\BIN

This directory contains BAYES0.EXE and LOOPSMTH.EXE.  Each of
these is a version of the Bayesian function interpolation
program.  LOOPSMTH is specialized for batch processing using the
batch files described under \LOOP below.  BAYES0 is interactive,
allowing exploration of different parts of the parameter space,
different scalings, etc.  

The program's algorithm is derived assuming f(p)=a*w(p), where w
is np-dimensional white noise.  The kernel a is taken to be
exp(-||p||~2~).  This is explained further in the U. of
Minnesota discussion paper on the theory underlying the
program.  

BAYES0:  The program will prompt you for the name of
the input file and give you options for how to proceed.  Its
input file has the same structure as the LOOP.DAT input file for
LOOPRATS:

    Line 1: A small positive real number defining the size of
    the change in the objective function at which convergence is
    assumed.

    Line 2: An integer NP, the number of parameters.
    
    Line 3: A vector of NP scale factors S(I), followed by a
    base function value B and by a normal change DN.  S(I) is
    the size of the change in P(I), the I'th parameter, needed
    to substantially weaken correlation of a priori beliefs
    about the function values at the original and the changed
    point.  For changes much smaller than S(I), one expects to
    find the function changes slowly and smoothly.  B should be
    below the likely maximum value of the function, but probably
    above a substantial fraction of the observed function values
    on the input file.  DN should be about .2 for log likelihood
    functions.  More generally, it is a guess as to the "normal"
    increase in the function value to be obtained with one
    iteration near the start of function maximization.  These
    "line 3" numbers can spill over onto other lines.  They are
    read with a single free format FORTRAN read statement.
    
    Line n:  After the line 3 numbers come observations, each
    observation beginning on a new line.  An observation
    contains NP real numbers giving a value for the P vector
    followed by the actual function value F(P) at that P.  

Getting the scale factors approximately right can be very
important for the performance of LOOP.  A good way to do this is
to run BAYES0 after you have used GRID and set up the LOOP.DAT
input file for LOOP.  Inside BAYES0 you can vary the scale
factors and base values and see what happens to the
corresponding likelihoods. These initial likelihoods are not
values of the likelihood function you are maximizing, but the
likelihood of the observed function values in LOOP.DAT, given
the program's Bayesian prior for the shape of the function you
are maximizing.  The scale parameters should be chosen to make
the likelihood of the data large.  Errors of 50% or so in the
scale parameters usually don't matter much.  With small numbers
of observations, occasionally the likelihood will increase
without bound as scale factors are increased.  Very large scale
factors, however, produce numerical problems in the program, and
in any case will eventually be seen to be unreasonable when more
data is entered.  

Numerical problems can be detected by evaluating the function
within BAYES0 at a data point.  The function value calculated
by the program is often not exactly the same as the input
function value (which can be seen with the program's display or
order commands).  If the disagreement is in the 4th or higher
decimal place, there is probably no problem.  If it is in a
lower decimal place, there is a problem, which usually can be
handled by making the scale factors a little smaller.  When
there are accuracy problems you cannot get rid of, making C (the
first element in the input file) somewhat larger than the numerical
errors in the function values may help give passable results.

Both BAYES0 and LOOPSMTH may produce error messages of the
form "Accuracy problems at diagonal element n."  If this occurs
during the search for a maximum of the interpolated function, it
is not an accuracy problem at all and can be ignored.  If it
occurs during the evaluation of the likelihood of the data at
the start of the program or after a change in scales or base
value, it may be a problem.  For one thing, it may indicate
numerical inaccuracy, which should be checked as indicated
above.  For another, likelihoods calculated for two different
scale settings which produce different lists of n's at which
accuracy problems are occurring are not comparable.  There is no
problem, however, if one observation is exactly the same as
another.  In this case the program drops the duplicate
observation, and it will always do so regardless of the scale
settings.

\LOOP

The directory LOOP contains MSDOS batch files to automatically
loop through runs of the program which evaluates the function. 
There are two commands to do this, GRID and LOOP, in the files
GRID.BAT and LOOP.BAT.  These files must be adapted to your own
directory names before they will work; they can easily be
adapted to run programs other than arma at each iteration.  If
you replace "arma" with another program name "X" everywhere in
both files, then this other program must read parameters from
X.par, write the function value to X.val.  However you can
easily vary these files once you understand how they work, which
you can discover from the section of the MSDOS manual on batch
file processing.

GRID:  The purpose of GRID is to execute RATS with a fixed grid
of parameter values.  It is needed because if LOOP starts with
only k points on its input file, its search will be confined to
the k-dimensional subspace spanned by these initial k points.
Thus it is always wise to run GRID for a group of n points,
where n is the length of the parmeter vector, to create the
initial input file for LOOP.  Also, you may notice in
explorations with BAYES0 that there is a section of parameter
space which LOOP has not explored which looks promising.  You
could run GRID for a grid in that region to force LOOP to take
account of it.  

To run GRID as it stands, you must have an executable file named
arma, which reads its parameter settings from arma.par and
writes a function value to arma.val.  You must provide a set of
parameter files arma1.par, arma2.par, etc.  GRID copies each
such file in succession to arma.par, evaluates the function, and
accumulates the parameter settings and function values in
GRID.DAT.  The second batch file, GRIDARMA.BAT, is actually an
integral part of GRID and must be in the directory from which
GRID.BAT is run.  

GRIDPAR: If you are using more than two or three parameters it
can be tedious to use an editor to prepare n copies of a
parameter file, each varying one parameter from some central
value; but such a set of files is usually what one wants to use
as input to GRID.  The program GRIDPAR reads a file whose name
it prompts you for (say NAMEX.PAR) containing some number n of
real numbers.  It then writes n files, with names NAME0.PAR,
NAME1.PAR, etc., with the i'th file having a value for the i'th
parameter 85% of the value it has in NAMEX.PAR, while all other
parameters are left as in NAMEX.PAR.  These are then ready for
input, possibly along with NAMEX (though usually you would
already have checked NAMEX with BAYES0), to GRID.  

LOOP: The purpose of LOOP is to search systematically for the
highest value of the likelihood.  It invokes LOOPARMA.BAT, which
in turn invokes the program LOOPSMTH to use Bayesian methods to
guess at the shape of the likelihood based on the points
calculated so far.  It uses the same function evaluation program
as does GRID.  It requires an input file named LOOP.DAT.  This
file is just like the GRID.DAT file created by the GRID command,
except that it has a few lines of extra information at the top.
It is in the same format as is required for input to BAYES0.

LOOPSMTH must also find a file named LOOP.YN which should
usually contain Y or N on the first line and N on the second.  If
it contains Y on the first line, the parameters are logged and
the scaling constants are in log units.  If it contains N the
scaling constants are in absolute units.  For logged parameters
of a likelihood scaling constants of 1 to 2 are usually OK. 

LOOPARMA.BAT is an integral part of LOOP.BAT and must reside in
the same directory from which LOOPARMA is run.

General Remarks:

Running either LOOP or GRID, you can adjust the number of
iterations by extending or contracting the list of iteration
id numbers on the first line.

\SOURCE

This directory contains all the fortran source code for BAYES0
and LOOPSMTH, together with two batch files which link them. 
You can recompile the program to make it handle bigger problems
or to work on a non-AT computer.  In its current form it handles
up to 120 observations on 15 parameters.  It has all been compiled
and linked successfully with Microsoft Fortran 5.0.