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Suppose I have a probability density function $\pi(x_1, \ldots, x_n)$, which is the density of a vector-valued random variable $X$ in $\mathbb{R}^n$. Assume that $\pi$ is strongly log-concave, i.e., the function $E(x_1, \ldots, x_n) = - \log \pi(x_1, \ldots, x_n)$ is strongly convex. This implies that $\pi$ possesses a unique global maximum $$ (x_1^\star, \ldots, x_n^\star) = \operatorname{argmax}_{(x_1, \ldots, x_n)} \pi(x_1, \ldots, x_n). $$

Now, suppose that I look at the marginal density for any coordinate, e.g., $$ \pi(x_k) = \int_{\mathbb{R}^{n-1}} \pi(x_1, \ldots, x_n) dx_1 \cdots dx_{k-1} dx_{k+1} \cdots dx_n, $$ for any $1 \leq k \leq n$. We then know that the marginal density is also log-concave and possesses a unique maximizer (e.g., see here). Let the mode of this marginal be denoted by $$ x^\dagger_k = \operatorname{argmax}_{x_k} \pi(x_k). $$

Question: Under the assumption that the joint density $\pi(x_1, \ldots, x_n)$ is strongly log-concave, does the mode of the joint necessarily agree with the mode of the marginals? By "agree", I am meaning whether $$ x_k^\star = x_k^\dagger $$ for each $k$?

I will note that without the assumption of log-concavity, the answer to this question is no, as discussed here. I am asking whether assuming log-concavity makes this statement true.

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    $\begingroup$ I believe some of the examples at stats.stackexchange.com/a/91944/919 work: a mixture of bivariate Normals whose modes are sufficiently close remains log concave but the mode of that mixture does not necessarily agree with the modes of its marginals. $\endgroup$
    – whuber
    Dec 5, 2022 at 19:30
  • $\begingroup$ Assuming the two modes of the mixture are not identical, doesn't this imply the joint density is not strongly log-concave? Since you cannot have both multi-modality and strong log-concavity? $\endgroup$ Dec 5, 2022 at 23:39
  • $\begingroup$ I didn't say the joint distribution would be multimodal: the "sufficiently close" condition is there in order to make the joint distribution unimodal as well as strongly log concave. $\endgroup$
    – whuber
    Dec 6, 2022 at 15:39

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