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I do not know if this question fits this stack, but I think it might suit here because it is something related to the radar sensing processing.

The question is about the difference between the cooperative/collaborative radar system when localization a target. For the papers and textbooks I studied, I find they all use the CRLB (Cramér–Rao lower bound, the inverse of Fisher information matrix) to describe the performance of the radar localization minimize mean-squared error, and for the calculation is like something that:

$$\frac{\sigma^2}{\sum\left(\frac{\partial \ln \textbf{p}(z|p)}{\partial p}\right)^2}.$$

It is the sum of the likelihood function from each radar received echo.

And if I calculate the likelihood for each radar $\textbf{p}(z_i|p)$ ($i$ is the index of radar), and then multiply them together to get the total likelihood $$\textbf{p}(z|p) = \prod \textbf{p}(z_i|p),$$ I can also get a value which can be used to show the performance of the radar localization.

I wonder why the papers and books prefer the previous one not the second one, and how to understand the relationship between CRLB and likelihood function.

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  • $\begingroup$ Could you elaborate how you get "a value which can be used to show the performance of the radar localization" from the likelihood? $\endgroup$
    – AlexTP
    Jan 6 at 14:38
  • $\begingroup$ @AlexTP The likelihood in radar localization, which is commonly named maximum likelihood(ML) estimator, is to find the θ^ which is the maximize the likelihood of the radar observed data set X. SInce the radar estimation has errors and noises, the ML estimators would say that the value θ^ is the result that "the most likely" result based what the radar observed, That's what my understanding of likelihood in radar. $\endgroup$ Jan 8 at 11:49
  • $\begingroup$ Ok, check my answer. $\endgroup$
    – AlexTP
    Jan 8 at 16:43

2 Answers 2

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They are not the same thing.

This is a longer version of PeterK's answer.

Let's start with the ML estimator, using your notation.

For a fixed target $p$ and assuming radar model that is characterized by probability density function (pdf) $$f_Z(z|p) = \textbf{p}(z|p),$$ the pdf of all radar $\mathbf{z}=\{z_i\}$ is $$f_\mathbf{Z}(\mathbf{z}|p)= \prod_i \textbf{p}(z_i|p)$$

The ML estimate is $$\hat{p}=\textrm{argmax}_{p \in \mathcal{P}}f_\mathbf{Z}(\mathbf{z}|p)$$

You see, the estimate depends on the measure $\mathbf{z}$ (and the embedded model $f_Z(z|p)$); let denote $\hat{p}(\mathbf{z})$. You know the error $$e(\mathbf{z})=\hat{p}(\mathbf{z})-p$$

Repeat the process, you get another set of radar measure $\mathbf{Z}=\mathbf{t}=\{t_i\}$, the same ML estimator returns $$\hat{p}(\mathbf{t})=\textrm{argmax}_{p \in \mathcal{P}}f_\mathbf{Z}(\mathbf{t}|p)$$ with error $$e(\mathbf{t})=\hat{p}(\mathbf{t})-p$$

The errors tell you good or bad the estimates (you need define "good/bad" though) but nothing about the performance of the estimator.

To assess the performance of the estimator, you repeat the process many many times to get $e(\mathbf{z}), e(\mathbf{t}), e(\mathbf{u}), e(\mathbf{w}), ...$ and, by assuming ergodicity, calculate the mean $\mu$ and the variance $\nu$ of the ML estimator's error (alternatively, you can derive $\mu$ and $\nu$ from the model $f_Z(z|p)$): they do not depend on the measure $\mathbf{z}$ or $\mathbf{t}$ and tell you the performance of the ML estimator.

Some day, you decide to try another estimator, e.g. moment estimator, and get another $\mu$ and $\nu$. Logically, you would prefer the estimator that gives you $\mu$ and $\nu$ as small as possible. Now while you can decide which estimator is better, you cannot tell whether it is the best. The CRLB chimes in here by telling you the lower bound of $\nu$ for a given model $f_Z(z|p)$.

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    $\begingroup$ Nice! Thank-you! $\endgroup$
    – Peter K.
    Jan 8 at 18:54
  • $\begingroup$ Thank you Alex! I see, if I use only the ML to describe the radar, I still need to give a definition of "good and bad". Within CRLB, the only thing I know from it is the lowest "error" of this radar system/estimator. I cannot tell if these radars could completely achieve this, but I can say that for one of the radars, its error might be the smallest if we use them to detect for infinite times. $\endgroup$ Jan 9 at 11:35
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One issue is that they are very different things.

The Cramér–Rao lower bound is:

the reciprocal of the Fisher information is a lower bound on its variance.

whereas $\textbf{p}(z|p)$ is a conditional probability function not a variance or a bound on a variance.

They are two very different things: one captures the variation of the estimate in a single number (usually a variance) and the other captures the variation as a function.


Looking at this answer on the CV.SE site and using their notation: think of $\textbf{p}(z|p)$ as $\mathscr{N}(0, \Sigma)$ and the CRLB as $\Sigma$ (though I'm conflating the conditional probability with a distribution).

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  • $\begingroup$ Hmm, the likelihood $p(z|p)$ is not a probability density function. Also, I think the last sentence easily makes readers confused as the variables whose variations are characterized by the likelihood and the CRLB are not the same thing. $\endgroup$
    – AlexTP
    Jan 6 at 17:47
  • $\begingroup$ @AlexTP Try that. Feel free to edit or make your own answer. :-) $\endgroup$
    – Peter K.
    Jan 6 at 17:51
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    $\begingroup$ I don't think I can do better, lol, without the OP clarification. I will wait for their answer to my comment. $\endgroup$
    – AlexTP
    Jan 6 at 18:31
  • $\begingroup$ @PeterK. Within the definition of these two, and take radar as an example, the radar observe a data set, then for Cramer-Rao lower bound, it shows the minimum variance of the data set within a given estimated value θ^, while for likelihood it is the one which gets the estimated value θ^ ? $\endgroup$ Jan 8 at 12:00
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    $\begingroup$ @PeterK. The CRLB also exists for biased estimators. The derivation is slightly more difficult, so it isn't just limited to unbiased estimators. $\endgroup$
    – David
    Jan 8 at 14:33

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