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Suppose $f$ is a probability density on $y \in \mathbb{R}$ and $\theta \in \mathbb{R}$ is some parameter.

Define $$S(\theta) := \log \big( \frac{\partial f(y; \theta)}{\partial \theta} \big)$$

Then I'm trying to show that $E[S(\theta)] = 0$. The derivation shown in my class is first noticing that, when $L(\theta; y) = f(y; \theta)$ and then seeing that:

\begin{align} 0 &= \frac{\partial 1}{\partial \theta}dy \\ &= \frac{\partial}{\partial \theta} \int f(y; \theta) dy \\ &= \int f(y; \theta) \frac{ \frac{ \partial L(\theta; y) }{\partial \theta} }{ L(\theta; y) } dy \\ &=\int f(y; \theta) \frac{ \partial \log(L(\theta; y)) }{ \partial \theta} dy\\ &= E[S(\theta)] \end{align}

I think that my lack of intuition for this derivation may come from a weak background in analysis, that is, I'm not so intuitively clear on what conditions are required for interchanging differentiation and integration and so perhaps without this intuition I don't see why this result is obvious.

So, I was trying to gain more intuition by proceeding more directly, starting with $E[S(\theta)]$ and then unpacking the result until I arrived at $$E[S(\theta)] = \int \frac{ \partial f(y; \theta) }{\partial \theta } dy$$

and from here, it is not obvious to me that I should be able to interchange integration and differentiation and so I don't know how to proceed directly.

Thanks in advance.

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  • $\begingroup$ The fact that the integral and derivative can be interchanged is part of the assumptions ("regularity conditions"). $\endgroup$ – StubbornAtom Jan 25 at 18:48
  • $\begingroup$ Sure, but why do we need this assumption? $\endgroup$ – libby Jan 25 at 20:05

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