Wednesday, December 25, 2024

3 Essential Ingredients For Exponential Family And Generalized Linear Models

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Generalized linear models were formulated by John Nelder and Robert Wedderburn as a way of unifying various other statistical models, including linear regression, logistic regression and Poisson regression.
In the cases of the exponential and gamma distributions, the domain of the canonical link function is not the same as the permitted range of the mean.
In this post, you will learn about the concepts of generalized linear models (GLM) with the help of Python examples.
The distribution in this case is written as
Or more compactly as
Or alternatively as
We use cumulative distribution functions (CDF) in order to encompass both discrete and continuous distributions.

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With these various pieces in place, we can frame our statistical problem as follows: given data \(\{(y_i, x_i)\}_{i=1}^n\), a link visit this page \(g\), and an exponential family density \(f\), estimate the unknown parameter \(\beta\). Link functions are one-to-one, continuous differentiable transformations, $g(.
In this case, H is also absolutely continuous and can be written

d

H
(
x
)
=
h
(
x
)

d

x

{\displaystyle \,{\rm {d\,}}H(x)=h(x)\,{\rm {d\,}}x\,}

so the formulas reduce to that of the previous paragraphs. , Xn) whose number of scalar components does not increase as the sample size n increases; the statistic T may be a vector or a single scalar number, but whatever it is, its size will neither grow nor shrink when more data are obtained.
Computing these formulas using integration would be much more difficult.
Exponential families form the basis for the distribution functions used in generalized linear models, a class of model that encompass many of the commonly used regression models in statistics.

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the Student’s t-distribution (compounding a normal distribution over a gamma-distributed precision prior), and the beta-binomial and Dirichlet-multinomial distributions. \) We can show through calculus (use the chain rule!) that \[ \nabla \mathcal{L}(\beta) = X^T \Delta s\] and \[ \nabla^2 \mathcal{L}(\beta) = – X^T (\Delta V \Delta – \DeltaH ) X.
\end{cases}
\] Also, define the vector \(s = [ y_1 – \mu_1, \dots, y_n – \mu_n]^T. When using the canonical link function,

b
(

)
=

=

X

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{\displaystyle b(\mu )=\theta =\mathbf {X} {\boldsymbol {\beta }}}

, which allows

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X

T

Y

{\displaystyle \mathbf {X} ^{\rm {T}}\mathbf {Y} }

to be a sufficient statistic for

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{\displaystyle {\boldsymbol {\beta }}}

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