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## 6.10 Type Inference

Stan is strongly statically typed, meaning that the implementation type of an expression can be resolved at compile time.

### Implementation Types

The primitive implementation types for Stan are

int, real, vector, row_vector,  matrix.

Every basic declared type corresponds to a primitive type; see the primitive type table for the mapping from types to their primitive types.

Primitive Type Table. The table shows the variable declaration types of Stan and their corresponding primitive implementation type. Stan functions, operators, and probability functions have argument and result types declared in terms of primitive types plus array dimensionality.

type primitive type
int int
real real
matrix matrix
cov_matrix matrix
corr_matrix matrix
cholesky_factor_cov matrix
cholesky_factor_corr matrix
vector vector
simplex vector
unit_vector vector
ordered vector
positive_ordered vector
row_vector row_vector

A full implementation type consists of a primitive implementation type and an integer array dimensionality greater than or equal to zero. These will be written to emphasize their array-like nature. For example, int[] has an array dimensionality of 1, int an array dimensionality of 0, and int[ , ,] an array dimensionality of 3. The implementation type matrix[ , , ] has a total of five dimensions and takes up to five indices, three from the array and two from the matrix.

Recall that the array dimensions come before the matrix or vector dimensions in an expression such as the following declaration of a three-dimensional array of matrices.

matrix[M, N] a[I, J, K];

The matrix a is indexed as a[i, j, k, m, n] with the array indices first, followed by the matrix indices, with a[i, j, k] being a matrix and a[i, j, k, m] being a row vector.

### Type Inference Rules

Stan’s type inference rules define the implementation type of an expression based on a background set of variable declarations. The rules work bottom up from primitive literal and variable expressions to complex expressions.

#### Literals

An integer literal expression such as 42 is of type int. Real literals such as 42.0 are of type real.

#### Variables

The type of a variable declared locally or in a previous block is determined by its declaration. The type of a loop variable is int.

There is always a unique declaration for each variable in each scope because Stan prohibits the redeclaration of an already-declared variables.3

#### Indexing

If x is an expression of total dimensionality greater than or equal to $$N$$, then the type of expression e[i1, ..., iN] is the same as that of e[i1]...[iN], so it suffices to define the type of a singly-indexed function. Suppose e is an expression and i is an expression of primitive type int. Then

• if e is an expression of array dimensionality $$K > 0$$, then e[i] has array dimensionality $$K-1$$ and the same primitive implementation type as e,

• if e has implementation type vector or row_vector of array dimensionality 0, then e[i] has implementation type real, and

• if e has implementation type matrix, then e[i] has type row_vector.

#### Function Application

If f is the name of a function and e1,...,eN are expressions for $$N \geq 0$$, then f(e1,...,eN) is an expression whose type is determined by the return type in the function signature for f given e1 through eN. Recall that a function signature is a declaration of the argument types and the result type.

In looking up functions, binary operators like real * real are defined as operator*(real,real) in the documentation and index.

In matching a function definition, arguments of type int may be promoted to type real if necessary (see the subsection on type promotion in the function application section for an exact specification of Stan’s integer-to-real type-promotion rule).

In general, matrix operations return the lowest inferable type. For example, row_vector * vector returns a value of type real, which is declared in the function documentation and index as real operator*(row_vector,vector).

1. Languages such as C++ and R allow the declaration of a variable of a given name in a narrower scope to hide (take precedence over for evaluation) a variable defined in a containing scope.