tf.contrib.distributions.SinhArcsinh

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The SinhArcsinh transformation of a distribution on (-inf, inf).

Inherits From: TransformedDistribution

tf.contrib.distributions.SinhArcsinh(
    loc, scale, skewness=None, tailweight=None, distribution=None,
    validate_args=False, allow_nan_stats=True, name='SinhArcsinh'
)

This distribution models a random variable, 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 random variable Z, we define the SinhArcsinh transformation of Z, Y, parameterized by (loc, scale, skewness, tailweight), via the relation:

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

loc Floating-point Tensor.
scale Tensor of same dtype as loc.
skewness Skewness parameter. Default is 0.0 (no skew).
tailweight Tailweight parameter. Default is 1.0 (unchanged tailweight)
distribution tf.Distribution-like instance. Distribution that is transformed to produce this distribution. Default is tfp.distributions.Normal(0., 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 SinhArcsinh 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.

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 Tensors 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) * (2 / F(2)). </td> </tr><tr> <td>name</td> <td> Name prepended to all ops created by thisDistribution. </td> </tr><tr> <td>parameters</td> <td> Dictionary of parameters used to instantiate thisDistribution. </td> </tr><tr> <td>reparameterization_type` Describes how samples from the distribution are reparameterized.

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

scale The LinearOperator scale in Y := loc + scale @ F(Z) * (2 / F(2)). </td> </tr><tr> <td>skewness</td> <td> Controls the skewness.Skewness > 0means right skew. </td> </tr><tr> <td>tailweight</td> <td> Controls the tail decay.tailweight > 1means faster than Normal. </td> </tr><tr> <td>validate_args</td> <td> Pythonbool` indicating possibly expensive checks are enabled.

Methods

batch_shape_tensor

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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

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cdf(
    value, name='cdf'
)

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.

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

copy

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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

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covariance(
    name='covariance'
)

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.

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

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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), (Shanon) 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 Python 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 (Shanon) cross entropy.

entropy

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entropy(
    name='entropy'
)

Shannon entropy in nats.

event_shape_tensor

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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

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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

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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

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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 (Shanon) cross entropy, and H[.] denotes (Shanon) entropy.

Args
other tfp.distributions.Distribution instance.
name Python 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

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log_cdf(
    value, name='log_cdf'
)

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.

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

log_prob

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log_prob(
    value, name='log_prob'
)

Log probability density/mass function.

Args
value float or double Tensor.
name Python str prepended to names of ops created by this function.

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

log_survival_function

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log_survival_function(
    value, name='log_survival_function'
)

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.

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

mean

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mean(
    name='mean'
)

Mean.

mode

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mode(
    name='mode'
)

Mode.

param_shapes

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@classmethod
param_shapes(
    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

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@classmethod
param_static_shapes(
    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

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prob(
    value, name='prob'
)

Probability density/mass function.

Args
value float or double Tensor.
name Python str prepended to names of ops created by this function.

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

quantile

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quantile(
    value, name='quantile'
)

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.

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

sample

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sample(
    sample_shape=(), seed=None, name='sample'
)

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 seed for RNG
name name to give to the op.

Returns
samples a Tensor with prepended dimensions sample_shape.

stddev

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stddev(
    name='stddev'
)

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.

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

survival_function

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survival_function(
    value, name='survival_function'
)

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.

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

variance

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variance(
    name='variance'
)

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.

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