- #1

fab13

- 312

- 6

Here a figure to illustrate my issue :

On the top is represented the normalized Likelihood of w0 parameter.

In all contours (with all triplot representing other parameters) and in all tripltot of thesis documents I have seen, the projection from the edge of 1 sigma contours on the likelihood intersects the likelihood at a height relatively low (on my scheme, roughly at 25%-30%, at first sight, of the maximum height of the likelihood).

However, one tells me that Likelihood should be intersected by the 1 sigma edge of joint distribution at roughly 70% of the maximum height of Likelihood (green bar and text on my figure)

For this, he justifies like this :

Concerning $$\Delta \chi^{2}$$, distribution function is a `\chi^2` law with 2 freedom degrees ; pdf is written as :

$$ f(\Delta\chi^{2})=\dfrac{1}{2} e^{-\dfrac{\Delta\chi^{2}}{2}} $$

So for a fixed `confidence level C.L`, we have :

$$1-CL= \int_{\Delta\chi^{2}_{CL}}^{+\infty}\dfrac{1}{2}e^{-\dfrac{\Delta\chi^{2}}{2}}\text{d}\chi^{2}$$

`$$=e^{-\dfrac{\Delta\chi_{CL}^{2}}{2}}` $$

and taking `CL=0.68`, we get :

$$ \Delta \chi ^{2}_{CL}=-2\ln(1-CL) $$

$$ \Delta \chi^{2}_{CL}=2.28 $$

And Finally, he concludes by saying that Maximum of Likelihood shoud fall from about 30% , i.e :

$$ e^{-\dfrac{(2.3)^2}{2}} = 0.31 $$

So I don't know why I get a falling of about 70% ~ (1-0.31) and not only of 31% ~ 0.3 like one says on my figure (red line on my figure above).

ps1 : I have seen an ineresting remark on https://docs.scipy.org/doc//numpy-1.10.4/reference/generated/numpy.random.normal.html which suggests a maximum at 60.7% of the max, which is not really what I expect (~ 70%).

ps2 : I have also found another interesting page, maybe more important since it talks about multivariate distribution :

https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.multivariate_normal.html

and here too, a justification of my reasoning :

If someone could explain me the trick to get an intersection at 70% of the maximal height of normalized Likelihood ...

Any help is welcome. [1]: https://i.stack.imgur.com/SXB1K.png

[2]: https://i.stack.imgur.com/2O8xi.jpg