax
tfd = tfp.distributions
dist = tfd.OrderedLogistic(cutpoints=[-2., 0., 2.], loc=0.)
dist.categorical_probs()
# ==> array([0.11920293, 0.38079706, 0.3807971 , 0.11920291], dtype=float32)
```

Create a batch of 3 4-class distributions via batching the location of the
underlying latent logistic distribution. Additionally, compared to the above
example, the cutpoints have moved closer together/to zero, thus the
probability of a latent draw falling in the inner two categories has shrunk
for the `loc = 0`

case:

```
dist = tfd.OrderedLogistic(cutpoints=[-1., 0., 1.], loc=[-1., 0., 1.])
dist.categorical_probs()
# ==> array([[0.5 , 0.23105855, 0.1497385 , 0.11920291],
[0.2689414 , 0.23105861, 0.23105855, 0.26894143],
[0.11920293, 0.14973842, 0.23105861, 0.5 ]], dtype=float32)
```

Some further functionallity:

```
dist = tfd.OrderedLogistic(cutpoints=[-1., 0., 2.], loc=0.)
dist.prob([0, 3])
# ==> array([0.2689414 , 0.11920291], dtype=float32)
dist.log_prob(1)
# ==> -1.4650838
dist.sample(3)
# ==> array([0, 1, 1], dtype=int32)
dist.entropy()
# ==> 1.312902
```

#### Args:

: A floating-point`cutpoints`

`Tensor`

with shape`[B1, ..., Bb, K]`

where`b >= 0`

indicates the number of batch dimensions. Each entry is then a`K`

-length vector of cutpoints. The vector of cutpoints should be non-decreasing, which is only checked if`validate_args=True`

.: A floating-point`loc`

`Tensor`

with shape`[B1, ..., Bb]`

where`b >= 0`

indicates the number of batch dimensions. The entries represent the mean(s) of the latent logistic distribution(s). Different batch shapes for`cutpoints`

and`loc`

are permitted, with the distribution`batch_shape`

being`tf.shape(loc[..., tf.newaxis] - cutpoints)[:-1]`

assuming the subtraction is a valid broadcasting operation.: The type of the event samples (default: int32).`dtype`

: Python`validate_args`

`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.: Python`allow_nan_stats`

`bool`

, default`True`

. When`True`

, statistics (e.g. mode) 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.: Python`name`

`str`

name prefixed to Ops created by this class.

#### Attributes:

: Python`allow_nan_stats`

`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.

: Shape of a single sample from a single event index as a`batch_shape`

`TensorShape`

.May be partially defined or unknown.

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

: Input argument`cutpoints`

`cutpoints`

.: The`dtype`

`DType`

of`Tensor`

s handled by this`Distribution`

.: Shape of a single sample from a single batch as a`event_shape`

`TensorShape`

.May be partially defined or unknown.

: Input argument`loc`

`loc`

.: Name prepended to all ops created by this`name`

`Distribution`

.: Dictionary of parameters used to instantiate this`parameters`

`Distribution`

.: Describes how samples from the distribution are reparameterized.`reparameterization_type`

Currently this is one of the static instances

`tfd.FULLY_REPARAMETERIZED`

or`tfd.NOT_REPARAMETERIZED`

.`trainable_variables`

: Python`validate_args`

`bool`

indicating possibly expensive checks are enabled.`variables`

## Methods

`__getitem__`

```
__getitem__(
slices
)
```

Slices the batch axes of this distribution, returning a new instance.

```
b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
b.batch_shape # => [3, 5, 7, 9]
b2 = b[:, tf.newaxis, ..., -2:, 1::2]
b2.batch_shape # => [3, 1, 5, 2, 4]
x = tf.random.stateless_normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.cholesky(cov)
loc = tf.random.stateless_normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape # => [4, 5, 3]
mvn.event_shape # => [2]
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape # => [4, 2, 3, 1]
mvn2.event_shape # => [2]
```

#### Args:

: slices from the [] operator`slices`

#### Returns:

: A new`dist`

`tfd.Distribution`

instance with sliced parameters.

`__iter__`

```
__iter__()
```

`batch_shape_tensor`

```
batch_shape_tensor(
name='batch_shape_tensor'
)
```

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 to give to the op`name`

#### Returns:

:`batch_shape`

`Tensor`

.

`categorical_log_probs`

```
categorical_log_probs()
```

Log probabilities for the `K+1`

ordered categories.

`categorical_probs`

```
categorical_probs()
```

Probabilities for the `K+1`

ordered categories.

`cdf`

```
cdf(
value, name='cdf', **kwargs
)
```

Cumulative distribution function.

Given random variable `X`

, the cumulative distribution function `cdf`

is:

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

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: a`cdf`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`copy`

```
copy(
**override_parameters_kwargs
)
```

Creates a deep copy of the distribution.

#### Args:

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

#### Returns:

: A new instance of`distribution`

`type(self)`

initialized from the union of self.parameters and override_parameters_kwargs, i.e.,`dict(self.parameters, **override_parameters_kwargs)`

.

`covariance`

```
covariance(
name='covariance', **kwargs
)
```

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:

: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: Floating-point`covariance`

`Tensor`

with shape`[B1, ..., Bn, k', k']`

where the first`n`

dimensions are batch coordinates and`k' = reduce_prod(self.event_shape)`

.

`cross_entropy`

```
cross_entropy(
other, name='cross_entropy'
)
```

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`

.

`other`

types with built-in registrations: `OrderedLogistic`

#### Args:

:`other`

`tfp.distributions.Distribution`

instance.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

:`cross_entropy`

`self.dtype`

`Tensor`

with shape`[B1, ..., Bn]`

representing`n`

different calculations of (Shannon) cross entropy.

`entropy`

```
entropy(
name='entropy', **kwargs
)
```

Shannon entropy in nats.

`event_shape_tensor`

```
event_shape_tensor(
name='event_shape_tensor'
)
```

Shape of a single sample from a single batch as a 1-D int32 `Tensor`

.

#### Args:

: name to give to the op`name`

#### Returns:

:`event_shape`

`Tensor`

.

`is_scalar_batch`

```
is_scalar_batch(
name='is_scalar_batch'
)
```

Indicates that `batch_shape == []`

.

#### Args:

: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

:`is_scalar_batch`

`bool`

scalar`Tensor`

.

`is_scalar_event`

```
is_scalar_event(
name='is_scalar_event'
)
```

Indicates that `event_shape == []`

.

#### Args:

: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

:`is_scalar_event`

`bool`

scalar`Tensor`

.

`kl_divergence`

```
kl_divergence(
other, name='kl_divergence'
)
```

Computes the Kullback--Leibler divergence.

Denote this distribution (`self`

) by `p`

and the `other`

distribution by
`q`

. Assuming `p, q`

are absolutely continuous with respect to reference
measure `r`

, the KL divergence is defined as:

```
KL[p, q] = E_p[log(p(X)/q(X))]
= -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
= H[p, q] - H[p]
```

where `F`

denotes the support of the random variable `X ~ p`

, `H[., .]`

denotes (Shannon) cross entropy, and `H[.]`

denotes (Shannon) entropy.

`other`

types with built-in registrations: `OrderedLogistic`

#### Args:

:`other`

`tfp.distributions.Distribution`

instance.: Python`name`

`str`

prepended to names of ops created by this function.

#### Returns:

:`kl_divergence`

`self.dtype`

`Tensor`

with shape`[B1, ..., Bn]`

representing`n`

different calculations of the Kullback-Leibler divergence.

`log_cdf`

```
log_cdf(
value, name='log_cdf', **kwargs
)
```

Log cumulative distribution function.

Given random variable `X`

, the cumulative distribution function `cdf`

is:

```
log_cdf(x) := Log[ P[X <= x] ]
```

Often, a numerical approximation can be used for `log_cdf(x)`

that yields
a more accurate answer than simply taking the logarithm of the `cdf`

when
`x << -1`

.

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: a`logcdf`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`log_prob`

```
log_prob(
value, name='log_prob', **kwargs
)
```

Log probability density/mass function.

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: a`log_prob`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`log_survival_function`

```
log_survival_function(
value, name='log_survival_function', **kwargs
)
```

Log survival function.

Given random variable `X`

, the survival function is defined:

```
log_survival_function(x) = Log[ P[X > x] ]
= Log[ 1 - P[X <= x] ]
= Log[ 1 - cdf(x) ]
```

Typically, different numerical approximations can be used for the log
survival function, which are more accurate than `1 - cdf(x)`

when `x >> 1`

.

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

`Tensor`

of shape `sample_shape(x) + self.batch_shape`

with values of type
`self.dtype`

.

`mean`

```
mean(
name='mean', **kwargs
)
```

Mean.

`mode`

```
mode(
name='mode', **kwargs
)
```

Mode.

`param_shapes`

```
@classmethod
param_shapes(
cls, sample_shape, name='DistributionParamShapes'
)
```

Shapes of parameters given the desired shape of a call to `sample()`

.

This is a class method that describes what key/value arguments are required
to instantiate the given `Distribution`

so that a particular shape is
returned for that instance's call to `sample()`

.

Subclasses should override class method `_param_shapes`

.

#### Args:

:`sample_shape`

`Tensor`

or python list/tuple. Desired shape of a call to`sample()`

.: name to prepend ops with.`name`

#### Returns:

`dict`

of parameter name to `Tensor`

shapes.

`param_static_shapes`

```
@classmethod
param_static_shapes(
cls, sample_shape
)
```

param_shapes with static (i.e. `TensorShape`

) shapes.

This is a class method that describes what key/value arguments are required
to instantiate the given `Distribution`

so that a particular shape is
returned for that instance's call to `sample()`

. Assumes that the sample's
shape is known statically.

Subclasses should override class method `_param_shapes`

to return
constant-valued tensors when constant values are fed.

#### Args:

:`sample_shape`

`TensorShape`

or python list/tuple. Desired shape of a call to`sample()`

.

#### Returns:

`dict`

of parameter name to `TensorShape`

.

#### Raises:

: if`ValueError`

`sample_shape`

is a`TensorShape`

and is not fully defined.

`prob`

```
prob(
value, name='prob', **kwargs
)
```

Probability density/mass function.

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: a`prob`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`quantile`

```
quantile(
value, name='quantile', **kwargs
)
```

Quantile function. Aka 'inverse cdf' or 'percent point function'.

Given random variable `X`

and `p in [0, 1]`

, the `quantile`

is:

```
quantile(p) := x such that P[X <= x] == p
```

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: a`quantile`

`Tensor`

of shape`sample_shape(x) + self.batch_shape`

with values of type`self.dtype`

.

`sample`

```
sample(
sample_shape=(), seed=None, name='sample', **kwargs
)
```

Generate samples of the specified shape.

Note that a call to `sample()`

without arguments will generate a single
sample.

#### Args:

: 0D or 1D`sample_shape`

`int32`

`Tensor`

. Shape of the generated samples.: Python integer or`seed`

`tfp.util.SeedStream`

instance, for seeding PRNG.: name to give to the op.`name`

: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: a`samples`

`Tensor`

with prepended dimensions`sample_shape`

.

`stddev`

```
stddev(
name='stddev', **kwargs
)
```

Standard deviation.

Standard deviation is defined as,

```
stddev = E[(X - E[X])**2]**0.5
```

where `X`

is the random variable associated with this distribution, `E`

denotes expectation, and `stddev.shape = batch_shape + event_shape`

.

#### Args:

: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: Floating-point`stddev`

`Tensor`

with shape identical to`batch_shape + event_shape`

, i.e., the same shape as`self.mean()`

.

`survival_function`

```
survival_function(
value, name='survival_function', **kwargs
)
```

Survival function.

Given random variable `X`

, the survival function is defined:

```
survival_function(x) = P[X > x]
= 1 - P[X <= x]
= 1 - cdf(x).
```

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

`Tensor`

of shape `sample_shape(x) + self.batch_shape`

with values of type
`self.dtype`

.

`variance`

```
variance(
name='variance', **kwargs
)
```

Variance.

Variance is defined as,

```
Var = E[(X - E[X])**2]
```

where `X`

is the random variable associated with this distribution, `E`

denotes expectation, and `Var.shape = batch_shape + event_shape`

.

#### Args:

: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: Floating-point`variance`

`Tensor`

with shape identical to`batch_shape + event_shape`

, i.e., the same shape as`self.mean()`

.