class_docstr: Marginal distribution of a Student's T process at finitely many points.
  
  A Student's T process (TP) is an indexed collection of random variables, any
  finite collection of which are jointly Multivariate Student's T. While this
  definition applies to finite index sets, it is typically implicit that the
  index set is infinite; in applications, it is often some finite dimensional
  real or complex vector space. In such cases, the TP may be thought of as a
  distribution over (real- or complex-valued) functions defined over the index
  set.
  
  Just as Student's T distributions are fully specified by their degrees of
  freedom, location and scale, a Student's T process can be completely specified
  by a degrees of freedom parameter, mean function and covariance function.
  Let `S` denote the index set and `K` the space in
  which each indexed random variable takes its values (again, often R or C).
  The mean function is then a map `m: S -> K`, and the covariance function,
  or kernel, is a positive-definite function `k: (S x S) -> K`. The properties
  of functions drawn from a TP are entirely dictated (up to translation) by
  the form of the kernel function.
  
  This `Distribution` represents the marginal joint distribution over function
  values at a given finite collection of points `[x[1], ..., x[N]]` from the
  index set `S`. By definition, this marginal distribution is just a
  multivariate Student's T distribution, whose mean is given by the vector
  `[ m(x[1]), ..., m(x[N]) ]` and whose covariance matrix is constructed from
  pairwise applications of the kernel function to the given inputs:
  
  ```none
      | k(x[1], x[1])    k(x[1], x[2])  ...  k(x[1], x[N]) |
      | k(x[2], x[1])    k(x[2], x[2])  ...  k(x[2], x[N]) |
      |      ...              ...                 ...      |
      | k(x[N], x[1])    k(x[N], x[2])  ...  k(x[N], x[N]) |
  ```
  
  For this to be a valid covariance matrix, it must be symmetric and positive
  definite; hence the requirement that `k` be a positive definite function
  (which, by definition, says that the above procedure will yield PD matrices).
  
  Note also we use a parameterization as suggested in [1], which requires `df`
  to be greater than 2. This allows for the covariance for any finite
  dimensional marginal of the TP (a multivariate Student's T distribution) to
  just be the PD matrix generated by the kernel.
  
  
  #### Mathematical Details
  
  The probability density function (pdf) is a multivariate Student's T whose
  parameters are derived from the TP's properties:
  
  ```none
  pdf(x; df, index_points, mean_fn, kernel) = MultivariateStudentT(df, loc, K)
  K = (df - 2) / df  * (kernel.matrix(index_points, index_points) +
       observation_noise_variance * eye(N))
  loc = (x - mean_fn(index_points))^T @ K @ (x - mean_fn(index_points))
  ```
  
  where:
  
  * `df` is the degrees of freedom parameter for the TP.
  * `index_points` are points in the index set over which the TP is defined,
  * `mean_fn` is a callable mapping the index set to the TP's mean values,
  * `kernel` is `PositiveSemidefiniteKernel`-like and represents the covariance
    function of the TP,
  * `observation_noise_variance` is a term added to the diagonal of the kernel
    matrix. In the limit of `df` to `inf`, this represents the observation noise
    of a gaussian likelihood.
  * `eye(N)` is an N-by-N identity matrix.
  
  #### Examples
  
  ##### Draw joint samples from a TP prior
  
  ```python
  import numpy as np
  import tensorflow.compat.v2 as tf
  import tensorflow_probability as tfp
  
  tfd = tfp.distributions
  psd_kernels = tfp.math.psd_kernels
  
  num_points = 100
  # Index points should be a collection (100, here) of feature vectors. In this
  # example, we're using 1-d vectors, so we just need to reshape the output from
  # np.linspace, to give a shape of (100, 1).
  index_points = np.expand_dims(np.linspace(-1., 1., num_points), -1)
  
  # Define a kernel with default parameters.
  kernel = psd_kernels.ExponentiatedQuadratic()
  
  tp = tfd.StudentTProcess(3., kernel, index_points)
  
  samples = tp.sample(10)
  # ==> 10 independently drawn, joint samples at `index_points`
  
  noisy_tp = tfd.StudentTProcess(
      df=3.,
      kernel=kernel,
      index_points=index_points)
  noisy_samples = noisy_tp.sample(10)
  # ==> 10 independently drawn, noisy joint samples at `index_points`
  ```
  
  ##### Optimize kernel parameters via maximum marginal likelihood.
  
  ```python
  # Suppose we have some data from a known function. Note the index points in
  # general have shape `[b1, ..., bB, f1, ..., fF]` (here we assume `F == 1`),
  # so we need to explicitly consume the feature dimensions (just the last one
  # here).
  f = lambda x: np.sin(10*x[..., 0]) * np.exp(-x[..., 0]**2)
  observed_index_points = np.expand_dims(np.random.uniform(-1., 1., 50), -1)
  # Squeeze to take the shape from [50, 1] to [50].
  observed_values = f(observed_index_points)
  
  amplitude = tfp.util.TransformedVariable(
      1., tfp.bijectors.Softplus(), dtype=np.float64, name='amplitude')
  length_scale = tfp.util.TransformedVariable(
      1., tfp.bijectors.Softplus(), dtype=np.float64, name='length_scale')
  
  # Define a kernel with trainable parameters.
  kernel = psd_kernels.ExponentiatedQuadratic(
      amplitude=amplitude,
      length_scale=length_scale)
  
  tp = tfd.StudentTProcess(3., kernel, observed_index_points)
  
  optimizer = tf_keras.optimizers.Adam()
  
  @tf.function
  def optimize():
    with tf.GradientTape() as tape:
      loss = -tp.log_prob(observed_values)
    grads = tape.gradient(loss, tp.trainable_variables)
    optimizer.apply_gradients(zip(grads, tp.trainable_variables))
    return loss
  
  for i in range(1000):
    nll = optimize()
    if i % 100 == 0:
      print("Step {}: NLL = {}".format(i, nll))
  print("Final NLL = {}".format(nll))
  ```
  
  #### References
  
  [1]: Amar Shah, Andrew Gordon Wilson, and Zoubin Ghahramani. Student-t
       Processes as Alternatives to Gaussian Processes. In _Artificial
       Intelligence and Statistics_, 2014.
       https://www.cs.cmu.edu/~andrewgw/tprocess.pdf
class_docstr_tok: ["Marginal","distribution","of","a","Student","'s","T","process","at","finitely","many","points",".","A","Student","'s","T","process","(","TP",")","is","an","indexed","collection","of","random","variables",",","any","finite","collection","of","which","are","jointly","Multivariate","Student","'s","T.","While","this","definition","applies","to","finite","index","sets",",","it","is","typically","implicit","that","the","index","set","is","infinite",";","in","applications",",","it","is","often","some","finite","dimensional","real","or","complex","vector","space",".","In","such","cases",",","the","TP","may","be","thought","of","as","a","distribution","over","(","real-","or","complex-valued",")","functions","defined","over","the","index","set",".","Just","as","Student","'s","T","distributions","are","fully","specified","by","their","degrees","of","freedom",",","location","and","scale",",","a","Student","'s","T","process","can","be","completely","specified","by","a","degrees","of","freedom","parameter",",","mean","function","and","covariance","function",".","Let","`","S","`","denote","the","index","set","and","`","K","`","the","space","in","which","each","indexed","random","variable","takes","its","values","(","again",",","often","R","or","C",")",".","The","mean","function","is","then","a","map","`","m",":","S","-",">","K","`",",","and","the","covariance","function",",","or","kernel",",","is","a","positive-definite","function","`","k",":","(","S","x","S",")","-",">","K","`",".","The","properties","of","functions","drawn","from","a","TP","are","entirely","dictated","(","up","to","translation",")","by","the","form","of","the","kernel","function",".","This","`","Distribution","`","represents","the","marginal","joint","distribution","over","function","values","at","a","given","finite","collection","of","points","`","[","x","[","1","]",",","...",",","x","[","N","]","]","`","from","the","index","set","`","S","`",".","By","definition",",","this","marginal","distribution","is","just","a","multivariate","Student","'s","T","distribution",",","whose","mean","is","given","by","the","vector","`","[","m","(","x","[","1","]",")",",","...",",","m","(","x","[","N","]",")","]","`","and","whose","covariance","matrix","is","constructed","from","pairwise","applications","of","the","kernel","function","to","the","given","inputs",":","``","`","none","|","k","(","x","[","1","]",",","x","[","1","]",")","k","(","x","[","1","]",",","x","[","2","]",")","...","k","(","x","[","1","]",",","x","[","N","]",")","|","|","k","(","x","[","2","]",",","x","[","1","]",")","k","(","x","[","2","]",",","x","[","2","]",")","...","k","(","x","[","2","]",",","x","[","N","]",")","|","|","...","...","...","|","|","k","(","x","[","N","]",",","x","[","1","]",")","k","(","x","[","N","]",",","x","[","2","]",")","...","k","(","x","[","N","]",",","x","[","N","]",")","|","``","`","For","this","to","be","a","valid","covariance","matrix",",","it","must","be","symmetric","and","positive","definite",";","hence","the","requirement","that","`","k","`","be","a","positive","definite","function","(","which",",","by","definition",",","says","that","the","above","procedure","will","yield","PD","matrices",")",".","Note","also","we","use","a","parameterization","as","suggested","in","[","1","]",",","which","requires","`","df","`","to","be","greater","than","2",".","This","allows","for","the","covariance","for","any","finite","dimensional","marginal","of","the","TP","(","a","multivariate","Student","'s","T","distribution",")","to","just","be","the","PD","matrix","generated","by","the","kernel",".","#","#","#","#","Mathematical","Details","The","probability","density","function","(","pdf",")","is","a","multivariate","Student","'s","T","whose","parameters","are","derived","from","the","TP","'s","properties",":","``","`","none","pdf","(","x",";","df",",","index_points",",","mean_fn",",","kernel",")","=","MultivariateStudentT","(","df",",","loc",",","K",")","K","=","(","df","-","2",")","\/","df","*","(","kernel.matrix","(","index_points",",","index_points",")","+","observation_noise_variance","*","eye","(","N",")",")","loc","=","(","x","-","mean_fn","(","index_points",")",")","^T","@","K","@","(","x","-","mean_fn","(","index_points",")",")","``","`","where",":","*","`","df","`","is","the","degrees","of","freedom","parameter","for","the","TP",".","*","`","index_points","`","are","points","in","the","index","set","over","which","the","TP","is","defined",",","*","`","mean_fn","`","is","a","callable","mapping","the","index","set","to","the","TP","'s","mean","values",",","*","`","kernel","`","is","`","PositiveSemidefiniteKernel","`","-like","and","represents","the","covariance","function","of","the","TP",",","*","`","observation_noise_variance","`","is","a","term","added","to","the","diagonal","of","the","kernel","matrix",".","In","the","limit","of","`","df","`","to","`","inf","`",",","this","represents","the","observation","noise","of","a","gaussian","likelihood",".","*","`","eye","(","N",")","`","is","an","N-by-N","identity","matrix",".","#","#","#","#","Examples","#","#","#","#","#","Draw","joint","samples","from","a","TP","prior","``","`","python","import","numpy","as","np","import","tensorflow.compat.v2","as","tf","import","tensorflow_probability","as","tfp","tfd","=","tfp.distributions","psd_kernels","=","tfp.math.psd_kernels","num_points","=","100","#","Index","points","should","be","a","collection","(","100",",","here",")","of","feature","vectors",".","In","this","#","example",",","we","'re","using","1-d","vectors",",","so","we","just","need","to","reshape","the","output","from","#","np.linspace",",","to","give","a","shape","of","(","100",",","1",")",".","index_points","=","np.expand_dims","(","np.linspace","(","-1.",",","1.",",","num_points",")",",","-1",")","#","Define","a","kernel","with","default","parameters",".","kernel","=","psd_kernels.ExponentiatedQuadratic","(",")","tp","=","tfd.StudentTProcess","(","3.",",","kernel",",","index_points",")","samples","=","tp.sample","(","10",")","#","==",">","10","independently","drawn",",","joint","samples","at","`","index_points","`","noisy_tp","=","tfd.StudentTProcess","(","df=3.",",","kernel=kernel",",","index_points=index_points",")","noisy_samples","=","noisy_tp.sample","(","10",")","#","==",">","10","independently","drawn",",","noisy","joint","samples","at","`","index_points","`","``","`","#","#","#","#","#","Optimize","kernel","parameters","via","maximum","marginal","likelihood",".","``","`","python","#","Suppose","we","have","some","data","from","a","known","function",".","Note","the","index","points","in","#","general","have","shape","`","[","b1",",","...",",","bB",",","f1",",","...",",","fF","]","`","(","here","we","assume","`","F","==","1","`",")",",","#","so","we","need","to","explicitly","consume","the","feature","dimensions","(","just","the","last","one","#","here",")",".","f","=","lambda","x",":","np.sin","(","10","*","x","[","...",",","0","]",")","*","np.exp","(","-x","[","...",",","0","]","*","*","2",")","observed_index_points","=","np.expand_dims","(","np.random.uniform","(","-1.",",","1.",",","50",")",",","-1",")","#","Squeeze","to","take","the","shape","from","[","50",",","1","]","to","[","50","]",".","observed_values","=","f","(","observed_index_points",")","amplitude","=","tfp.util.TransformedVariable","(","1.",",","tfp.bijectors.Softplus","(",")",",","dtype=np.float64",",","name='amplitude","'",")","length_scale","=","tfp.util.TransformedVariable","(","1.",",","tfp.bijectors.Softplus","(",")",",","dtype=np.float64",",","name='length_scale","'",")","#","Define","a","kernel","with","trainable","parameters",".","kernel","=","psd_kernels.ExponentiatedQuadratic","(","amplitude=amplitude",",","length_scale=length_scale",")","tp","=","tfd.StudentTProcess","(","3.",",","kernel",",","observed_index_points",")","optimizer","=","tf_keras.optimizers.Adam","(",")","@","tf.function","def","optimize","(",")",":","with","tf.GradientTape","(",")","as","tape",":","loss","=","-tp.log_prob","(","observed_values",")","grads","=","tape.gradient","(","loss",",","tp.trainable_variables",")","optimizer.apply_gradients","(","zip","(","grads",",","tp.trainable_variables",")",")","return","loss","for","i","in","range","(","1000",")",":","nll","=","optimize","(",")","if","i","%","100","==","0",":","print","(","``","Step","{","}",":","NLL","=","{","}","''",".format","(","i",",","nll",")",")","print","(","``","Final","NLL","=","{","}","''",".format","(","nll",")",")","``","`","#","#","#","#","References","[","1","]",":","Amar","Shah",",","Andrew","Gordon","Wilson",",","and","Zoubin","Ghahramani",".","Student-t","Processes","as","Alternatives","to","Gaussian","Processes",".","In","_Artificial","Intelligence","and","Statistics_",",","2014.","https",":","\/\/www.cs.cmu.edu\/~andrewgw\/tprocess.pdf"]
docstr: Instantiate a StudentTProcess Distribution.
  
  Args:
    df: Positive Floating-point `Tensor` representing the degrees of freedom.
      Must be greater than 2.
    kernel: `PositiveSemidefiniteKernel`-like instance representing the
      TP's covariance function.
    index_points: (Nested) `float` `Tensor` representing finite (batch of)
      vector(s) of points in the index set over which the TP is defined. Shape
      (or shape of each nested component) has the form
      `[b1, ..., bB, e, f1, ..., fF]` where `F` is the number of feature
      dimensions and must equal `kernel.feature_ndims` (or its corresponding
      nested component) and `e` is the number (size) of index points in each
      batch. Ultimately this distribution corresponds to a `e`-dimensional
      multivariate Student's T. The batch shape must be broadcastable with
      `kernel.batch_shape` and any batch dims yielded by `mean_fn`.
    mean_fn: Python `callable` that acts on `index_points` to produce a (batch
      of) vector(s) of mean values at `index_points`. Takes a (nested)
      `Tensor` of shape `[b1, ..., bB, e, f1, ..., fF]` and returns a `Tensor`
      whose shape is broadcastable with `[b1, ..., bB, e]`.
      Default value: `None` implies constant zero function.
    observation_noise_variance: `float` `Tensor` representing (batch of)
      scalar variance(s) of the noise in the Normal likelihood
      distribution of the model. If batched, the batch shape must be
      broadcastable with the shapes of all other batched parameters
      (`kernel.batch_shape`, `index_points`, etc.).
      Default value: `0.`
    marginal_fn: A Python callable that takes a location, covariance matrix,
      optional `validate_args`, `allow_nan_stats` and `name` arguments, and
      returns a multivariate Student T subclass of `tfd.Distribution`.
      Default value: `None`, in which case a Cholesky-factorizing function
      is created using `make_cholesky_factored_marginal_fn` and the
      `jitter` argument.
    cholesky_fn: Callable which takes a single (batch) matrix argument and
      returns a Cholesky-like lower triangular factor.  Default value: `None`,
      in which case `make_cholesky_with_jitter_fn` is used with the `jitter`
      parameter. At most one of `cholesky_fn` and `marginal_fn` should be set.
    jitter: `float` scalar `Tensor` added to the diagonal of the covariance
      matrix to ensure positive definiteness of the covariance matrix.
      This argument is ignored if `cholesky_fn` is set.
      Default value: `1e-6`.
    always_yield_multivariate_student_t: Deprecated and ignored.
    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.
      Default value: `False`.
    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.
      Default value: `False`.
    name: Python `str` name prefixed to Ops created by this class.
      Default value: "StudentTProcess".
  
  Raises:
    ValueError: if `mean_fn` is not `None` and is not callable.
docstr_tok: ["Instantiate","a","StudentTProcess","Distribution",".","Args",":","df",":","Positive","Floating-point","`","Tensor","`","representing","the","degrees","of","freedom",".","Must","be","greater","than","2.","kernel",":","`","PositiveSemidefiniteKernel","`","-like","instance","representing","the","TP","'s","covariance","function",".","index_points",":","(","Nested",")","`","float","`","`","Tensor","`","representing","finite","(","batch","of",")","vector","(","s",")","of","points","in","the","index","set","over","which","the","TP","is","defined",".","Shape","(","or","shape","of","each","nested","component",")","has","the","form","`","[","b1",",","...",",","bB",",","e",",","f1",",","...",",","fF","]","`","where","`","F","`","is","the","number","of","feature","dimensions","and","must","equal","`","kernel.feature_ndims","`","(","or","its","corresponding","nested","component",")","and","`","e","`","is","the","number","(","size",")","of","index","points","in","each","batch",".","Ultimately","this","distribution","corresponds","to","a","`","e","`","-dimensional","multivariate","Student","'s","T.","The","batch","shape","must","be","broadcastable","with","`","kernel.batch_shape","`","and","any","batch","dims","yielded","by","`","mean_fn","`",".","mean_fn",":","Python","`","callable","`","that","acts","on","`","index_points","`","to","produce","a","(","batch","of",")","vector","(","s",")","of","mean","values","at","`","index_points","`",".","Takes","a","(","nested",")","`","Tensor","`","of","shape","`","[","b1",",","...",",","bB",",","e",",","f1",",","...",",","fF","]","`","and","returns","a","`","Tensor","`","whose","shape","is","broadcastable","with","`","[","b1",",","...",",","bB",",","e","]","`",".","Default","value",":","`","None","`","implies","constant","zero","function",".","observation_noise_variance",":","`","float","`","`","Tensor","`","representing","(","batch","of",")","scalar","variance","(","s",")","of","the","noise","in","the","Normal","likelihood","distribution","of","the","model",".","If","batched",",","the","batch","shape","must","be","broadcastable","with","the","shapes","of","all","other","batched","parameters","(","`","kernel.batch_shape","`",",","`","index_points","`",",","etc.",")",".","Default","value",":","`","0.","`","marginal_fn",":","A","Python","callable","that","takes","a","location",",","covariance","matrix",",","optional","`","validate_args","`",",","`","allow_nan_stats","`","and","`","name","`","arguments",",","and","returns","a","multivariate","Student","T","subclass","of","`","tfd.Distribution","`",".","Default","value",":","`","None","`",",","in","which","case","a","Cholesky-factorizing","function","is","created","using","`","make_cholesky_factored_marginal_fn","`","and","the","`","jitter","`","argument",".","cholesky_fn",":","Callable","which","takes","a","single","(","batch",")","matrix","argument","and","returns","a","Cholesky-like","lower","triangular","factor",".","Default","value",":","`","None","`",",","in","which","case","`","make_cholesky_with_jitter_fn","`","is","used","with","the","`","jitter","`","parameter",".","At","most","one","of","`","cholesky_fn","`","and","`","marginal_fn","`","should","be","set",".","jitter",":","`","float","`","scalar","`","Tensor","`","added","to","the","diagonal","of","the","covariance","matrix","to","ensure","positive","definiteness","of","the","covariance","matrix",".","This","argument","is","ignored","if","`","cholesky_fn","`","is","set",".","Default","value",":","`","1e-6","`",".","always_yield_multivariate_student_t",":","Deprecated","and","ignored",".","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",".","Default","value",":","`","False","`",".","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",".","Default","value",":","`","False","`",".","name",":","Python","`","str","`","name","prefixed","to","Ops","created","by","this","class",".","Default","value",":","``","StudentTProcess","''",".","Raises",":","ValueError",":","if","`","mean_fn","`","is","not","`","None","`","and","is","not","callable","."]
code: def __init__(
      self,
      df,
      kernel,
      index_points=None,
      mean_fn=None,
      observation_noise_variance=0.0,
      marginal_fn=None,
      cholesky_fn=None,
      jitter=1e-6,
      always_yield_multivariate_student_t=None,
      validate_args=False,
      allow_nan_stats=False,
      name="StudentTProcess",
  ):
      """Instantiate a StudentTProcess Distribution.
  
      Args:
        df: Positive Floating-point `Tensor` representing the degrees of freedom.
          Must be greater than 2.
        kernel: `PositiveSemidefiniteKernel`-like instance representing the
          TP's covariance function.
        index_points: (Nested) `float` `Tensor` representing finite (batch of)
          vector(s) of points in the index set over which the TP is defined. Shape
          (or shape of each nested component) has the form
          `[b1, ..., bB, e, f1, ..., fF]` where `F` is the number of feature
          dimensions and must equal `kernel.feature_ndims` (or its corresponding
          nested component) and `e` is the number (size) of index points in each
          batch. Ultimately this distribution corresponds to a `e`-dimensional
          multivariate Student's T. The batch shape must be broadcastable with
          `kernel.batch_shape` and any batch dims yielded by `mean_fn`.
        mean_fn: Python `callable` that acts on `index_points` to produce a (batch
          of) vector(s) of mean values at `index_points`. Takes a (nested)
          `Tensor` of shape `[b1, ..., bB, e, f1, ..., fF]` and returns a `Tensor`
          whose shape is broadcastable with `[b1, ..., bB, e]`.
          Default value: `None` implies constant zero function.
        observation_noise_variance: `float` `Tensor` representing (batch of)
          scalar variance(s) of the noise in the Normal likelihood
          distribution of the model. If batched, the batch shape must be
          broadcastable with the shapes of all other batched parameters
          (`kernel.batch_shape`, `index_points`, etc.).
          Default value: `0.`
        marginal_fn: A Python callable that takes a location, covariance matrix,
          optional `validate_args`, `allow_nan_stats` and `name` arguments, and
          returns a multivariate Student T subclass of `tfd.Distribution`.
          Default value: `None`, in which case a Cholesky-factorizing function
          is created using `make_cholesky_factored_marginal_fn` and the
          `jitter` argument.
        cholesky_fn: Callable which takes a single (batch) matrix argument and
          returns a Cholesky-like lower triangular factor.  Default value: `None`,
          in which case `make_cholesky_with_jitter_fn` is used with the `jitter`
          parameter. At most one of `cholesky_fn` and `marginal_fn` should be set.
        jitter: `float` scalar `Tensor` added to the diagonal of the covariance
          matrix to ensure positive definiteness of the covariance matrix.
          This argument is ignored if `cholesky_fn` is set.
          Default value: `1e-6`.
        always_yield_multivariate_student_t: Deprecated and ignored.
        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.
          Default value: `False`.
        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.
          Default value: `False`.
        name: Python `str` name prefixed to Ops created by this class.
          Default value: "StudentTProcess".
  
      Raises:
        ValueError: if `mean_fn` is not `None` and is not callable.
      """
      parameters = dict(locals())
      with tf.name_scope(name) as name:
          input_dtype = dtype_util.common_dtype(
              dict(
                  kernel=kernel,
                  index_points=index_points,
              ),
              dtype_hint=nest_util.broadcast_structure(
                  kernel.feature_ndims, tf.float32
              ),
          )
  
          # If the input dtype is non-nested float, we infer a single dtype for the
          # input and the float parameters, which is also the dtype of the STP's
          # samples, log_prob, etc. If the input dtype is nested (or not float), we
          # do not use it to infer the STP's float dtype.
          if not tf.nest.is_nested(
              input_dtype
          ) and dtype_util.is_floating(input_dtype):
              dtype = dtype_util.common_dtype(
                  dict(
                      kernel=kernel,
                      index_points=index_points,
                      observation_noise_variance=observation_noise_variance,
                      jitter=jitter,
                      df=df,
                  ),
                  dtype_hint=tf.float32,
              )
              input_dtype = dtype
          else:
              dtype = dtype_util.common_dtype(
                  dict(
                      df=df,
                      observation_noise_variance=observation_noise_variance,
                      jitter=jitter,
                  ),
                  dtype_hint=tf.float32,
              )
  
          if index_points is not None:
              index_points = nest_util.convert_to_nested_tensor(
                  index_points,
                  dtype=input_dtype,
                  name="index_points",
                  convert_ref=False,
                  allow_packing=True,
              )
          df = tensor_util.convert_nonref_to_tensor(
              df, dtype=dtype, name="df"
          )
          observation_noise_variance = (
              tensor_util.convert_nonref_to_tensor(
                  observation_noise_variance,
                  dtype=dtype,
                  name="observation_noise_variance",
              )
          )
          jitter = tensor_util.convert_nonref_to_tensor(
              jitter, dtype=dtype, name="jitter"
          )
  
          self._kernel = kernel
          self._index_points = index_points
          # Default to a constant zero function, borrowing the dtype from
          # index_points to ensure consistency.
          mean_fn = stochastic_process_util.maybe_create_mean_fn(
              mean_fn, dtype
          )
          self._df = df
          self._observation_noise_variance = observation_noise_variance
          self._mean_fn = mean_fn
          self._jitter = jitter
          self._cholesky_fn = cholesky_fn
          if marginal_fn is not None and cholesky_fn is not None:
              raise ValueError(
                  "At most one of `marginal_fn` and `cholesky_fn` should be set."
              )
          if marginal_fn is None:
              if self._cholesky_fn is None:
                  self._cholesky_fn = (
                      cholesky_util.make_cholesky_with_jitter_fn(jitter)
                  )
              self._marginal_fn = make_cholesky_factored_marginal_fn(
                  self._cholesky_fn
              )
          else:
              self._marginal_fn = marginal_fn
  
          self._always_yield_multivariate_student_t = (
              always_yield_multivariate_student_t
          )
          with tf.name_scope("init"):
              super(StudentTProcess, self).__init__(
                  dtype=dtype,
                  reparameterization_type=reparameterization.FULLY_REPARAMETERIZED,
                  validate_args=validate_args,
                  allow_nan_stats=allow_nan_stats,
                  parameters=parameters,
                  name=name,
              )
  
imports: import functools
  import warnings
  import numpy
  import tensorflow.compat.v2
  from tensorflow_probability.python.bijectors import identity
  from tensorflow_probability.python.bijectors import softplus
  from tensorflow_probability.python.distributions import cholesky_util
  from tensorflow_probability.python.distributions import distribution
  from tensorflow_probability.python.distributions import multivariate_student_t
  from tensorflow_probability.python.distributions.internal import stochastic_process_util
  from tensorflow_probability.python.internal import assert_util
  from tensorflow_probability.python.internal import batch_shape_lib
  from tensorflow_probability.python.internal import distribution_util
  from tensorflow_probability.python.internal import dtype_util
  from tensorflow_probability.python.internal import nest_util
  from tensorflow_probability.python.internal import parameter_properties
  from tensorflow_probability.python.internal import prefer_static
  from tensorflow_probability.python.internal import reparameterization
  from tensorflow_probability.python.internal import tensor_util
  from tensorflow_probability.python.math import linalg
  from tensorflow_probability.python.math import special
  from tensorflow.python.util import deprecation
  from tensorflow.python.util import nest
label_desc: Use image node_id 1 to create a new StudentTProcess object from inherited base classes: distribution with example: obj = StudentTProcess(df, kernel, index_points, mean_fn, observation_noise_variance, marginal_fn, cholesky_fn, jitter, always_yield_multivariate_student_t, validate_args, allow_nan_stats, name)