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General Statistical Models

  General Statistical Models The time has come to learn some theory. This is a preview of STAT 5101–5102. We don’t need to learn much theory. We will proceed with the general strategy of all introductory statistics: don’t derive anything, just tell you stuff. 3.1  Probability Models 3.1.1  Kinds of Probability Theory There are two kinds of probability theory. There is the kind you will learn in STAT 5101-5102 (or 4101–4102, which is more or less the same except for leaving out the multivariable stuff). The material covered goes back hundreds of years, some of it discovered in the early 1600’s. And there is the kind you would learn in MATH 8651–8652 if you could take it, which no undergraduate does (it is a very hard Ph. D. level math course). The material covered goes back to 1933. It is all “new math”. These two kinds of probability can be called  classical  and  measure-theoretic , respectively. The old kind (classical) is still very useful. For every 1000 people who know a lot of pr
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Solution of a linear system

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  Solution of a linear system [ edit ] The steepest descent algorithm applied to the  Wiener filter [11] Gradient descent can be used to solve a system of linear equations � � − � = 0 reformulated as a quadratic minimization problem. If the system matrix  �  is real  symmetric  and  positive-definite , an objective function is defined as the quadratic function, with minimization of � ( � ) = � � � � − 2 � � � , so that ∇ � ( � ) = 2 ( � � − � ) . For a general real matrix  � ,  linear least squares  define � ( � ) = ‖ � � − � ‖ 2 . In traditional linear least squares for real  �  and  �  the  Euclidean norm  is used, in which case ∇ � ( � ) = 2 � � ( � � − � ) . The  line search  minimization, finding the locally optimal step size  �  on every iteration, can be performed analytically for quadratic functions, and explicit formulas for the locally optimal  �  are known. [5] [12] For example, for real  symmetric  and  positive-definite  matrix  � , a simple algorithm can be as follows, [5
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