Is the Likelihood Function a Multivariate Gaussian Near a Minimum?

[kelly0303]

Hello! I am reading Data Reduction and Error Analysis by Bevington, 3rd Edition and in Chapter 8.1, Variation of $\chi^2$ Near a Minimum he states that for enough data the likelihood function becomes a Gaussian function of each parameter, with the mean being the value that minimizes the chi-square: $$P(a_j)=Ae^{-(a_j-a_j')^2/2\sigma_j^2}$$ where $A$ is a function of the other parameters, but not $a_j$. Is this the general formula or is it a simplification where the correlation between the parameters is zero? From some examples later I guess this is just a particular case and I assume the most general formula would be a multivariate gaussian, but he doesn't explicitly state this anywhere. Can someone tell me what's the actual formula? Also, can someone point me towards a proof of this? Thank you!

 

[mfb]

It is true if you only consider changes in a_j and fix all other parameters to their value at the minimum, but not otherwise.

 

[kelly0303]

It is true if you only consider changes in a_j and fix all other parameters to their value at the minimum, but not otherwise.

Thank you! Does it become a multivariate Gaussian, tho, away from the minimum (if there are correlations)? Or does this formula apply only at the minimum value?

 

[mfb]

A multivariate Gaussian can describe the function in a small region around the minimum.