GM-RKB: Statistical Model

A Statistical Model is a Set of Probability Functions defined by a 5-Tuple Statistical Model Parameter Vector (S_θ, B_θ, P_θ, f_θ, W, P).

AKA: Stochastic Model, Statistical Function, Probabilistic Model, Probability Model, Statistical Model Family, Family of Models, Statistical Metamodel.
Context:
- It can represent a Stochastic System.
- It can be grounded in Statistical Theory that describes a Statistical Modeling System, such as a Probabilistic Graphical Model or Logistic Regression.
- It can be:
  - a Discrete Statistical Model.
  - a Continuous Statistical Model.
- It can be:
  - a Parametric Statistical Model.
  - a Nonparametric Statistical Model.
- It can be:
  - a True Statistical Model of some Stochastic Process.
  - a False Statistical Model of some Stochastic Process.
- It can be:
  - a Correctly Specified Statistical Model for a Statistical Modeling Task.
  - a Misspecified Statistical Model for a Statistical Modeling Task.
- It can be Produced by a Probabilistic Modeling Algorithm/Statistical Modeling Algorithm.
- It can assign a Likelihood Estimate to Events.
Example(s):
See: Deterministic Model, Heuristic Model, Predictive Function, Statistical Learning, Bayesian Model, Stochastic Calculus.

References

2009

(Wikipedia, 2009) http://en.wikipedia.org/wiki/Statistical_model
- A statistical model is a set of mathematical equations which describe the behavior of an object of study in terms of random variables and their associated probability distributions. If the model has only one equation it is called a single-equation model, whereas if it has more than one equation, it is known as a multiple-equation model.
- In mathematical terms, a statistical model is frequently thought of as a pair <math> (Y, P) </math> where <math> Y </math> is the set of possible observations and <math> P </math> the set of possible probability distributions on <math> Y </math>. It is assumed that there is a distinct element of <math> P </math> which generates the observed data. Statistical inference enables us to make statements about which element(s) of this set are likely to be the true one.
- Three notions are sufficient to describe all statistical models.
  - We choose a statistical unit, such as a person, to observe directly. Multiple observations of the same unit over time is called longitudinal research. Observations of multiple statistical attributes is a common way of studying relationships among the attributes of a single unit.
  - Our interest may be in a statistical population (or set) of similar units rather than in any individual unit. Survey sampling offers an example of this type of modeling.
  - Our interest may focus on a statistical assembly where we examine functional subunits of the statistical unit. For example, Physiology modeling probes the organs which compose the unit. A common model for this type of research is the stimulus-response model.
http://www.nature.com/nrg/journal/v5/n4/glossary/nrg1318_glossary.html
- Probabilistic Model: A model in which the data are modelled as random variables, the probability distribution of which depends on parameter values. ...
http://www.cdc.gov/drugresistance/community/program-planner/Glossary-Eval-Res.htm
- A model that is normally based on previous research and permits transformation of a specific impact measure into another specific impact measure ...

2008

(Georgii, 2008) => Hans-Otto Georgii. (2008). "Stochastics: introduction to probability theory and statistics." Walter de Gruyter. ISBN 3110191458 (alternate, search)
- A statistical model is a triple (X F, P_v : v ∈ ϴ) consisting of a sample space X, a σ-algebra F on X, and a class {P_v : v ∈ ϴ} of (at least two) probability measures on (X, F), which are indexed by an Index Set ϴ.

2007

American Meteorology Society. (2007). "Glossary of Meteorology" http://amsglossary.allenpress.com/glossary/browse?s=s&p=102
- stochastic model — A model of a system that includes some sort of random forcing. In many cases, stochastic models are used to simulate deterministic systems that include smaller- scale phenomena that cannot be accurately observed or modeled. As such, these small-scale phenomena are effectively unpredictable. A good stochastic model manages to represent the average effect of unresolved phenomena on larger-scale phenomena in terms of a random forcing.

2006

(Cox, 2006) => David R. Cox. (2006). "Principles of Statistical Inference." Cambridge University Press. ISBN 9780521685672 (alternate, search)
- Key ideas about probability models and the objectives of statistical analysis are introduced. The differences between frequentist and Bayesian analyses are illustrated in a very special case.
- We use throughout the notation that observable random variables are represented by capital letters and observations by the corresponding lower case letters.
- A model, or strictly a family of models, specifies the density of Y to be
  - f_Y(y : z: θ), (1.1)
- Where θ ⊂ Ω_θ is unknown. The distribution may depend also on design features of the study that generated the data. We typically simplify the notation to f_Y(y:θ), although the explanatory variables z are frequently essential in specific applications.
- To chose the model appropriately is crucial to fruitful application.

2004

(Isaev, 2004) => Alexander Isaev. (2004). "Introduction to Mathematical Methods in Bioinformatics." Springer. ISBN 3540219730 (alternate, search)
- Definition 8.1. A statistical model is a family of probability spaces {(S_θ, B_θ, P_θ)} and a family of random variables {f_θ} with common range W &sub' R, each defined on the respective space for θ ∈ P, where P is an index set. The variable θ denotes the parameters of the model, the set is called the range of the model, and the index set P the parameter space of the model. Hence a statistical model can be thought of as a family {(S_θ, B_θ, P_θ, f_θ, W, P)}.
- When studying and applying statistical models, one is primarily interested in the distributions of f_θ. Therefore, often statistical models are not specified in full as in Definition 8.1, but only the family {F_θ = F_{f θ}}, ....

2003

(Davison, 2003) => Anthony C. Davison. (2003). "Statistical Models." Cambridge University Press. ISBN 0521773393 (alternate, search)
- Models and likelihood are the backbone of modern statistics and data analysis.
- A statistical model is a probability distribution constructed to enable inference to be drawn or decisions made from data.
- The huge variety of such problems makes it hard to develop a single over-arching theory, but nevertheless common strands appear. Uniting them is the idea of a statistical model.
- The key feature of a statistical model is that variability is represented using probability distributions, which form the building-blocks from which the model is constructed. Typically it must accommodate both random and systematic variation. The randomness inherent in the probability distribution accounts for apparently haphazard scatter in the data, and systematic pattern is supposed to be generated by structure in the model. The art of modelling lies in finding a balance that enables the questions at hand to be answered or new ones posed. The complexity of the model will depend on the problem at hand and the answer required, so different models and analyses may be appropriate for a single set of data.

2002

(McCullagh, 2002) => Peter McCullagh. (2002). "What is a Statistical Model" In: The Annals of Statistics, 30(5).
- ...
(Pitt & al, 2002) => Mark A. Pitt, In Jae Myung, and Shaobo Zhang. (2002). "Toward a Method of Selecting Among Computational Models of Cognition." In: Psychological Review, 109(3). [** ... From a statistical standpoint, data are a sample generated from a true but unknown [[Probability Function|probability distribution, which is the regularity underlying the data. A statistical model is defined as a collection of probability distributions defined on experimental data and indexed by the model’s parameter vector, whose values range over the parameter space of the model. If the model contains as a special case the probability distribution that generated the data (i.e., the “true” model), then the model is said to be correctly specified; otherwise it is misspecified. ...

1987

(Hogg & Ledolter) => Robert V. Hogg and Johannes Ledolter. (1987). "Engineering Statistics. Macmillan Publishing Company.
- In applied mathematics we are usually concerned with either deterministic or probabilistic models, although in many instances these are intertwined. ... a deterministic model because everything is known once ... conditions are specified.