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Consider two random variables $\theta$ and $x$ whose supports are $[0,1]$ with the joint distribution function $f(x,\theta).$ Consider a conditional expectation $E[\theta|x]$. Suppose I have some information about $\theta$ which is given as some measurable set $S$. I want to consider a condition that the rate of change of the conditional expectation with respect to $x$ is nonzero for almost every $x$ for each $S$. That is $ \frac{\partial}{\partial x}E[\theta|x, S] \ne 0 $ for every measurable set $S$. Does this condition makes sense? Can there be any sufficient condition in terms of $f(x, \theta)$?

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What is the utility of this condition? Also do you mean $E[x| \theta]$? – PEV Apr 3 '11 at 6:30
In the last line $f(x,\theta)$ should be $f(x|\theta)$. More importantly, it seems a condition such as the one you are after should involve the (absolute) distribution of $\theta$. But you only introduce the conditional distribution of $x$ conditionally on $\theta$. Any hypothesis on the distribution of $\theta$? – Did Apr 3 '11 at 7:52
Yes you can make assumption on the distribution of $\theta$. – Thales Apr 3 '11 at 11:09
It implies that, whatever information you obtain, when $x$ is different, its estimate of $\theta$ is different. – Thales Apr 3 '11 at 11:28
It means $E[\theta|x].$ – Thales Apr 3 '11 at 11:50
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