<programming> An algorithm for ascribing types to expressions in some language, based on the types of the constants of the language and a set of type inference rules such as f :: A -> B, x :: A --------------------- (App) f x :: B This rule, called "App" for application, says that if expression f has type A -> B and expression x has type A then we can deduce that expression (f x) has type B. The expressions above the line are the premises and below, the conclusion. An alternative notation often used is: G |- x : A where "|-" is the turnstile symbol (LaTeX vdash) and G is a type assignment for the free variables of expression x. The above can be read "under assumptions G, expression x has type A". (As in Haskell, we use a double "::" for type declarations and a single ":" for the infix list constructor, cons). Given an expression plus (head l) 1 we can label each subexpression with a type, using type variables X, Y, etc. for unknown types: (plus :: Int -> Int -> Int) (((head :: [a] -> a) (l :: Y)) :: X) (1 :: Int) We then use unification on type variables to match the partial application of plus to its first argument against the App rule, yielding a type (Int -> Int) and a substitution X = Int. Re-using App for the application to the second argument gives an overall type Int and no further substitutions. Similarly, matching App against the application (head l) we get Y = [X]. We already know X = Int so therefore Y = [Int]. This process is used both to infer types for expressions and to check that any types given by the user are consistent. See also generic type variable, principal type. (1995-02-03)

Type inference refers to the automatic detection of the type of an expression in a formal language. These include programming languages and mathematical type systems, but also natural languages in some branches of computer science and linguistics.

Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G.