tf.contrib.distributions.StudentT

class tf.contrib.distributions.StudentT

See the guide: Statistical Distributions (contrib) > Univariate (scalar) distributions

Student's t distribution with degree-of-freedom parameter df.

Mathematical details

Write sigma for the scale and mu for the mean (both are scalars). The PDF of this distribution is:

f(x) = (1 + y**2 / df)**(-0.5 (df + 1)) / Z
where,
y(x) = (x - mu) / sigma
Z    = abs(sigma) sqrt(df pi) Gamma(0.5 df) / Gamma(0.5 (df + 1))

Notice that sigma has semantics more similar to standard deviation than variance. (Recall that the variance of the Student's t-distribution is sigma**2 df / (df - 2) when df > 2.)

Examples

Examples of initialization of one or a batch of distributions.

# Define a single scalar Student t distribution.
single_dist = tf.contrib.distributions.StudentT(df=3)

# Evaluate the pdf at 1, returning a scalar Tensor.
single_dist.pdf(1.)

# Define a batch of two scalar valued Student t's.
# The first has degrees of freedom 2, mean 1, and scale 11.
# The second 3, 2 and 22.
multi_dist = tf.contrib.distributions.StudentT(df=[2, 3],
                                               mu=[1, 2.],
                                               sigma=[11, 22.])

# Evaluate the pdf of the first distribution on 0, and the second on 1.5,
# returning a length two tensor.
multi_dist.pdf([0, 1.5])

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

Arguments are broadcast when possible.

# Define a batch of two Student's t distributions.
# Both have df 2 and mean 1, but different scales.
dist = tf.contrib.distributions.StudentT(df=2, mu=1, sigma=[11, 22.])

# Evaluate the pdf of both distributions on the same point, 3.0,
# returning a length 2 tensor.
dist.pdf(3.0)

Properties

allow_nan_stats

Python boolean 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 boolean.

df

Degrees of freedom in these Student's t distribution(s).

dtype

The DType of Tensors handled by this Distribution.

is_continuous

is_reparameterized

mu

Locations of these Student's t distribution(s).

name

Name prepended to all ops created by this Distribution.

parameters

Dictionary of parameters used to instantiate this Distribution.

sigma

Scaling factors of these Student's t distribution(s).

validate_args

Python boolean indicated possibly expensive checks are enabled.

Methods

__init__(df, mu, sigma, validate_args=False, allow_nan_stats=True, name='StudentT')

Construct Student's t distributions.

The distributions have degree of freedom df, mean mu, and scale sigma.

The parameters df, mu, and sigma must be shaped in a way that supports broadcasting (e.g. df + mu + sigma is a valid operation).

Args:

• df: Numeric Tensor. The degrees of freedom of the distribution(s). df must contain only positive values.
• mu: Numeric Tensor. The mean(s) of the distribution(s).
• sigma: Numeric Tensor. The scaling factor(s) for the distribution(s). Note that sigma is not technically the standard deviation of this distribution but has semantics more similar to std. deviation than variance.
• validate_args: Boolean, default False. Whether to assert that df > 0 and sigma > 0. If validate_args is False and inputs are invalid, correct behavior is not guaranteed.
• allow_nan_stats: Boolean, default True. If False, raise an exception if a statistic (e.g. mean/mode/etc...) is undefined for any batch member. If True, batch members with valid parameters leading to undefined statistics will return NaN for this statistic.
• name: The name to give Ops created by the initializer.

Raises:

• TypeError: if mu and sigma are different dtypes.

batch_shape(name='batch_shape')

Shape of a single sample from a single event index as a 1-D Tensor.

The product of the dimensions of the batch_shape is the number of independent distributions of this kind the instance represents.

Args:

• name: name to give to the op

Returns:

• batch_shape: Tensor.

cdf(value, name='cdf', **condition_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: The name to give this op. **condition_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(**override_parameters_kwargs)

Creates a deep copy of the distribution.

Note: the copy distribution may continue to depend on the original intialization arguments.

Args:

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

Returns:

• distribution: A new instance of type(self) intitialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

entropy(name='entropy')

Shannon entropy in nats.

event_shape(name='event_shape')

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.

get_batch_shape()

Shape of a single sample from a single event index as a TensorShape.

Same meaning as batch_shape. May be only partially defined.

Returns:

• batch_shape: TensorShape, possibly unknown.

get_event_shape()

Shape of a single sample from a single batch as a TensorShape.

Same meaning as event_shape. May be only partially defined.

Returns:

• event_shape: TensorShape, possibly unknown.

is_scalar_batch(name='is_scalar_batch')

Indicates that batch_shape == [].

Args:

• name: The name to give this op.

Returns:

• is_scalar_batch: Boolean scalar Tensor.

is_scalar_event(name='is_scalar_event')

Indicates that event_shape == [].

Args:

• name: The name to give this op.

Returns:

• is_scalar_event: Boolean scalar Tensor.

log_cdf(value, name='log_cdf', **condition_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: The name to give this op. **condition_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_pdf(value, name='log_pdf', **condition_kwargs)

Log probability density function.

Args:

• value: float or double Tensor.
• name: The name to give this op. **condition_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.

Raises:

• TypeError: if not is_continuous.

log_pmf(value, name='log_pmf', **condition_kwargs)

Log probability mass function.

Args:

• value: float or double Tensor.
• name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

Raises:

• TypeError: if is_continuous.

log_prob(value, name='log_prob', **condition_kwargs)

Log probability density/mass function (depending on is_continuous).

Args:

• value: float or double Tensor.
• name: The name to give this op. **condition_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(value, name='log_survival_function', **condition_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: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

mean(name='mean')

Mean.

Additional documentation from StudentT:

The mean of Student's T equals mu if df > 1, otherwise it is NaN. If self.allow_nan_stats=True, then an exception will be raised rather than returning NaN.

mode(name='mode')

Mode.

param_shapes(cls, sample_shape, name='DistributionParamShapes')

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

Subclasses should override static 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(cls, sample_shape)

param_shapes with static (i.e. TensorShape) shapes.

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.

pdf(value, name='pdf', **condition_kwargs)

Probability density function.

Args:

• value: float or double Tensor.
• name: The name to give this op. **condition_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.

Raises:

• TypeError: if not is_continuous.

pmf(value, name='pmf', **condition_kwargs)

Probability mass function.

Args:

• value: float or double Tensor.
• name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

Raises:

• TypeError: if is_continuous.

prob(value, name='prob', **condition_kwargs)

Probability density/mass function (depending on is_continuous).

Args:

• value: float or double Tensor.
• name: The name to give this op. **condition_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.

sample(sample_shape=(), seed=None, name='sample', **condition_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. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

• samples: a Tensor with prepended dimensions sample_shape.

std(name='std')

Standard deviation.

survival_function(value, name='survival_function', **condition_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: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

Tensorof shapesample_shape(x) + self.batch_shapewith values of typeself.dtype`.

variance(name='variance')

Variance.

Additional documentation from StudentT:

The variance for Student's T equals

df / (df - 2), when df > 2
infinity, when 1 < df <= 2
NaN, when df <= 1

Defined in tensorflow/contrib/distributions/python/ops/student_t.py.