Linear Predictors

Why

A simple class of predictors when the input and output sets are vector spaces is the class of linear predictors.

Definition

A linear predictor (or linear model or deterministic linear model) is a predictor which is a linear function of its inputs.

Such a model is simple to implement and interpretable, at the cost of flexibility.

$\R ^d$ Example

Let $X = \R ^d$ be a set of inputs and $Y = \R $ a set of outputs. The linear functions on $\R ^d$ are in one-to-one correspondence with vectors in $\R ^d$.

A linear function $f: \R ^d \to \R $ over the vector space $(\R ^d, \R )$ has a set of parameters $w \in \R ^d$ so that

\[ f(x) = \sum_{i} w_i x_i = w^\top x. \]