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I'm searching for a book who treat GLM in a formal way, with a measure theoretic approach. Someone could help me?

i try to be more specific

Suppose $ (\Omega,\mathcal{F},\mathbb{P})$ is a probability space, $\vec{X} : \Omega \rightarrow \mathbb{R}^n$ random vector and $Y:\Omega \rightarrow \mathbb{R}$ random variable

i can write $Y(\omega) = \mathbb{E}[Y|\vec{X}](\omega) + \mathcal{E}(\omega)$ where $\mathbb{E}[Y|\vec{X}]$ is the only(a.s) r.variable that satisfy

$$\int_{B}\mathbb{E}[Y|\vec{X}]d\mathbb{P}=\int_{B}Yd\mathbb{P},\forall B \in \sigma(\vec{X})$$

by doob-dynkin lemma we can demonstrate that, for some $g: \mathbb{R}^n \rightarrow \mathbb{R}$ measurable function,

$$\mathbb{E}[Y|\vec{X}](\omega)=g(\vec{X}(\omega))=\mathbb{E}[Y|\vec{X}=\vec{X}(\omega)]$$

where $\mathbb{E}[Y|\vec{X}=\vec{X}(\omega)]=\int_{\Omega}Yd\mathbb{P}(*|\vec{X}=\vec{X}(\omega))$

ok now my opinion about GLM is to specify the following equations:

  • $\mathbb{P}(Y \in A|\vec{X}=\vec{x})=\int_{A}f_{Y|\vec{X}}(y|\vec{x}; \theta,\varphi)dy, \forall A \in \mathcal{B}(\mathbb{R})$
  • $f_{Y|\vec{X}}$ belongs to exponential family
  • $\mathbb{E}_{\vec{\beta}}[Y|\vec{X}]=g(\vec{X}*\vec{\beta})$

this means that $Y=g(\vec{X}*\vec{\beta})+\mathcal{E}$

$\forall \omega \in \vec{X}=\vec{x}$ holds $Y(\omega)=g(\vec{x}*\vec{\beta})+\mathcal{E}(\omega)$

suppose $\vec{x}_1,...,\vec{x}_m$ are values of $\vec{X}$ that for at least one $\omega_i \in \vec{X}=\vec{x}_i$, $y_i=Y(\omega_i)$ is observable

Now i always see that $Y_i$ is the $i$- observation, maybe defined as $Y(\omega_i)$, and specify the joint distribution saying that these are independent, but.. what? is there a passage to product measure? we can build a set of independent r.v defining on the product space the variable $$Y_i(\vec{\omega})=Y(\omega_i)\hspace{1em} \mathrm{and} \hspace{1em} \vec{X}_i (\vec{\omega})=\vec{X}(\omega_i)$$

now $\{(Y_i,\vec{X}_i)\}_{i=1,...,m}$ is an independent set of r.v and holds

$$\mathbb{P}((Y_1,...,Y_m) \in A_1 \times...\times A_m | \vec{X}_1=\vec{x}_1,...,\vec{X}_m=\vec{x}_m)=\prod_{i=1}^{m}\mathbb{P}(Y_i \in A_i |\vec{X}_i=\vec{x}_i)$$

so the joint conditional density is $$f(\vec{y}|\vec{x}_1,...,\vec{x}_m;\vec{\theta},\varphi)=\prod_{i=1}^{m}f_{Y|\vec{X}}(y_i|\vec{x}_i; \theta_i,\varphi)$$

this is some use of measure theory and this is what i need to find in a book.

anto_zoolander
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    I don't think there's a consensus on what the [best book on GLM](https://stats.stackexchange.com/q/209531/930) is, but maybe you could clarify what you expect from a measure theoretic approach to GLM. – chl Oct 16 '20 at 18:22
  • I need a book who specify how i can build a glm and use regression. Also i search for formally use of random variables, conditional expectation, conditional distribution and so on. I always see only the equation $y_i = g(x_i* \beta) + \epsilon_i$ but nobody says that the equation comes out from random variables, conditional expectation and so on – anto_zoolander Oct 16 '20 at 18:57
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    In this case it looks to me this has nothing to do with measure theory and the link provided above might be a good start. – chl Oct 16 '20 at 19:04
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    I second the idea that you’re not looking for something involving measure theory. Perhaps you could say what measure theory concepts you’d want to tie to GLMs. – Dave Oct 16 '20 at 19:45
  • i edit my post to be more specific – anto_zoolander Oct 16 '20 at 21:34

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