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I am looking for some examples that Cross Validated community encountered in their work when they are modeling conditional distributions.

So far all the data sets I work with I end up with unimodal conditional distributions.

Can you give some real life examples where you encountered data sets that led you to conditional distributions that are not unimodal?

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    $\begingroup$ Lawyers' salaries is always a fun example: nalp.org/class_of_2014_salary_curve $\endgroup$
    – Bill
    Nov 28, 2017 at 20:31
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    $\begingroup$ What do you mean here by "conditional distributions", @Cagdas? How is it different from asking for real life examples for distributions without using the word "conditional"? $\endgroup$
    – amoeba
    Nov 28, 2017 at 22:11
  • $\begingroup$ @amoeba Usually for predictive purposes my understanding is that conditional distributions are usually modeled to be unimodal. It gives a concentration around the point estimate. $\endgroup$
    – Cagdas Ozgenc
    Nov 29, 2017 at 7:43
  • $\begingroup$ @Cagdas If you have an issue directly relevant to the administration of the site, consider raising it on meta. It doesn't belong here. $\endgroup$
    – Glen_b
    Nov 30, 2017 at 6:59
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    $\begingroup$ This site works better when you try to learn from it rather than getting angry at it. The first reason for downvoting--"this question does not show any research effort"--looks applicable here, given there are millions of readily available datasets on the internet and many of them have the properties you seek. Indeed, in any multiple regression model with a categorical regressor, if you omit that regressor in the fit then you are creating a mixture response and should expect to see multimodality if the effect is sufficiently large. Surely you have already seen some datasets like that. $\endgroup$
    – whuber ♦
    Nov 30, 2017 at 16:05

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Saying that there is a conditional multimodal distribution is somewhat like trying to prove a negative. For example, a bimodal distribution is really strong evidence, in most cases, of hidden variable bias. The heights of people is a common bimodal distribution but the hidden variable there is gender.

In other words, for me to say that a distribution is conditionally multimodal would be to affirmatively state that there was not hidden variable that could be the underlying cause.

Do multimodal distributions occur naturally? Yes. Are they conditionally multimodal? Probably not.

For some phenomena to be naturally bimodal (to use the simplest multimodal distribution) would mean that there would have to exist some threshold x where conditional population y behaves differently upon surpassing that threshold.

Even in the above lawyer salary, the hidden variable bias is the firm that hires them. In essence, the conditional distribution is unimodal once firm type is controlled for.

Another example of a multimodal distribution that is only so due to being an unconditional distribution: school district total revenue. Condition on locale (rural, urban, suburban) and you find you have three unimodal distributions.

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  • $\begingroup$ Basically what you are saying is that the result of a sound modeling (by accounting for the possible factors) will not yield a multimodal conditional distribution. Is that right? Can you support this with some theoretical justification (other than common sense)? $\endgroup$
    – Cagdas Ozgenc
    Nov 29, 2017 at 7:47
  • $\begingroup$ Depends a lot on when the prediction is done. If you are thinking about a law career, you decide (to go to law school) before you know the stuff that determines which of those two modes you are going to be near---though, to be fair, going to a top 14 law school helps with getting into the higher one, so you know that. The law salary thing is kind of a big deal. In my experience, prospective lawyers tend to be risk averse folks, and the fact that "mean/median lawyer salary" hides this gigantic risk of which mode you are going to be in is important to them. $\endgroup$
    – Bill
    Nov 29, 2017 at 14:30
  • $\begingroup$ My fall back to not being able to prove a negative is usually cantor's diagonal proof. Its a nice way of showing that even if you have an infinite amount of information, you don't know everything. If you are asking for a formal proof that a given distribution with n peaks can be conditional, that I cannot help you with. I will leave that to the mathematical statisticians on this board. $\endgroup$
    – JWH2006
    Nov 29, 2017 at 18:32
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    $\begingroup$ @Bill, that's the point of hidden variable bias. The distribution was bimodal because there was an unknown variable influencing the distribution when the prediction was made. Even risk averseness is a hidden variable, that when controlled for, would push the distribution back to unimodal. $\endgroup$
    – JWH2006
    Nov 29, 2017 at 18:35
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    $\begingroup$ @JWH2006 The whole distribution, unimodal, bimodal, trimodel, or otherwise is hidden variables. There is nothing special about bimodality. $\endgroup$
    – Bill
    Nov 30, 2017 at 19:02
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There are many examples:

  1. Here is a post about the price of books for sale on Amazon.com.
  2. The number of cars that cross the George Washington bridge plotted by the time of day. There will be peaks around 8:00am and 6:00pm, rush hours, with fewer cars in the hours in between.
  3. Some types of cancer when plotted as a function of age.
  4. Also, similar to the traffic on the GW bridge example, on google maps if you look at a popular restaurant in Manhattan and look at the "how busy the restaurant is" plot which plots by time of day, you usually see 2 or 3 peaks. I've linked one spot that usually has peaks on Saturdays and other days of the week. Some slow days during the week only have 1 peak though.
  5. Lawyer's salaries are bimodal. My conjecture is a public sector vs private sector cause of the salary difference. If this is the case you probably see this in other professions too. (this one is taken from comment in OP)
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Any experiment that is a mixture of causes with similar effects can have a multimodal conditional distribution. As a simplified case of some of the projects I've worked on, let's say you observe an empty bottle on the shore of a river. You know from the distribution of river speeds (and associated durations) that either (a) the river flows fast for a short time (e.g., storm runoff events) or (b) rather slowly for a long time (dry weather flow), but not much in between. In that case, knowing the bottle was in the water for 24 hours (e.g., by noting the amount of leakage past the cork) will give a roughly bi-modal distribution of possible locations where it could have entered the river (the degree that it actually has two modes depends on the autocorrelation of the river flow regimes and the range of durations for each regime -- the more correlated and longer the duration, the stronger the bi-modal behavior).

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  • $\begingroup$ Have you modeled it as a mixture of parametric unimodal distributions or using non-parametric methods? $\endgroup$
    – Cagdas Ozgenc
    Nov 27, 2017 at 7:24
  • $\begingroup$ @CagdasOzgenc we used ECDFs of the two flow regimes plus a autoregressive time series of bernoulli random variables to model the transitions. $\endgroup$
    – user145807
    Nov 27, 2017 at 14:29

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