Timeline for Convexity of the exponential of the negative Renyi entropy
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jan 24, 2022 at 21:02 | comment | added | Iosif Pinelis | @Hans : Yes, you were right. Sorry about that. This is now fixed. | |
| Jan 24, 2022 at 21:02 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
edited body
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| Jan 24, 2022 at 20:56 | comment | added | Iosif Pinelis | @Hans : Yes, I have the same 1st derivative. | |
| Jan 24, 2022 at 20:24 | comment | added | Hans | I just checked my calculation. It seems my statement is correct. As a reference, the first derivative from my calculation is $$\frac d{dt}M(p+th)=\Big(1+\frac1r\Big)\Big(\sum_i(p_i+th_i)^r\Big)^{\frac1r-1}\sum_i(p_i+th_i)^rh_i.$$ | |
| Jan 24, 2022 at 20:09 | comment | added | Iosif Pinelis | @Hans : I think there is no typo there. Can you re-check your calculation? | |
| Jan 24, 2022 at 19:24 | comment | added | Hans | You are again right, Iosif. There may be one typo in your last expression though. It seems the factor $1+r$ ought to be outside of the left-most large parenthesis. | |
| Jan 24, 2022 at 16:06 | comment | added | Iosif Pinelis | @Hans : Yes, the same method works for $r>1$ -- then we can also use the Cauchy--Schwarz inequality, with weights $p_i^{r-1}$. | |
| Jan 24, 2022 at 15:46 | vote | accept | Hans | ||
| Jan 24, 2022 at 15:46 | comment | added | Hans | Thank you, Iosif, as usual. I actually thought of this conventional direct method of proving the second derivative being positive late last night, but decided to write it up later this morning... I am glad though you came up with the same method. Sometimes we overlook the obvious... By the way, for $r>1$, we can prove it the way I have mentioned in my question, but can you prove it by the second derivative method as well? Now we have to compare the two terms of opposite signs. | |
| Jan 24, 2022 at 13:28 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 52 characters in body
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| Jan 24, 2022 at 13:22 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |