A question about equation(1) to equation(2).

Question

A question about equation(1) to equation(2).

TheCuriousJoe opened this issue 2 years ago · comments

Hi HanSheng, very excellent work!
In equation (2), likelihood fuction p(X|y) is defined as $https://latex.codecogs.com/png.image?\huge \dpi{50}p(X|y)=e^{-\frac{1}{2}\sum_{i=1}^{N}||f(y_i)||^2}$ .
We can view p(X|y) as the joint probability of p(X1|y), ... , p(X2|y). So $https://latex.codecogs.com/png.image?\huge \dpi{50}p(X_i|y)=e^{-\frac{1}{2}{}||f(y_i)||^2}$ .
||fi(y)|| is reprojective error and its value from 0 to infinity. Let x be ||fi(y)||, $https://latex.codecogs.com/png.image?\huge \dpi{50}\int_{0}^{+\infty}e^{-\frac{1}{2}x^2}=\frac{\sqrt{2\pi}}{2}\neq 1$ . So p(Xi|y) is not a pdf.
Is my induction correct? If my induction is correct, statement p(X|y) is not proper here. And following equation can't use Bayes theorem to get p(y|X).
Perhaps i'm splitting hairs, but it really confused me.

Hansheng Chen · Answer 1 · Tue Sep 20 2022 18:35:57 GMT+0800 (China Standard Time)

Yes the likelihood is not a PDF. Likelihood has nothing to do with density. Only after applying the Bayesian rule can the posterior become a PDF.

wddr955 · Answer 2 · Tue Sep 20 2022 19:01:10 GMT+0800 (China Standard Time)

Yes the likelihood is not a PDF. Likelihood has nothing to do with density. Only after applying the Bayesian rule can the posterior become a PDF.

The likelihood function is not as normal as usual. Can u give me more detail about how to get equation(2) from equation(1)? Or there is not a strictly math induction?

Hansheng Chen · Answer 3 · Tue Sep 20 2022 19:14:27 GMT+0800 (China Standard Time)

If you strictly consider the likelihood to be the conditional probability density of X, then Eq. (2) is incomplete because a normalization factor is missing. But this is OK because it's being used as a likelihood and its absolute scale doesn't matter.

wddr955 · Answer 4 · Wed Sep 21 2022 16:06:41 GMT+0800 (China Standard Time)

Thanks a lot. Excellent work again and wish I could figure out all the equations and intuitions.