# User-Defined Functions

This chapter explains functions from a user perspective with examples; see the language reference for a full specification. User-defined functions allow computations to be encapsulated into a single named unit and invoked elsewhere by name. Similarly, functions allow complex procedures to be broken down into more understandable components. Writing modular code using descriptively named functions is easier to understand than a monolithic program, even if the latter is heavily commented.1

## Basic functions

Here’s an example of a skeletal Stan program with a user-defined relative difference function employed in the generated quantities block to compute a relative differences between two parameters.

functions {
real relative_diff(real x, real y) {
real abs_diff;
real avg_scale;
abs_diff = abs(x - y);
avg_scale = (abs(x) + abs(y)) / 2;
return abs_diff / avg_scale;
}
}
// ...
generated quantities {
real rdiff;
rdiff = relative_diff(alpha, beta);
}

The function is named relative_diff, and is declared to have two real-valued arguments and return a real-valued result. It is used the same way a built-in function would be used in the generated quantities block.

### User-defined functions block

All functions are defined in their own block, which is labeled functions and must appear before all other program blocks. The user-defined functions block is optional.

### Function bodies

The body (the part between the curly braces) contains ordinary Stan code, including local variables. The new function is used in the generated quantities block just as any of Stan’s built-in functions would be used.

### Return statements

Return statements, such as the one on the last line of the definition of relative_diff above, are only allowed in the bodies of function definitions. Return statements may appear anywhere in a function, but functions with non-void return types must end in a return statement.

### Reject and error statements

The Stan reject statement provides a mechanism to report errors or problematic values encountered during program execution. It accepts any number of quoted string literals or Stan expressions as arguments. This statement is typically embedded in a conditional statement in order to detect bad or illegal outcomes of some processing step.

If an error is indicative of a problem from which it is not expected to be able to recover, Stan provides a fatal_error statement.

#### Catching errors

Rejection is used to flag errors that arise in inputs or in program state. It is far better to fail early with a localized informative error message than to run into problems much further downstream (as in rejecting a state or failing to compute a derivative).

The most common errors that are coded is to test that all of the arguments to a function are legal. The following function takes a square root of its input, so requires non-negative inputs; it is coded to guard against illegal inputs.

real dbl_sqrt(real x) {
if (!(x >= 0)) {
reject("dblsqrt(x): x must be positive; found x = ", x);
}
return 2 * sqrt(x);
}

The negation of the positive test is important, because it also catches the case where x is a not-a-number value. If the condition had been coded as (x < 0) it would not catch the not-a-number case, though it could be written as (x < 0 || is_nan(x)). The positive infinite case is allowed through, but could also be checked with the is_inf(x) function. The square root function does not itself reject, but some downstream consumer of dbl_sqrt(-2) would be likely to raise an error, at which point the origin of the illegal input requires detective work. Or even worse, as Matt Simpson pointed out in the GitHub comments, the function could go into an infinite loop if it starts with an infinite value and tries to reduce it by arithmetic, likely consuming all available memory and crashing an interface. Much better to catch errors early and report on their origin.

The effect of rejection depends on the program block in which the rejection is executed. In transformed data, rejections cause the program to fail to load. In transformed parameters or in the model block, rejections cause the current state to be rejected in the Metropolis sense.2

In generated quantities there is no way to recover and generate the remaining parameters, so rejections cause subsequent values to be reported as NaNs. Extra care should be taken in calling functions which may reject in the generated quantities block.

### Type declarations for functions

Function argument and return types for vector and matrix types are not declared with their sizes, unlike type declarations for variables. Function argument type declarations may not be declared with constraints, either lower or upper bounds or structured constraints like forming a simplex or correlation matrix, (as is also the case for local variables); see the table of types in the reference manual for full details.

For example, here’s a function to compute the entropy of a categorical distribution with simplex parameter theta.

real entropy(vector theta) {
return sum(theta .* log(theta));
}

Although theta must be a simplex, only the type vector is used.3

Upper or lower bounds on values or constrained types are not allowed as return types or argument types in function declarations.

### Array types for function declarations

Array arguments have their own syntax, which follows that used in this manual for function signatures. For example, a function that operates on a two-dimensional array to produce a one-dimensional array might be declared as follows.

array[] real baz(array[,] real x);

The notation [ ] is used for one-dimensional arrays (as in the return above), [ , ] for two-dimensional arrays, [ , , ] for three-dimensional arrays, and so on.

Functions support arrays of any type, including matrix and vector types. As with other types, no constraints are allowed.

### Data-only function arguments

A function argument which is a real-valued type or a container of a real-valued type, i.e., not an integer type or integer array type, can be qualified using the prefix qualifier data. The following is an example of a data-only function argument.

real foo(real y, data real mu) {
return -0.5 * (y - mu)^2;
}

This qualifier restricts this argument to being invoked with expressions which consist only of data variables, transformed data variables, literals, and function calls. A data-only function argument cannot involve real variables declared in the parameters, transformed parameters, or model block. Attempts to invoke a function using an expression which contains parameter, transformed parameters, or model block variables as a data-only argument will result in an error message from the parser.

Use of the data qualifier must be consistent between the forward declaration and the definition of a functions.

This qualifier should be used when writing functions that call the built-in ordinary differential equation (ODE) solvers, algebraic solvers, or map functions. These higher-order functions have strictly specified signatures where some arguments of are data only expressions. (See the ODE solver chapter for more usage details and the functions reference manual for full definitions.) When writing a function which calls the ODE or algebraic solver, arguments to that function which are passed into the call to the solver, either directly or indirectly, should have the data prefix qualifier. This allows for compile-time type checking and increases overall program understandability.

## Functions as statements

In some cases, it makes sense to have functions that do not return a value. For example, a routine to print the lower-triangular portion of a matrix can be defined as follows.

functions {
void pretty_print_tri_lower(matrix x) {
if (rows(x) == 0) {
print("empty matrix");
return;
}
print("rows=", rows(x), " cols=", cols(x));
for (m in 1:rows(x)) {
for (n in 1:m) {
print("[", m, ",", n, "]=", x[m, n]);
}
}
}
}

The special symbol void is used as the return type. This is not a type itself in that there are no values of type void; it merely indicates the lack of a value. As such, return statements for void functions are not allowed to have arguments, as in the return statement in the body of the previous example.

Void functions applied to appropriately typed arguments may be used on their own as statements. For example, the pretty-print function defined above may be applied to a covariance matrix being defined in the transformed parameters block.

transformed parameters {
cov_matrix[K] Sigma;
// ... code to set Sigma ...
pretty_print_tri_lower(Sigma);
// ...
}

## Functions accessing the log probability accumulator

Functions whose names end in _lp are allowed to use sampling statements and target += statements; other functions are not. Because of this access, their use is restricted to the transformed parameters and model blocks.

Here is an example of a function to assign standard normal priors to a vector of coefficients, along with a center and scale, and return the translated and scaled coefficients; see the reparameterization section for more information on efficient non-centered parameterizations

functions {
vector center_lp(vector beta_raw, real mu, real sigma) {
beta_raw ~ std_normal();
sigma ~ cauchy(0, 5);
mu ~ cauchy(0, 2.5);
return sigma * beta_raw + mu;
}
// ...
}
parameters {
vector[K] beta_raw;
real mu_beta;
real<lower=0> sigma_beta;
// ...
}
transformed parameters {
vector[K] beta;
// ...
beta = center_lp(beta_raw, mu_beta, sigma_beta);
// ...
}

## Functions acting as random number generators

A user-specified function can be declared to act as a (pseudo) random number generator (PRNG) by giving it a name that ends in _rng. Giving a function a name that ends in _rng allows it to access built-in functions and user-defined functions that end in _rng, which includes all the built-in PRNG functions. Only functions ending in _rng are able access the built-in PRNG functions. The use of functions ending in _rng must therefore be restricted to transformed data and generated quantities blocks like other PRNG functions; they may also be used in the bodies of other user-defined functions ending in _rng.

For example, the following function generates an $$N \times K$$ data matrix, the first column of which is filled with 1 values for the intercept and the remaining entries of which have values drawn from a standard normal PRNG.

matrix predictors_rng(int N, int K) {
matrix[N, K] x;
for (n in 1:N) {
x[n, 1] = 1.0;  // intercept
for (k in 2:K) {
x[n, k] = normal_rng(0, 1);
}
}
return x;
}

The following function defines a simulator for regression outcomes based on a data matrix x, coefficients beta, and noise scale sigma.

vector regression_rng(vector beta, matrix x, real sigma) {
vector[rows(x)] y;
vector[rows(x)] mu;
mu = x * beta;
for (n in 1:rows(x)) {
y[n] = normal_rng(mu[n], sigma);
}
return y;
}

These might be used in a generated quantity block to simulate some fake data from a fitted regression model as follows.

parameters {
vector[K] beta;
real<lower=0> sigma;
// ...
}
generated quantities {
matrix[N_sim, K] x_sim;
vector[N_sim] y_sim;
x_sim = predictors_rng(N_sim, K);
y_sim = regression_rng(beta, x_sim, sigma);
}

A more sophisticated simulation might fit a multivariate normal to the predictors x and use the resulting parameters to generate multivariate normal draws for x_sim.

## User-defined probability functions

Probability functions are distinguished in Stan by names ending in _lpdf for density functions and _lpmf for mass functions; in both cases, they must have real return types.

Suppose a model uses several standard normal distributions, for which there is not a specific overloaded density nor defaults in Stan. So rather than writing out the location of 0 and scale of 1 for all of them, a new density function may be defined and reused.

functions {
real unit_normal_lpdf(real y) {
return normal_lpdf(y | 0, 1);
}
}
// ...
model {
alpha ~ unit_normal();
beta ~ unit_normal();
// ...
}

The ability to use the unit_normal function as a density is keyed off its name ending in _lpdf (names ending in _lpmf for probability mass functions work the same way).

In general, if foo_lpdf is defined to consume $$N + 1$$ arguments, then

y ~ foo(theta1, ..., thetaN);

can be used as shorthand for

target += foo_lpdf(y | theta1, ..., thetaN);

As with the built-in functions, the suffix _lpdf is dropped and the first argument moves to the left of the tilde symbol (~) in the distribution statement.

Functions ending in _lpmf (for probability mass functions), behave exactly the same way. The difference is that the first argument of a density function (_lpdf) must be continuous (not an integer or integer array), whereas the first argument of a mass function (_lpmf) must be discrete (integer or integer array).

As described in the reference manual function overloading is permitted in Stan, beginning in version 2.29.

This means multiple functions can be defined with the same name as long as they accept different numbers or types of arguments. User-defined functions can also overload Stan library functions.

### Warning on usage

Overloading is a powerful productivity tool in programming languages, but it can also lead to confusion. In particular, it can be unclear at first glance which version of a function is being called at any particular call site, especially with type promotion allowed between scalar types. Because of this, it is a programming best practice that overloaded functions maintain the same meaning across definitions.

For example, consider a function triple which has the following three signatures

real triple(real x);
complex triple(complex x);
array[] real triple(array[] real);

One should expect that all overloads of this function perform the same basic task. This should lead to definitions of these functions which would satisfy the following assumptions that someone reading the program would expect

// The function does what it says
triple(3.0) == 9.0
// It is defined reasonably for different types
triple(to_complex(3.0)) == to_complex(triple(3.0))
// A container version of this function works by element
triple({3.0, 4.0})[0] == triple({3.0, 4.0}[0])

Note that none of these properties are enforced by Stan, they are mentioned merely to warn against uses of overloading which cause confusion.

### Function resolution

Stan resolves overloaded functions by the number and type of arguments passed to the function. This can be subtle when multiple signatures with the same number of arguments are present.

Consider the following function signatures

real foo(int a, real b);
real foo(real a, real b);

Given these, the function call foo(1.5, 2.5) is unambiguous - it must resolve to the second signature. But, the function call foo(1, 1.5) could be valid for either under Stan’s promotion rules, which allow integers to be promoted to real numbers.

To resolve this, Stan selects the signature which requires the fewest number of promotions for a given function call. In the above case, this means the call foo(1, 1.5) would select the first signature, because it requires 0 promotions (the second signature would require 1 promotion).

Furthermore, there must be only one such signature, e.g., the minimum number of promotions must be a unique minimum. This requirement forbids certain kinds of overloading. For example, consider the function signatures

real bar(int x, real y);
real bar(real x, int y);

These signatures do not have a unique minimum number of promotions for the call bar(1, 2). Both signatures require one int to real promotion, and so it cannot be determined which is correct. Stan will produce a compilation error in this case.

Promotion from integers to complex numbers is considered to be two separate promotions, first from int to real, then from real to complex. This means that integer arguments will “prefer” a signature with real types over complex types.

For example, consider the function signatures

real pop(real x);
real pop(complex x);

Stan will select the first signature when pop is called with an integer argument such as pop(0).

## Documenting functions

Functions will ideally be documented at their interface level. The Stan style guide for function documentation follows the same format as used by the Doxygen (C++) and Javadoc (Java) automatic documentation systems. Such specifications indicate the variables and their types and the return value, prefaced with some descriptive text.

For example, here’s some documentation for the prediction matrix generator.

/**
* Return a data matrix of specified size with rows
* corresponding to items and the first column filled
* with the value 1 to represent the intercept and the
* remaining columns randomly filled with unit-normal draws.
*
* @param N Number of rows corresponding to data items
* @param K Number of predictors, counting the intercept, per
*          item.
* @return Simulated predictor matrix.
*/
matrix predictors_rng(int N, int K) {
// ...

The comment begins with /**, ends with */, and has an asterisk (*) on each line. It uses @param followed by the argument’s identifier to document a function argument. The tag @return is used to indicate the return value. Stan does not (yet) have an automatic documentation generator like Javadoc or Doxygen, so this just looks like a big comment starting with /* and ending with */ to the Stan parser.

For functions that raise exceptions, exceptions can be documented using @throws.4

For example,

 /** ...
* @param theta
* @throws If any of the entries of theta is negative.
*/
real entropy(vector theta) {
// ...
}

Usually an exception type would be provided, but these are not exposed as part of the Stan language, so there is no need to document them.

## Summary of function types

Functions may have a void or non-void return type and they may or may not have one of the special suffixes, _lpdf, _lpmf, _lp, or _rng.

### Void vs. non-void return

Only functions declared to return void may be used as statements. These are also the only functions that use return statements with no arguments.

Only functions declared to return non-void values may be used as expressions. These functions require return statements with arguments of a type that matches the declared return type.

### Suffixed or non-suffixed

Only functions ending in _lpmf or _lpdf and with return type real may be used as probability functions in distribution statements.

Only functions ending in _lp may access the log probability accumulator through distribution statements or target += statements. Such functions may only be used in the transformed parameters or model blocks.

Only functions ending in _rng may access the built-in pseudo-random number generators. Such functions may only be used in the generated quantities block or transformed data block, or in the bodies of other user-defined functions ending in _rng.

## Recursive functions

Stan supports recursive function definitions, which can be useful for some applications. For instance, consider the matrix power operation, $$A^n$$, which is defined for a square matrix $$A$$ and positive integer $$n$$ by $A^n = \begin{cases} \textrm{I} & \quad\text{if } n = 0, \text{ and} \\ A \, A^{n-1} & \quad\text{if } n > 0. \end{cases}$

where $$\textrm{I}$$ is the identity matrix. This definition can be directly translated to a recursive function definition.

matrix matrix_pow(matrix a, int n) {
if (n == 0) {
return diag_matrix(rep_vector(1, rows(a)));
} else {
return a *  matrix_pow(a, n - 1);
}
}

It would be more efficient to not allow the recursion to go all the way to the base case, adding the following conditional clause.

else if (n == 1) {
return a;
}

## Truncated random number generation

### Generation with inverse CDFs

To generate random numbers, it is often sufficient to invert their cumulative distribution functions. This is built into many of the random number generators. For example, to generate a standard logistic variate, first generate a uniform variate $$u \sim \textsf{uniform}(0, 1)$$, then run through the inverse cumulative distribution function, $$y = \textrm{logit}(u)$$. If this were not already built in as logistic_rng(0, 1), it could be coded in Stan directly as

real standard_logistic_rng() {
real u = uniform_rng(0, 1);
real y = logit(u);
return y;
}

Following the same pattern, a standard normal RNG could be coded as

real standard_normal_rng() {
real u = uniform_rng(0, 1);
real y = inv_Phi(u);
return y;
}

that is, $$y = \Phi^{-1}(u)$$, where $$\Phi^{-1}$$ is the inverse cumulative distribution function for the standard normal distribution, implemented in the Stan function inv_Phi.

In order to generate non-standard variates of the location-scale variety, the variate is scaled by the scale parameter and shifted by the location parameter. For example, to generate $$\textsf{normal}(\mu, \sigma)$$ variates, it is enough to generate a uniform variate $$u \sim \textsf{uniform}(0, 1)$$, then convert it to a standard normal variate, $$z = \Phi(u)$$, where $$\Phi$$ is the inverse cumulative distribution function for the standard normal, and then, finally, scale and translate it, $$y = \mu + \sigma \times z$$. In code,

real my_normal_rng(real mu, real sigma) {
real u = uniform_rng(0, 1);
real z = inv_Phi(u);
real y = mu + sigma * z;
return y;
}

A robust version of this function would test that the arguments are finite and that sigma is non-negative, e.g.,

  if (is_nan(mu) || is_inf(mu)) {
reject("my_normal_rng: mu must be finite; ",
"found mu = ", mu);
}
if (is_nan(sigma) || is_inf(sigma) || sigma < 0) {
reject("my_normal_rng: sigma must be finite and non-negative; ",
"found sigma = ", sigma);
}

### Truncated variate generation

Often truncated uniform variates are needed, as in survival analysis when a time of death is censored beyond the end of the observations. To generate a truncated random variate, the cumulative distribution is used to find the truncation point in the inverse CDF, a uniform variate is generated in range, and then the inverse CDF translates it back.

#### Truncating below

For example, the following code generates a $$\textsf{Weibull}(\alpha, \sigma)$$ variate truncated below at a time $$t$$,5

real weibull_lb_rng(real alpha, real sigma, real t) {
real p = weibull_cdf(lt | alpha, sigma);   // cdf for lb
real u = uniform_rng(p, 1);               // unif in bounds
real y = sigma * (-log1m(u))^inv(alpha);  // inverse cdf
return y;
}

#### Truncating above and below

If there is a lower bound and upper bound, then the CDF trick is used twice to find a lower and upper bound. For example, to generate a $$\textsf{normal}(\mu, \sigma)$$ truncated to a region $$(a, b)$$, the following code suffices,

real normal_lub_rng(real mu, real sigma, real lb, real ub) {
real p_lb = normal_cdf(lb | mu, sigma);
real p_ub = normal_cdf(ub | mu, sigma);
real u = uniform_rng(p_lb, p_ub);
real y = mu + sigma * inv_Phi(u);
return y;
}

To make this more robust, all variables should be tested for finiteness, sigma should be tested for positiveness, and lb and ub should be tested to ensure the upper bound is greater than the lower bound. While it may be tempting to compress lines, the variable names serve as a kind of chunking of operations and naming for readability; compare the multiple statement version above with the single statement

  return mu + sigma * inv_Phi(uniform_rng(normal_cdf(lb | mu, sigma),
normal_cdf(ub | mu, sigma)));

for readability. The names like p indicate probabilities, and p_lb and p_ub indicate the probabilities of the bounds. The variable u is clearly named as a uniform variate, and y is used to denote the variate being generated itself.

## Footnotes

1. The main problem with comments is that they can be misleading, either due to misunderstandings on the programmer’s part or because the program’s behavior is modified after the comment is written. The program always behaves the way the code is written, which is why refactoring complex code into understandable units is preferable to simply adding comments.↩︎

2. Just because this makes it possible to code a rejection sampler does not make it a good idea. Rejections break differentiability and the smooth exploration of the posterior. In Hamiltonian Monte Carlo, it can cause the sampler to be reduced to a diffusive random walk.↩︎

3. A range of built-in validation routines is coming to Stan soon! Alternatively, the reject statement can be used to check constraints on the simplex.↩︎

4. As of Stan 2.9.0, the only way a user-defined producer will raise an exception is if a function it calls (including distribution statements) raises an exception via the reject statement.↩︎

5. The original code and impetus for including this in the manual came from the Stan forums post; by user lcomm, who also explained truncation above and below.↩︎