    GP-SPEC:  Specify a Gaussian process model, or display existing spec.

    Gp-spec creates a log file containing a specification of a Gaussian
    process model and the associated priors over hyperparameters.  When
    invoked with just a log file as argument, it displays the
    specifications of the Gaussian process model stored in that log file.

    Usage:

        gp-spec log-file N-inputs N-outputs
          [ const-part [ linear-part [ jitter-part ] ] ] { flag } [ spread ]
          { / scale-prior relevance-prior [ power ] { flag } }

    or:

        gp-spec log-file

    N-inputs and N-outputs are the numbers of input variables and output
    variables in the model - ie, the dimensionality of the domain and the
    range of the functions that the Gaussian process defines a
    distribution over.  For the Gaussian processes allowed here, the
    functions from the inputs to the various output variables are
    independent, given particular values for the hyperparameters.  The
    covariance functions for the each output all have the same form, and
    share the same hyperparameters.

    The const-part argument gives the prior for the hyperparameter
    controlling the constant part of the covariance function, in the form
    described in prior.doc.  A value of c for this hyperparameter adds a
    term of c^2 to the covariance between all pairs of input points.  The
    prior for this hyperparameter can have only one level.  If this
    argument is "-", or missing altogether, the covariance function will
    not have a constant part.

    The linear-part argument gives the prior for the hyperparameters, s_i,
    controlling the linear part of the covariance function.  This part of
    the covariance between inputs x and x' is

           SUM_i x_i x'_i s_i^2

    This prior can have up to two levels, allowing for a common
    hyperparameter that controls the priors for the coefficients, s_i,
    associated with the various inputs.  The "x" option may be used in
    order to have the width of this prior scale with the number of inputs,
    as described in prior.doc.  The linear part may be "-", or missing, in
    which case the covariance function will not have a linear part.

    The jitter-part argument gives the prior for a hyperparameter whose
    square is added to the covariance of a training or test case with
    itself.  Such a contribution may be desirable for modeling reasons, or
    in order to improve the numerical methods.  If a regression model with
    noise is being used, it is usually not necessary to include such a
    contribution to the covariance; if jitter-part is included, it
    effectively adds to the noise level (this could be useful if you want
    to constrain the noise to be at least some amount).  For a
    classification model, a jitter-part should usually be included, as
    otherwise the updates of the underlying function values may be very
    slow, and numerical problems can arise.  If the jitter-part is "-" or
    missing, it is taken to be zero.

    Zero or more additional terms in the covariance function may be
    specified using further groups of arguments.  A group for which
    "power" is absent or positive results in a term in the expression for
    the covariance between a particular output at inputs x and x' that has
    the form:

          v^2 * exp( - SUM_i (w_i |x_i - x'_i|)^R )

    The power, R, in this expression must be in the interval (0,2].  It is
    given by the last argument in a group, with the default being R=2.
    The first argument in the group gives the prior for the scale
    hyperparameter, v, which must have only one level.  The prior for the
    w_i, which determine the relevance of the various inputs, can have up
    to two levels, allowing for a common hyperparameter and for
    hyperparameters for each input, i.  Here again, the "x" option may be
    used to automatically scale the prior.

    A group can also have a "power" of -1, in which case it produces a
    term in the covariance function of the form

          v^2 / PROD_i (1 + w_i^2 (x_i - x'_i)^2 )

    The v and w hyperparameters play the same roles as described above.

    Optional flags may be appended to the specifications of the linear
    and other parts of the covariance.  They have the form:

          <flag>[:[-]<input>{,<input>}]

    where <flag> is one of the flag names below, and <input> is the number
    of an input (starting with 1) that the flag applies to.  If the "-" is
    present, the flag applies to all inputs EXCEPT those mentioned.  If
    only the flag name is given, it applies to all inputs.  The possible
    flags are as follows:

        delta     Use a "delta" distance for these inputs, in which the
                  distance is 0 if x_i=x'_i and 1 otherwise.  Not allowed
                  for the linear part of the covariance.

        omit      Ignore these inputs when computing the covariance
                  (ie, don't include them in the sum above).

        spread    Spread out the relevance parameters for these inputs.
                  This is presently allowed only for the linear part, and
                  causes this term in the covariance function to become

                     SUM_i x_i x'_i SUM_j s_j^2

                  The sum over j includes only i when i is not marked for
                  spreading.  When i is being spread, j includes all
                  indexes from i-spread to i+spread that are marked for
                  spreading.  Here, "spread" is the spreading width, which
                  must be specified after the flags.

    The "delta" flag is useful when the inputs are categorical.  The
    "omit" flag is useful in setting up additive models, in which
    different covariance parts correspond to additive components, with
    each component looking at only a subset of the inputs.  The "spread"
    flag is useful for data such as spectra where nearby inputs are
    likely to be of similar relevance.

    Note that the prior for the const-part corresponds closely to the
    prior for output biases in neural network models, and that the prior
    for the linear-part corresponds to the prior for input-output weights
    in a network.  The priors for v and w_i correspond to the priors in
    neural network models on the hidden-output and the input-hidden
    weights.  (The neural network priors can go to more levels, however.)

    To use a Gaussian process to model data, additional information must
    be specified as well, as described in model-spec.doc.  Depending on
    the model used, state information in addition to the hyperparameter
    values for the Gaussian process may need to be kept (case-specific
    function values and/or case-specific noise variances).

                Copyright (c) 1995-2003 by Radford M. Neal