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)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)