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Let's say a data matrix $\bf{X} \in \mathbb{R}^{N \times D}$ has $D$ random variables each with $N$ observations. So $j$th column of $\bf{X}$ is $N$ observations of $j$th random variable.

Suppose that plotting histograms for each columns results in unimodal distribution. Is it possible that the joint distribution of $D$ random variables to have multimodal distribution in $D$ dimensional space?

Is multimodality in at least one random variable a necessary condition for a joint multimodality?

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No, unimodality of the margins doesn't imply unimodality of the joint -- it's quite possible to be unimodal on the margins and multimodal on the joint.

Consider the following mixture of independent unit-variance normals:

0.25 probability of a component centered at (-2.5,-2.5), 0.50 centered at (0,0) and 0.25 centered at (2.5,2.5).

The margins are unimodal. The bivariate distribution is not:

Bivariate normal mixture witree modes

Here's the marginal density for both $X$ and $Y$:

Density for marginal distribution of X

... which is unimodal. Note that the bumps in the bivariate density are clearly separated along the diagonal but overlap enough in the two axis directions that the peaks "blend in" -- there are no dips (antimodes) in the marginal distributions. [This effect is more marked in higher dimensions, as we can keep the three bumps the same distance apart in relation to the margins, while the diagonal distance grows like $\sqrt{d}$.]

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  • $\begingroup$ Thank you for your answer. What I meant by unimodality is in each dimension. In your example, the mixture distribution is trimodal in x dimension, and in y dimension as well. I am interested in a distribution which is unimodal in each dimension separately (e.g., x, y and other dimensions) but multimodal in high dimension. $\endgroup$ – zcadqe Jul 28 '17 at 2:20
  • $\begingroup$ You're mistaken. This example is definitely unimodal in the x and y margins. (What did you do to check the marginal density before posting the above comment?). My example will scale up to any higher dimension in obvious ways (Edit: I have now included the marginal density in the post) $\endgroup$ – Glen_b Jul 28 '17 at 4:09

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