The following result is well-known:



> 
> Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have
>  $$H \left( (1 - \lambda)(x\_1,y\_1) + \lambda (x\_2,y\_2) \right) \geq H(x\_1,y\_1)^{1 - \lambda} H(x\_2,y\_2)^{\lambda},$$
>  and let $M(y)$ denote the marginal distribution obtained by integrating over $x$
> $$M(y) = \int\_{\mathbb{R}^m} H(x,y) \, dx.$$
>  Let $y\_1$ $y\_2 \in \mathbb R^n$ and $\lambda \in (0,1)$ be given. Then the Prékopa–Leindler inequality applies. It can be written in terms of $M$ as
>  $$M((1-\lambda) y\_1 + \lambda y\_2) \geq M(y\_1)^{1-\lambda} M(y\_2)^\lambda$$
>  which is the log-concavity for $M$.
> 
> 
> 


Now, I wanted to understand this for a very simple example where $f: \mathbb R^2 \rightarrow \mathbb R:$


$$e^{-g(y)} = \int\_{\mathbb R} e^{-f(y,z)} \ dz.$$


Then, I want to prove that $g''\ge 0$ if $f$ satisfies $D^2f > 0$ globally as a matrix. We assume for simplicity that $f$ is such that the above integral is well-defined.


It is easy to see that 


$$g''(y) = \langle D\_{yy}f \rangle\_z - \operatorname{ Var}\_z (D\_{y}f)$$


where $\langle \cdot \rangle\_z$ is the expected value $$ \langle F \rangle\_z(y) := \frac{\int\_{\mathbb R} F(y,z) e^{-f(y,z)} \ dz}{ \int\_{\mathbb R} e^{-f(y,z)} \ dz}
$$ and $\operatorname{ Var}\_z$ is the variance with respect to the probability measure with density $p(z) \propto e^{-f(y,z)}$. 


However, it is not at all clear to me from this representation why $g''\ge 0$ holds.


**Is there a pedestrian way to see this from the above expression for the second derivative?**


I am looking for a more "Calculus" based derivation (using the 2nd derivative) of this inequality than the usual convex-combinatorial arguments.



Simple proof of Prékopa's Theorem: log-concavity is preserved by marginalization
By the [Brascamp–Lieb concentration inequality](https://en.wikipedia.org/wiki/Brascamp%E2%80%93Lieb_inequality#The_concentration_inequality), we have $$
 \operatorname{ Var}\_z (D\_{y}f) \le \langle (D\_{zz}f)^{-1} (D\_{zy} f)^2 \rangle\_z \;,
$$ and hence, $$
g''(y) \ge \langle (D\_{zz}f^{-1}) (D\_{zz} f D\_{yy} f - (D\_{z y} f)^2 ) \rangle\_z = \langle (D\_{zz}f^{-1}) \det D^2 f \rangle\_z > 0 
$$ since $D^2 f$ is globally positive definite (by assumption), which implies that $\det D^2 f > 0$ and $D\_{zz} f > 0$. 


A much more general version of this result can be found in Theorem 4.2 of Brascamp and Lieb's original paper (cited below), where they regard this result as a sharpened version of [Prékopa's theorem](https://en.wikipedia.org/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality#Applications_in_probability_and_statistics). See Section 4 of:


*Brascamp, Herm Jan; Lieb, Elliott H.*, [**On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation**](http://dx.doi.org/10.1016/0022-1236(76)90004-5), J. Funct. Anal. 22, 366-389 (1976). [ZBL0334.26009](https://zbmath.org/?q=an:0334.26009).


