tf.compat.v1.distributions.StudentT Student's t-distribution. Inherits From: Distribution 
tf.compat.v1.distributions.StudentT(
    df, loc, scale, validate_args=False, allow_nan_stats=True,
    name='StudentT'
)
 This distribution has parameters: degree of freedom df, location loc, and scale. Mathematical details The probability density function (pdf) is, pdf(x; df, mu, sigma) = (1 + y**2 / df)**(-0.5 (df + 1)) / Z
where,
y = (x - mu) / sigma
Z = abs(sigma) sqrt(df pi) Gamma(0.5 df) / Gamma(0.5 (df + 1))
 where:  
loc = mu, 
scale = sigma, and, 
Z is the normalization constant, and, 
Gamma is the gamma function.  The StudentT distribution is a member of the location-scale family, i.e., it can be constructed as, X ~ StudentT(df, loc=0, scale=1)
Y = loc + scale * X
 Notice that scale has semantics more similar to standard deviation than variance. However it is not actually the std. deviation; the Student's t-distribution std. dev. is scale sqrt(df / (df - 2)) when df > 2. Samples of this distribution are reparameterized (pathwise differentiable). The derivatives are computed using the approach described in (Figurnov et al., 2018). Examples Examples of initialization of one or a batch of distributions. import tensorflow_probability as tfp
tfd = tfp.distributions

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

# Evaluate the pdf at 1, returning a scalar Tensor.
single_dist.prob(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 = tfd.StudentT(df=[2, 3], loc=[1, 2.], scale=[11, 22.])

# Evaluate the pdf of the first distribution on 0, and the second on 1.5,
# returning a length two tensor.
multi_dist.prob([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 = tfd.StudentT(df=2, loc=1, scale=[11, 22.])

# Evaluate the pdf of both distributions on the same point, 3.0,
# returning a length 2 tensor.
dist.prob(3.0)
 Compute the gradients of samples w.r.t. the parameters: df = tf.constant(2.0)
loc = tf.constant(2.0)
scale = tf.constant(11.0)
dist = tfd.StudentT(df=df, loc=loc, scale=scale)
samples = dist.sample(5)  # Shape [5]
loss = tf.reduce_mean(tf.square(samples))  # Arbitrary loss function
# Unbiased stochastic gradients of the loss function
grads = tf.gradients(loss, [df, loc, scale])
 References: Implicit Reparameterization Gradients: Figurnov et al., 2018 (pdf)
 


 Args
  df   Floating-point Tensor. The degrees of freedom of the distribution(s). df must contain only positive values.  
  loc   Floating-point Tensor. The mean(s) of the distribution(s).  
  scale   Floating-point Tensor. The scaling factor(s) for the distribution(s). Note that scale is not technically the standard deviation of this distribution but has semantics more similar to standard deviation than variance.  
  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.   
 


 Raises
  TypeError   if loc and scale are different dtypes.   
 


 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. 
 
  df   Degrees of freedom in these Student's t distribution(s).  
  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   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.  
  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   Scaling factors of these Student's t distribution(s).  
  validate_args   Python bool indicating possibly expensive checks are enabled.    Methods 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'
)
 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 View source 
copy(
    **override_parameters_kwargs
)
 Creates a deep copy of the distribution. 
Note: the copy distribution may continue to depend on the original initialization arguments.

 


 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'
)
 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 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), (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 View source 
entropy(
    name='entropy'
)
 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 (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 View source 
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 View source 
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 View source 
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 View source 
mean(
    name='mean'
)
 Mean. Additional documentation from StudentT: The mean of Student's T equals loc if df > 1, otherwise it is NaN. If self.allow_nan_stats=True, then an exception will be raised rather than returning NaN. mode View source 
mode(
    name='mode'
)
 Mode. param_shapes View source 
@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 View source 
@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 View source 
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 View source 
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 View source 
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 View source 
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 View source 
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 View source 
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. 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

 


 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().