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Question

To work our way up to GLMs, we will begin by defining exponential family distributions. We say that a class of distributions is in the exponential family if it can be written in the form

$$p(y ; \eta)=b(y) \exp \left(\eta^{T} T(y)-a(\eta)\right)$$

Here, $$\eta$$ is called the natural parameter (also called the canonical parameter) of the distribution; T(y) is the [...] (for the distributions we consider, it will often be the case that T(y) = y); and $$a(\eta)$$ is the log partition function. The quantity $$e^{-a(\eta)}$$ essentially plays the role of a normalization constant, that makes sure the distribution $$p(y;\eta)$$ sums/integrates over y to 1.

sufficient statistic

Question

To work our way up to GLMs, we will begin by defining exponential family distributions. We say that a class of distributions is in the exponential family if it can be written in the form

$$p(y ; \eta)=b(y) \exp \left(\eta^{T} T(y)-a(\eta)\right)$$

Here, $$\eta$$ is called the natural parameter (also called the canonical parameter) of the distribution; T(y) is the [...] (for the distributions we consider, it will often be the case that T(y) = y); and $$a(\eta)$$ is the log partition function. The quantity $$e^{-a(\eta)}$$ essentially plays the role of a normalization constant, that makes sure the distribution $$p(y;\eta)$$ sums/integrates over y to 1.

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Question

To work our way up to GLMs, we will begin by defining exponential family distributions. We say that a class of distributions is in the exponential family if it can be written in the form

$$p(y ; \eta)=b(y) \exp \left(\eta^{T} T(y)-a(\eta)\right)$$

Here, $$\eta$$ is called the natural parameter (also called the canonical parameter) of the distribution; T(y) is the [...] (for the distributions we consider, it will often be the case that T(y) = y); and $$a(\eta)$$ is the log partition function. The quantity $$e^{-a(\eta)}$$ essentially plays the role of a normalization constant, that makes sure the distribution $$p(y;\eta)$$ sums/integrates over y to 1.

sufficient statistic

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