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# tfp.distributions.VectorSinhArcsinhDiag

The (diagonal) SinhArcsinh transformation of a distribution on `R^k`.

Inherits From: `TransformedDistribution`

``````tfp.distributions.VectorSinhArcsinhDiag(
loc=None, scale_diag=None, scale_identity_multiplier=None, skewness=None,
tailweight=None, distribution=None, validate_args=False, allow_nan_stats=True,
name='VectorSinhArcsinhDiag'
)
``````

This distribution models a random vector `Y = (Y1,...,Yk)`, making use of a `SinhArcsinh` transformation (which has adjustable tailweight and skew), a rescaling, and a shift.

The `SinhArcsinh` transformation of the Normal is described in great depth in Sinh-arcsinh distributions. Here we use a slightly different parameterization, in terms of `tailweight` and `skewness`. Additionally we allow for distributions other than Normal, and control over `scale` as well as a 'shift' parameter `loc`.

#### Mathematical Details

Given iid random vector `Z = (Z1,...,Zk)`, we define the VectorSinhArcsinhDiag transformation of `Z`, `Y`, parameterized by `(loc, scale, skewness, tailweight)`, via the relation (with `@` denoting matrix multiplication):

``````Y := loc + scale @ F(Z)
F(Z) := Sinh( (Arcsinh(Z) + skewness) * tailweight ) * (2 / F_0(2))
F_0(Z) := Sinh( Arcsinh(Z) * tailweight )
``````

This distribution is similar to the location-scale transformation `L(Z) := loc + scale @ Z` in the following ways:

• If `skewness = 0` and `tailweight = 1` (the defaults), `F(Z) = Z`, and then `Y = L(Z)` exactly.
• `loc` is used in both to shift the result by a constant factor.
• The multiplication of `scale` by `2 / F_0(2)` ensures that if `skewness = 0` `P[Y - loc <= 2 * scale] = P[L(Z) - loc <= 2 * scale]`. Thus it can be said that the weights in the tails of `Y` and `L(Z)` beyond `loc + 2 * scale` are the same.

This distribution is different than `loc + scale @ Z` due to the reshaping done by `F`:

• Positive (negative) `skewness` leads to positive (negative) skew.
• positive skew means, the mode of `F(Z)` is 'tilted' to the right.
• positive skew means positive values of `F(Z)` become more likely, and negative values become less likely.
• Larger (smaller) `tailweight` leads to fatter (thinner) tails.
• Fatter tails mean larger values of `|F(Z)|` become more likely.
• `tailweight < 1` leads to a distribution that is 'flat' around `Y = loc`, and a very steep drop-off in the tails.
• `tailweight > 1` leads to a distribution more peaked at the mode with heavier tails.

To see the argument about the tails, note that for `|Z| >> 1` and `|Z| >> (|skewness| * tailweight)**tailweight`, we have `Y approx 0.5 Z**tailweight e**(sign(Z) skewness * tailweight)`.

To see the argument regarding multiplying `scale` by `2 / F_0(2)`,

``````P[(Y - loc) / scale <= 2] = P[F(Z) * (2 / F_0(2)) <= 2]
= P[F(Z) <= F_0(2)]
= P[Z <= 2]  (if F = F_0).
``````

#### Args:

• `loc`: Floating-point `Tensor`. If this is set to `None`, `loc` is implicitly `0`. When specified, may have shape `[B1, ..., Bb, k]` where `b >= 0` and `k` is the event size.
• `scale_diag`: Non-zero, floating-point `Tensor` representing a diagonal matrix added to `scale`. May have shape `[B1, ..., Bb, k]`, `b >= 0`, and characterizes `b`-batches of `k x k` diagonal matrices added to `scale`. When both `scale_identity_multiplier` and `scale_diag` are `None` then `scale` is the `Identity`.
• `scale_identity_multiplier`: Non-zero, floating-point `Tensor` representing a scale-identity-matrix added to `scale`. May have shape `[B1, ..., Bb]`, `b >= 0`, and characterizes `b`-batches of scale `k x k` identity matrices added to `scale`. When both `scale_identity_multiplier` and `scale_diag` are `None` then `scale` is the `Identity`.
• `skewness`: Skewness parameter. floating-point `Tensor` with shape broadcastable with `event_shape`.
• `tailweight`: Tailweight parameter. floating-point `Tensor` with shape broadcastable with `event_shape`.
• `distribution`: `tf.Distribution`-like instance. Distribution from which `k` iid samples are used as input to transformation `F`. Default is `tfd.Normal(loc=0., scale=1.)`. Must be a scalar-batch, scalar-event distribution. Typically `distribution.reparameterization_type = FULLY_REPARAMETERIZED` or it is a function of non-trainable parameters. WARNING: If you backprop through a VectorSinhArcsinhDiag sample and `distribution` is not `FULLY_REPARAMETERIZED` yet is a function of trainable variables, then the gradient will be incorrect!
• `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.

#### Attributes:

• `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.

• `bijector`: Function transforming x => y.

• `distribution`: Base distribution, p(x).

• `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.

• `loc`: The `loc` in `Y := loc + scale @ F(Z)`.

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

• `name_scope`: Returns a `tf.name_scope` instance for this class.

• `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`.

• `scale`: The `LinearOperator` `scale` in `Y := loc + scale @ F(Z)`.

• `skewness`: Controls the skewness. `Skewness > 0` means right skew.

• `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
assert list(a.submodules) == [b, c]
assert list(b.submodules) == [c]
assert list(c.submodules) == []
``````
• `tailweight`: Controls the tail decay. `tailweight > 1` means faster than Normal.
• `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.

#### Raises:

• `ValueError`: if at most `scale_identity_multiplier` is specified.

## Methods

### `__getitem__`

View source

``````__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.normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.cholesky(cov)
loc = tf.random.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`: slices from the [] operator

#### Returns:

• `dist`: A new `tfd.Distribution` instance with sliced parameters.

### `__iter__`

View source

``````__iter__()
``````

### `batch_shape_tensor`

View source

``````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`: name to give to the op

#### Returns:

• `batch_shape`: `Tensor`.

### `cdf`

View source

``````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`.
• `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

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

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(
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:

• `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

``````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`.

#### Returns:

• `cross_entropy`: `self.dtype` `Tensor` with shape `[B1, ..., Bn]` representing `n` different calculations of (Shannon) cross entropy.

### `entropy`

View source

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

Shannon entropy in nats.

### `event_shape_tensor`

View source

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

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

#### Args:

• `name`: name to give to the op

#### Returns:

• `event_shape`: `Tensor`.

### `is_scalar_batch`

View source

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

Indicates that `batch_shape == []`.

#### Args:

• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `is_scalar_batch`: `bool` scalar `Tensor`.

### `is_scalar_event`

View source

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

Indicates that `event_shape == []`.

#### Args:

• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `is_scalar_event`: `bool` scalar `Tensor`.

### `kl_divergence`

View source

``````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.

#### Returns:

• `kl_divergence`: `self.dtype` `Tensor` with shape `[B1, ..., Bn]` representing `n` different calculations of the Kullback-Leibler divergence.

### `log_cdf`

View source

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

#### Returns:

• `logcdf`: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `log_prob`

View source

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

Log probability density/mass function.

#### 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:

• `log_prob`: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `log_survival_function`

View source

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

#### Returns:

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

### `mean`

View source

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

Mean.

### `mode`

View source

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

Mode.

### `param_shapes`

View source

``````@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`: name to prepend ops with.

#### Returns:

`dict` of parameter name to `Tensor` shapes.

### `param_static_shapes`

View source

``````@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:

• `ValueError`: if `sample_shape` is a `TensorShape` and is not fully defined.

### `prob`

View source

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

Probability density/mass function.

#### 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:

• `prob`: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `quantile`

View source

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

#### Returns:

• `quantile`: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `sample`

View source

``````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:

• `sample_shape`: 0D or 1D `int32` `Tensor`. Shape of the generated samples.
• `seed`: Python integer or `tfp.util.SeedStream` instance, for seeding PRNG.
• `name`: name to give to the op.
• `**kwargs`: Named arguments forwarded to subclass implementation.

#### Returns:

• `samples`: a `Tensor` with prepended dimensions `sample_shape`.

### `stddev`

View source

``````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:

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

#### Returns:

• `stddev`: Floating-point `Tensor` with shape identical to `batch_shape + event_shape`, i.e., the same shape as `self.mean()`.

### `survival_function`

View source

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

#### Returns:

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

### `variance`

View source

``````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:

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

#### Returns:

• `variance`: Floating-point `Tensor` with shape identical to `batch_shape + event_shape`, i.e., the same shape as `self.mean()`.

### `with_name_scope`

``````@classmethod
with_name_scope(
cls, method
)
``````

Decorator to automatically enter the module name scope.

``````class MyModule(tf.Module):
@tf.Module.with_name_scope
def __call__(self, x):
if not hasattr(self, 'w'):
self.w = tf.Variable(tf.random.normal([x.shape[1], 64]))
return tf.matmul(x, self.w)
``````

Using the above module would produce `tf.Variable`s and `tf.Tensor`s whose names included the module name:

``````mod = MyModule()
mod(tf.ones([8, 32]))
# ==> <tf.Tensor: ...>
mod.w
# ==> <tf.Variable ...'my_module/w:0'>
``````

#### Args:

• `method`: The method to wrap.

#### Returns:

The original method wrapped such that it enters the module's name scope.