# Issue about the percentage of falling of height of Likelihood

• I
• fab13
In summary, the author argues that the projected 1 sigma edges of the contours in a 2D joint distribution should intersect the associated likelihood at a height of roughly 30% of the maximum height of the likelihood. However, this is not what is observed in most cases. The author provides a small calculus to show that the height of the likelihood at which this should occur is 2.28.
fab13
I have currently an issue about the height at which the projection of 1 sigma edges in 2D contour should intersect the associated Likelihood.

Here a figure to illustrate my issue :

At bottom left is represented the joint distribution (shaded blue = contours at 2 sigma (95% C.L) and classic blue = contours at 1 sigma (68% C.L) of the 2 parameters considered (w0 and wa).
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

Isn't this just the difference between a 1- and 2-tailed test?

The 2 degrees of freedom are the key point here. "1 sigma" covers a larger likelihood range. There are two distributions where you can (and expect to) "lose" likelihood relative to the peak. The outer edge might be less likely in one distribution but it's then the peak of the other distribution (conditional if they are correlated).

So, it is impossible to justify this "lose" lileklihood of ~ 30% with 1C.L of 2D contours ? if yes, what is expected as percentage of "losing" factor if I take the projection from the 1 C.L of 2D contours on 1D Likelihood ?

Thanks

I don't understand what you are asking, but check the chi2 distribution.

yes, I did above a small calculus which represents the ##\Delta\chi2## for 2 degrees of freedom when we are at 1 C.L. It gives a ##\Delta\chi2 = 2.28##.

Such way that I have a height of ##exp(-\Delta\chi2/2) = 0.3198##. It would correspond to a falling of ~ 0.32 from the maximal height, do you agree ?

Otherwise, which value (height) is expected for intersection when I project the 1 C.L 2D contours on the Likelihood (like for example on my figure at the beginning of the post) ?

Best regards

Finally, is thre anyone who tells me if, on my initial top figure of the post, the vertical bar represents what is expected, i.e only roughly 30% of the max height of Likelihood ?

## 1. What is the issue about the percentage of falling of height of Likelihood?

The issue about the percentage of falling of height of Likelihood refers to the likelihood of an object or person falling from a certain height and the potential consequences of this event. This is a concern in various fields such as engineering, construction, and safety regulations.

## 2. Why is the percentage of falling of height of Likelihood important to study?

Studying the percentage of falling of height of Likelihood is important because it helps us understand the potential risks and dangers associated with falling from a certain height. This information can be used to develop safety measures and guidelines to prevent accidents and injuries.

## 3. How is the percentage of falling of height of Likelihood calculated?

The percentage of falling of height of Likelihood is typically calculated by considering factors such as the height of the fall, the weight and size of the object or person falling, and the surface or ground on which the fall occurs. Mathematical equations and models are often used to determine the likelihood of a fall and the potential impact of the fall.

## 4. What are some factors that can affect the percentage of falling of height of Likelihood?

Some factors that can affect the percentage of falling of height of Likelihood include the height of the fall, the speed and trajectory of the fall, the weight and size of the object or person falling, and the surface or ground on which the fall occurs. Other factors such as weather conditions, surface friction, and human error can also play a role.

## 5. How can the percentage of falling of height of Likelihood be reduced?

The percentage of falling of height of Likelihood can be reduced by implementing safety measures such as guardrails, safety harnesses, and warning signs. Proper training and education on fall prevention and safety precautions can also help reduce the likelihood of falls. In addition, regular maintenance and inspections of structures and equipment can help identify and address potential hazards that could lead to falls.

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