Formal Definition
A Statistical model, is a collection of probability distribution functions or probability density functions (collectively referred to as distributions for brevity). A parametric model is a collection of distributions, each of which is indexed by a unique finite-dimensional parameter:, where is a parameter and is the feasible region of parameters, which is a subset of d-dimensional Euclidean space. A statistical model may be used to describe the set of distributions from which one assumes that a particular data set is sampled. For example, if one assumes that data arise from a univariate Gaussian distribution, then one has assumed a Gaussian model: .
A non-parametric model is a set of probability distributions with infinite dimensional parameters, and might be written as . A semi-parametric model also has infinite dimensional parameters, but is not dense in the space of distributions. For example, a mixture of Gaussians with one Gaussian at each data point is dense is the space of distributions. Formally, if d is the dimension of the parameter, and n is the number of samples, if as and as, then the model is semi-parametric.
Read more about this topic: Statistical Model
Famous quotes containing the words formal and/or definition:
“Good gentlemen, look fresh and merrily.
Let not our looks put on our purposes,
But bear it as our Roman actors do,
With untired spirits and formal constancy.”
—William Shakespeare (1564–1616)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)