FORMULA: Syntax for arithmetic formulas. Arithmetic formulas are used to specify a distribution (see dist.doc), and by the calculator program (see calc.doc). These formulas are written in pretty much the usual way, with the following conventions: 1) A variable name is either a single lower-case letter (a-z) or a lower-case letter followed by a single digit (0-9). Note, however, that when a formula is used to specify a distribution, the allowed state variables are restricted (see dist.doc). 2) Function names start with an upper-case letter (eg, Sin or Sqrt). 3) As usual, addition and subtraction (+ and -) have lowest precedence, with multiplication and division (* and /) having higher precedence, and exponentiation (^) having highest precedence. Evaluation is otherwise left to right. 4) The multiplication operator can be omitted. Note, however, that x6 is a variable name, not x*6 (but x 6 and 6x are products). 5) Only positive integer constants are allowed as exponents (eg, x^2 is allowed, but not x^y or x^0.5). 6) Three kinds of brackets can be used: (), [], and {}. There's no difference in their effects, but they must be properly matched. 7) Scientific notation uses "E", not "e" - eg, 2.1E-10, not 2.1e-10. The constant Pi=3.14159... is defined. Use Exp(1), defined below, to get the value of e=2.71828... A number of functions are built in; others can be defined by external programs, as described below. The following univariate functions are built in: Sin Cos Tan Log Exp Abs Theta Delta Sinh Cosh Tanh Sqrt LGamma Frac Sign These compute the obvious things, except as follows: LGamma Computes the log of the Gamma function. Frac Computes the fractional part of a number (eg, Frac(2.7) is 0.7, Frac(3.0) is 0, and Frac(-0.1) is 0.9). Sign Finds the sign of its argument, -1 or +1, or 0. Theta Has the value 1 for non-negative arguments, 0 for negative arguments. Delta Has the value 1 for an argument of zero, 0 for non-zero arguments. A special function LogSumExp, which takes an indefinite number of arguments, is also provided. It computes the log of the sum of the exponentials of its arguments. This is done in such a way as to avoid overflow whenever the final result is within range. For example LogSumExp(1000,2000,1500) evaluates to 2000 (or very close to it), without overflow occurring. The following functions return minus the log of the probability density for a quantity with respect to some distribution: Normal Gaussian ExpGamma ExpGamma2 Bernoulli The Normal (equivalently, Gaussian) function computes minus the log of the probability density for the normal distribution with the given mean and variance. That is, Normal(x,m,v) is equivalent to Log(2*Pi*v)/2 + (x-m)^2/(2*v) The ExpGamma function computes minus the log of the probability density for a quantity whose exponential has the Gamma distribution with the given shape and mean. ExpGamma(x,s,b) is equivalent to - s*Log(b) + LGamma(s) + Exp(x)*b - s*x ExpGamma2(x,a,m) is an alternative parameterization, defined by ExpGamma2 (x, a, m) = ExpGamma (x, a/2, (a/2)/m) This is closer to the way priors are specified for neural network and other models. For example, the log of a hyperparameter whose prior is specified by "5:3" has the ExpGamma2(3,1/5^2) distribution. Bernoulli(x,p) returns -Log(p) if x is non-zero, and -Log(1-p) if x is zero. If the first parameter of one of these density functions is a variable, it can be written as follows instead: x ~ Normal(m,v) is equivalent to Normal(x,m,v) x ~ Gaussian(m,v) is equivalent to Gaussian(x,m,v) x ~ ExpGamma(s,b) is equivalent to ExpGamma(x,s,b) x ~ ExpGamma2(a,m) is equivalent to ExpGamma2(x,a,m) x ~ Bernoulli(p) is equivalent to Bernoulli(x,p) Use of this form can allow random number generation by some programs. Errors in computations (eg, division by zero) are handled in whatever way C handles them. Multiplications and divisions in which the first operand evaluates to zero evaluate to zero without evaluating the second operand being evaluated. For example, 0*Log(0) will evaluate to zero, whereas Log(0)*0 might generate an error. (However, this shortcut does not apply when derivatives are also being computed.) When formulas are given as arguments to commands, they must often be quoted, to prevent the shell from interpreting the special characters they contain. The detailed syntax of formulas is as follows, with | indicating alternatives, [] indicating optional parts, and {} indicating parts that can occur zero or more times). <formula> ::= [ <term> ] { <plusminus> <term> } <plusminus> ::= "+" | "-" <term> ::= <factor> { <timesdiv> <factor> } <timesdiv> ::= [ "*" ] | "/" <factor> ::= <prefactor> [ "^" <integer> ] <prefactor> ::= "(" <formula> ")" | "[" <formula> "]" | "{" <formula> "}" | <variable> | <constant> | <number> | <ufunction> <prefactor> | <dfunction> <arglist> | <variable> ~ <dfunction> <arglist> | "LogSumExp" <arglist> <arglist> ::= "(" <formula> { "," <formula> } ")" | "[" <formula> { "," <formula> } "]" | "{" <formula> { "," <formula> } "}" <ufunction> ::= <one of the univariate functions built in or defined externally> <dfunction> ::= <one of the log density functions built in or defined externally> <variable> ::= <lower-case-letter> [ <digit> ] <constant> ::= Pi <number> ::= <integer> [ "." { <digit> } ] [ <exponent> ] | "." <digit> { <digit> } [ <exponent> ] <exponent> ::= "E" [ <plusminus> ] <integer> <integer> ::= <digit> { <digit> } Spaces may be inserted anywhere except inside variable names, function names, or numbers. See calc.doc for some examples of formulas. Defining new functions. A new function can be defined by writing a program for computing its value, typically in C. New distributions can also be defined in this way. However, at present, gradients for these functions are not handled, and random generation from distributions is also not implemented. Such an externally-defined function or distribution must have a name consisting of an upper-case letter followed by zero or more upper-case or lower-case letters or digits. A program with this same name must exist in the current directory. When the value of the function is first needed, this program will be started as a subprocess connected via pipes replacing standard input and output. The program should consist of a loop that reads a request to evaluate the function for specific values of its arguments from standard input, and then writes the corresponding function value to standard output. The program should exit when it encounters EOF on standard input. Each function evaluation request will start with the following header, defined in extfunc.h in the util directory: typedef struct { enum { Value, Value_and_gradient, Random_variate} want; int n_args; } ext_header; This should be read from standard input using fread. The 'want' field says what type of operation has been requested. Currently, only 'Value' is used. The 'n_args' field says how many arguments were passed to the function. After the header has been read, this many double-precision floating-point arguments should be read from standard input using fread. The value of the function with these arguments should then be written to standard output as a double-precision floating-point number using fwrite, and the output forced out using fflush. Note that this procedure makes use of Unix-specific facilities. Copyright (c) 1995-2004 by Radford M. Neal