I already asked this question in the "Mathematics" stackexchange, but apparently did not find the right audience, so I am duplicating my question here, hoping someone might be of help.

My question is: what is the relationship between the Cramér-Rao Lower Bounds (CRLB) and the uncertainty propagation formula?

I would have imagined that for a function $f$ of normally distributed random variables $x_i\sim N(\mu_i,\sigma_i^2)$, those two results might be similar if not equal.

The uncertainty propagation gives:

$$\Delta f^2 = \sum_i \left\vert\frac{\partial f}{\partial x_i}\right\vert^2\sigma_i^2$$

The CRLB can be written as:

$$\text{CRLB} = \left[\sum_i \frac{1}{\sigma_i^2}\left(\frac{\partial f}{\partial x_i}\right)^2 \right]^{-1}$$

Am I missing an underlying hypothesis?



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