# Plot indicating Tower Property of conditional expectation

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 $$\mathcal{H}$$.

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