DIST-SPEC:  Specify a distribution to sample from.

Dist-spec creates a log file containing the formulas that specify a
distribution to sample from.  The distribution may be specified by
giving a formula for the energy (minus the log probability density).
Alternatively, the distribution may be the Bayesian posterior
distribution resulting from a specified prior and likelihood.  When
invoked with just the log file as argument, dist-spec displays the
specification stored in an existing log file.

Usage:

    dist-spec log-file { var=formula } energy [ -zero-temper ]

or:

    dist-spec log-file { var=formula } prior likelihood [ -read-prior ]

or:

    dist-spec log-file 

Distributions are over a set of real-valued state variables, which
consist of one of the letters 'u', 'v', 'w', 'x', 'y', or 'z',
possibly followed by a single digit ('0' to '9').  

Constants may be defined immediately after the log-file argument.
These have names of the same form as state variables, but starting
with other lower-case letters.  They may be used in the formulas for
the energy, minus-log-prior, or minus-log-likelihood.  Constant
definitions must not refer to state variables, or to inputs or
targets, but may refer to constants defined earlier.

In the first form of the dist-spec command, the distribution is given
by an "energy" function, specified by an arithmetic formula involving
the state variables that evaluates to minus the log of the probability
density (plus any arbitrary constant).

In the second form, the energy function is minus the log of the
posterior density for a Bayesian model, whose parameters are a subset
of the state variables (of the same form as described above).  The
model is specified by giving formulas for minus the log of the prior
density and for minus the log likelihood for a single observation.

The likelihood may be given as a single formula, in which any data
values are written explicitly, as constants.  In this case, no
data-spec command should be given later.  More typically, the formula
given for the likelihood is for one of a set of independent
observations, which are read from the files specified by a later
data-spec command (see data-spec.doc).  The inputs for an observation
are referred to by the names i0, i1, etc., with i being a synonym for
i0.  The targets are referred to by t0, t1, etc., with t being a
synonym for t0.  There can be at most ten inputs or targets (ie, up to
i9 and t9).  These variables are rebound to the inputs and targets for
each observation in turn, and the formula for minus the log likelihood
is re-evaluated for each observation.  Note that, by convention, the
model describes the conditional distribution of the targets (aka the
response variables), given values for the inputs (aka the predictor
variables, or covariates), if any.  If no data-spec command is given
when the likelihood refers to "i" or "t" variables, it is assumed that
there are no observations, in which case the posterior distribution is
the same as the prior.

For the syntax of formulas, see formula.doc.  Although all state
variables are unbounded real values, the effect of a non-negative
variable can be obtained by always referring to an appropriately
transformed version of the variable (eg, Abs(x) or Exp(x)), but note
that one must then include a term in the energy equal to minus the log
of the Jacobian of the transformation (though this term is zero for
Abs(x)).

For distributions specified by a single energy function, the behaviour
of tempering methods is controlled by whether the -zero-temper option
is specified.  If -zero-temper is not specified, the energy at inverse
temperature b is b*E + (1-b)*N, where E is the specified energy, and N
is minus the log of the Gaussian probability density function for the
state variables, with all variables being independent, and having mean
zero and variance one.  That is,

    N = (D/2) * log(2*Pi) + (1/2) * SUM_i q_i^2

where D is the number of state variables, and q_i is the value of the
i'th variable.  If -zero-temper is specified the energy at an inverse
temperature other than one is found by simply multiplying the energy
specified in dist-spec by the inverse temperature - ie, it is what
would be obtained if N above were zero rather than minus the log of
the Gaussian probability density.  Note that annealed importance
sampling cannot be used when -zero-temper is specified, since the
distribution at inverse temperature zero is improper.

For Bayesian posterior distributions, the various distributions used
in tempering methods are found by multiplying the log likelihood terms
by the inverse temperature.  The distribution at inverse temperature
one is thus the usual posterior, whereas that at inverse temperature
zero is the prior.  

The 'dist' programs known how to sample from the prior distribution
only when it is specified as a sum of terms of the form v~D(...),
where D is one of the parameterized distributions as described in
formula.doc, and where the parameters of the distribution refer only
to variables whose distributions are given in earlier terms.  Sampling
from the prior is needed to perform Annealed Importance Sampling, and
for sampling initial states in xxx-circ.  Sampling from the prior can
also be done just for its own interest, or to initialize an ordinary
Markov chain sampling run.  See dist-gen.doc for how to do this.  If
the prior does not have the form that allows automatic sampling, prior
generation can be done only if the -read-prior option is given, in
which case points from the prior are obtained by reading from standard
input, which must be a file of points, or a pipe from a separate
program for sampling from the prior.  See dist-mc.doc for details.

            Copyright (c) 1998-2000 by Radford M. Neal