# For log-concave densities, are joint and marginal modes consistent?

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.

• 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.
– whuber
Commented Dec 5, 2022 at 19:30
• 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? Commented Dec 5, 2022 at 23:39
• 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.
– whuber
Commented Dec 6, 2022 at 15:39
• Not related to consistency, but I had a similar question regarding the "benefits" of (log-)concavity in Bayesian estimation. Commented Jul 7 at 16:59