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During my work I encountered a plot, in which we have two curves indicating:

  1. Mean value of observations of dependent variable $Y$ (in my case it was annual frequency of claims), calculated for diffent values of some other variable, let's say $Z$;
  2. Mean predicted value $\hat{Y}$, which is an evaluation in data point of estimate of conditional expectation $\mathbb{E}(Y|\mathbf{X})$ for some vector of variable $\mathbf{X}$, calculated for different values of $Z$.

Plot described above serves for checking goodness-of-fit: if two curves don't coincide on some part of a range of variable $Z$, then we should consider adding this variable to our vector $\mathbf{X}$ in some form, since our statistical model does not capture variability of $Y$ across range of $Z$.

To gain more insight into this tool, I translated its content into probabilistic language:

  • (1) would be a nonparametric estimate of conditional mean $\mathbb{E}(Y|Z=\cdot)$;
  • (2) would be an estimate of $\mathbb{E}(\mathbb{E}(Y|\mathbf{X})|Z=\cdot)$.

After this step I realized, that desirable "closeness" of two aforementioned quantities looks the same as Tower Property for conditional expectations, which, in a language of sub-sigma-algebras $\mathcal{G}\subseteq\mathcal{H}$, states that:

$$ \mathbb{E}(\mathbb{E}(Y|\mathcal{H})|\mathcal{G})=\mathbb{E}(Y|\mathcal{G})\quad a.s.\quad (*) $$

This observation provides a reformulation of a meaning of plot: "closeness" of $\mathbb{E}(Y|Z=\cdot)$ and $\mathbb{E}(\mathbb{E}(Y|\mathbf{X})|Z=\cdot)$ says that $Z$ provides no additional information- its predictive power is already contained in $\mathbf{X}$.

But here I started to wonder, if I am fully right: truth of condition $\mathcal{G}\subseteq\mathcal{H}$ is sufficient, but maybe not necessary for $(*)$ to hold.

Here arise my questions:

  1. Is it true that $(*)\Rightarrow\mathcal{G}\subseteq\mathcal{H}$? If no, what is an example of situation, where $(*)$ hold, but $\mathcal{G}$ is not a subset of $\mathcal{H}$?
  2. Do you have an experience with plots like one described above? Do you agree with my interpretation? Any additions and corrections are welcome.

Edit
(1) My progress after comments of @whuber: assuming that $(*)$ is true, by linearity of conditional expectation we have: $$ \mathbb{E}(\mathbb{E}(Y|\mathcal{H})-Y|\mathcal{G})=0\quad a.s. $$ By defining equations of conditional expectation, for all $G\in\mathcal{G}$: $$ \mathbb{E}([\mathbb{E}(Y|\mathcal{H})-Y]\mathbf{1}_G)=\mathbb{E}(0\mathbf{1}_G)=0\\\mathbb{E}(\mathbb{E}(Y|\mathcal{H})\mathbf{1}_G)=\mathbb{E}(Y\mathbf{1}_G)=\mathbb{E}(\mathbb{E}(Y|\mathcal{G})\mathbf{1}_G). $$ Here I am struggling again since I cannot establish from equation above any relation between $\mathcal{G}$ and $\mathcal{H}$.

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  • $\begingroup$ I can't even make sense of $(*)$ unless $\mathcal G\subseteq\mathcal H.$ What definition of conditional expectation are you using? $\endgroup$
    – whuber ♦
    Mar 14 at 22:44
  • $\begingroup$ @whuber I am using classical one: for integrable $Y$ it is a $\mathcal{G}$-measurable random variable, which has equal integrals with $Y$, on $G\in\mathcal{G}$. $\endgroup$
    – Mentossinho
    Mar 15 at 15:08
  • $\begingroup$ Then why is there any question here? The implication is immediate. $\endgroup$
    – whuber ♦
    Mar 15 at 16:37
  • $\begingroup$ @whuber I added "edit" section to my question. $\endgroup$
    – Mentossinho
    Mar 15 at 20:00
  • $\begingroup$ The very notation implies $E(Y\mid\mathcal H)$ is $\mathcal G$-measurable, whence $\mathcal G$ is a subalgebra of $\mathcal H.$ $\endgroup$
    – whuber ♦
    Mar 15 at 20:38

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