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Consider a multivariate random variable $x$ with density function $P_x(\theta)$ for a scalar parameter $\theta$. Assume the Fisher information $J_x(\theta)$ is known.

Now, for a transformation (deterministic mapping) $y=F(x)$ can you express $J_y(\theta)$ in terms of $J_x(\theta)$ and $F$?

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  • $\begingroup$ It is possible to prove that $I_X(\theta)-I_Y(\theta)$ is positive semidefinite in general. $\endgroup$
    – Zen
    Commented Nov 14, 2013 at 0:56
  • $\begingroup$ And, if $Y$ is sufficient, then $I_X(\theta)=I_Y(\theta)$. $\endgroup$
    – Zen
    Commented Nov 14, 2013 at 0:58
  • $\begingroup$ Regarding your original question, think about this: if I tell you the value of $\xi=\mathrm{E}[g(X)]$, can you really find, in general, an expression for $\mathrm{E}[g(h(X))]$ in terms of $\xi$ and $h$ only? $\endgroup$
    – Zen
    Commented Nov 14, 2013 at 1:04

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