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tf.contrib.distributions.VectorSinhArcsinhDiag

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

Inherits From: `TransformedDistribution`

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) * (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`. 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 `tfp.distributions.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.

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

`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) * (2 / F(2)). </td> </tr><tr> <td>```name```</td> <td> Name prepended to all ops created by this```Distribution```. </td> </tr><tr> <td>```parameters```</td> <td> Dictionary of parameters used to instantiate this```Distribution```. </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 > 0```means right skew. </td> </tr><tr> <td>```tailweight```</td> <td> Controls the tail decay.```tailweight > 1```means faster than Normal. </td> </tr><tr> <td>```validate_args```</td> <td> Python```bool` indicating possibly expensive checks are enabled.

Methods

`batch_shape_tensor`

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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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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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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.

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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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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Shannon entropy in nats.

`event_shape_tensor`

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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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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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Indicates that `event_shape == []`.

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

Returns
`is_scalar_event` `bool` scalar `Tensor`.