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Let the random variable $Y = X_1+X_2$, where $X_1$ follows an unknown distribution and $Y$ has finite variance.

Assuming as measurement of normality the entropy, is it correct to support that the entropy of $Y$ is maximized if and only if $X_2$ follows the normal distribution?

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  • $\begingroup$ For a very short counterexample, suppose $Y$ has finite variance=1 and $X_1\sim |\mathcal{N}(0,1)|$, i.e., the absolute value of a normally distributed r.v. Then the entropy would be maximized if $X_2\sim -|\mathcal{N}(0,1)|$, i.e., the negative of the absolute value of a normally distributed r.v. $\endgroup$ – Bill Bradley Nov 18 at 20:18
  • $\begingroup$ @BillBradley : How do you prove this? Also, if I understood your example correctly, then the variance of $Y$ will be $2-4/\pi\ne1$ (if that matters). $\endgroup$ – Iosif Pinelis Nov 18 at 21:16
  • $\begingroup$ @IosifPinelis Ah, you're right! I had my head screwed on sideways; I was thinking of adding the distributions, not the r.v.s themselves. $\endgroup$ – Bill Bradley Nov 19 at 2:11
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I interpret the problem as follows:

Let $Y=X_1+X_2$, where $X_1$ and $X_2$ are independent random variables (r.v.'s). We want to maximize the entropy of $Y$ with respect $X_2$ given a finite variance of $Y$, with the r.v. $X_1$ fixed. Is it true that the entropy of $Y$ is maximized if and only if $X_2$ has a normal distribution?

The answer to this question is no. Indeed, suppose that $P(X_1=1)=P(X_1=-1)=1/2$ and $Var\,Y=1+s^2$ for some real $s>0$, so that $Var X_2=s^2$. Suppose that $X_2$ is normally distributed, with a pdf $g$. Since the entropy of a pdf is invariant with respect to shifts, without loss of generality $X_2\sim N(0,s^2)$.

Let now $h$ be another pdf with mean $0$ and variance $s^2$. For $t\in[0,1]$, let $$f_t:=(1-t)g+th,$$ which is the pdf of a r.v. $X_{2,t}$ with variance $s^2$ such that $X_{2,t}$ is independent of $X_1$. So, the pdf $p_t$ of $Y_t:=X_1+X_{2,t}$ (with the same variance, $1+s^2$, as $Y$) is given by the formula $$p_t(y)=\tfrac12\,f_t(y+1)+\tfrac12\,f_t(y-1) $$ for real $y$. The entropy of $Y_t$ is $$H(t):=-\int_{\mathbb R}p_t(y)\ln p_t(y)\, dy, $$ and $H(0)$ is the entropy of $Y$. Next, $$H'(0)=-\int_{\mathbb R}[1+\ln p_0(y)]\,\frac\partial{\partial t}\,p_t(y)\Big|_{t=0}\, dy \\ =\int_{\mathbb R}\left[\frac{g(y+1)+g(y-1)}2-\frac{h(y+1)+h(y-1)}2\right]\, \ln\frac{g(y+1)+g(y-1)}2\, dy. $$ To disprove the conjecture, it is enough to find a suitable pdf $h$ such that $H'(0)>0$. In view of Chebyshev's integral inequality, it is then natural to choose $h$ which is almost mutually singular with $g$, so that $h$ is small where $g$ is large, and vice versa. With this in mind, let $h$ be the half-and-half mixture of the pdf's of $N(s\sqrt{1-u^2},u^2s^2)$ and $N(-s\sqrt{1-u^2},u^2s^2)$ with $u\in(0,1)$, so that the mean and variance of $h$ are indeed $0$ and $s^2$, respectively. Then $H'(0)=0.05121\ldots>0$ if $s=1/2$ and $u=1/20$, and we are done.

Here are graphs of the almost mutually singular $g$ (blue) and $h$ (yellow):

enter image description here

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  • $\begingroup$ Thank you for your rigorus proof. I evaluated numerically your example and indeed for $t=0.0028$ the entropy is maximized. However, the half-and-half mixture has variance $0.2381 \neq s^2$. Maybe a correction is needed. I am wondering if there is a explicit method to approximate the optimal $X_2$ given $X_1$. Also if there is a bound for the optimality of the gaussian rv since the maximum is extremely close to $H(0)$ in this example. $\endgroup$ – Ioannis Papoutsidakis Nov 19 at 14:02
  • $\begingroup$ @IoannisPapoutsidakis : The second moment of each of the two distributions $N(\pm s\sqrt{1-u},u^2s^2)$ is $(\pm s\sqrt{1-u})^2+(us)^2=s^2$. So, the second moment of any mixture of these two distributions is also $s^2$. Since the mean of the half-and-half mixture is $0$, its variance is indeed $s^2$. As for the explicit optimal solution or the bound for the optimality, I don't know; you may want to ask those questions in (a) separate post(s). $\endgroup$ – Iosif Pinelis Nov 19 at 15:07
  • $\begingroup$ I think there is a typo. Because $(\pm s \sqrt{1-u})^2+(us)^2 = s^2-us^2+u^2s^2$. Maybe the mean should be defined as $ \pm s \sqrt{1-u^2}$. $\endgroup$ – Ioannis Papoutsidakis Nov 19 at 16:16
  • $\begingroup$ @IoannisPapoutsidakis : Thank you for spotting the mistake. It is now corrected. The main conclusion is not affected. $\endgroup$ – Iosif Pinelis Nov 19 at 18:14

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