tfp.distributions.TruncatedNormal

Class `TruncatedNormal`

The Truncated Normal distribution.

Inherits From: `Distribution`

Mathematical details

The truncated normal is a normal distribution bounded between `low` and `high` (the pdf is 0 outside these bounds and renormalized).

Samples from this distribution are differentiable with respect to `loc`, `scale` as well as the bounds, `low` and `high`, i.e., this implementation is fully reparameterizeable.

For more details, see here.

Mathematical Details

The probability density function (pdf) of this distribution is:

``````  pdf(x; loc, scale, low, high) =
{ (2 pi)**(-0.5) exp(-0.5 y**2) / (scale * z) for low <= x <= high
{ 0                                    otherwise
y = (x - loc)/scale
z = NormalCDF((high - loc) / scale) - NormalCDF((lower - loc) / scale)
``````

where:

• `NormalCDF` is the cumulative density function of the Normal distribution with 0 mean and unit variance.

This is a scalar distribution so the event shape is always scalar and the dimensions of the parameters defined the batch_shape.

Examples

``````
tfd = tfp.distributions
# Define a batch of two scalar TruncatedNormals which modes at 0. and 1.0
dist = tfd.TruncatedNormal(loc=[0., 1.], scale=1.0,
low=[-1., 0.],
high=[1., 1.])

# Evaluate the pdf of the distributions at 0.5 and 0.8 respectively returning
# a 2-vector tensor.
dist.prob([0.5, 0.8])

# Get 3 samples, returning a 3 x 2 tensor.
dist.sample()
``````

`__init__`

``````__init__(
loc,
scale,
low,
high,
validate_args=False,
allow_nan_stats=True,
name='TruncatedNormal'
)
``````

Construct TruncatedNormal.

All parameters of the distribution will be broadcast to the same shape, so the resulting distribution will have a batch_shape of the broadcast shape of all parameters.

Args:

• `loc`: Floating point tensor; the mean of the normal distribution(s) ( note that the mean of the resulting distribution will be different since it is modified by the bounds).
• `scale`: Floating point tensor; the std deviation of the normal distribution(s).
• `low`: `float` `Tensor` representing lower bound of the distribution's support. Must be such that `low < high`.
• `high`: `float` `Tensor` representing upper bound of the distribution's support. Must be such that `low < high`.
• `validate_args`: Python `bool`, default `False`. When `True` distribution parameters are checked at run-time.
• `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.

Properties

`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.

Returns:

• `allow_nan_stats`: Python `bool`.

`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.

Returns:

• `batch_shape`: `TensorShape`, possibly unknown.

`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.

Returns:

• `event_shape`: `TensorShape`, possibly unknown.

`name`

Name prepended to all ops created by this `Distribution`.

`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`.

Returns:

An instance of `ReparameterizationType`.

`scale`

Distribution parameter for the scale.

`validate_args`

Python `bool` indicating possibly expensive checks are enabled.

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.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  # => 
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape  # => [4, 2, 3, 1]
mvn2.event_shape  # => 
``````

Args:

• `slices`: slices from the [] operator

Returns:

• `dist`: A new `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`: name to give to the op

Returns:

• `batch_shape`: `Tensor`.

`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`.
• `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`

``````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`

``````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`

``````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`

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

Returns:

• `event_shape`: `Tensor`.

`is_scalar_batch`

``````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`

``````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`

``````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`

``````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`

``````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`

``````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`

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

Mean.

`mode`

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

Mode.

`param_shapes`

``````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`

``````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`

``````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`

``````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`

``````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 seed for RNG
• `name`: name to give to the op.
• `**kwargs`: Named arguments forwarded to subclass implementation.

Returns:

• `samples`: a `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:

• `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`

``````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`

``````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()`.