# Distributions over continuous distributions [closed]

I know a few stochastic processes that are described as 'distributions over distributions'. For example, the Dirichlet process draws discrete distributions from a default model (see e.g., p3 of these notes) .

The discrete distributions could be made continuous by assuming some kinds of kernels. E.g., if you have a discrete distribution with a probability $p_i$ associated to a variable $x_i$, you could make a continuous distribution from Gaussian kernels, $$p(x) = \sum_i p_i \cdot g(x | \mu = x_i, \sigma^2 = 1)$$ where $g$ is a Gaussian.

Furthermore, although the discrete distributions from a Dirichlet process are a 'small' set of the continuous ones, it can be argued that they are a 'large' set in the weak or pointwise sense (see e.g., p4 of these notes). Thus, a Dirichlet process could be used to relax a parametric model in a non-parametric way.

However, are there any distributions over continuous distributions? That is, is there e.g. a stochastic process like the Dirichlet one that covers a set of continuous distributions in a strong rather than weak sense? Is there e.g. a meaningful continuum limit of the Dirichlet process that could be used for this purpose? Are there other possibilities?

## closed as unclear what you're asking by whuber♦Aug 9 '18 at 14:31

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• Every Bayesian model concerns a "distribution of distributions," whence any Bayesian model of a continuous distribution--such as the textbook Normal distribution--will serve to answer your question. – whuber Aug 7 '18 at 11:52
• Can you elaborate? I don’t see how a Normal distribution could be an answer to my question. – innisfree Aug 8 '18 at 1:31
• Let $X$ be a random variable with a Normal distribution and let $Y$ be a Normal distribution of unit variance and mean $X$: That's what might be understood by a "distribution of distributions." If that's not your understanding, then please edit your question to clarify what you mean. – whuber Aug 8 '18 at 21:43
• I see, but the example is obviously parametric and only covers the set of Gaussian distributions of unit variance. Let me clarify – innisfree Aug 8 '18 at 23:48
• Could you explain what you mean by this edit? Since you are referring to two levels of distribution--a prior distribution that governs the generation of posterior distributions--which of these is supposed to be "non-parametric"? I'm unsure even what you might mean by "non-parametrically." The problem is that the language of your post seems to be all over the place in referring to "distributions," "kernels," and a "stochastic process." It is very hard to determine what subject it's trying to focus on or what it's trying to ask. – whuber Aug 9 '18 at 14:31