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I have been reading about copula models. Essentially, copula models seem to be a creative method for creating a joint probability distribution from several variables, in which each individual variable can belong to a different probability distribution.

My Question: How are copula models actually used in the real world?

For example, suppose you have data corresponding to 5 variables $X_1, X_2, X_3, X_4$ and $X_5$. Let's say that you fit a copula model to these data.

Given values of $X_2, X_3, X_4$ and $X_5$, suppose you want to find the most probable value of $X_1$. At first glance, it seems like you could consider this as a conditional probability distribution $X_1 | X_2 = x_2, X_3 = x_3, X_4 = x_4, X_5 = x_5\dots$ therefore, you could use the Metropolis Hastings algorithm to draw many random samples from the above conditional probability distribution function, and then take the expected value of these samples as your answer?

Have I understood this correctly? Is this the main application of copula models?

Note: I have read about copula regression that relates more closely to GLM regression models – but it seems like the approach I outlined above might be closer to the intended use for copulas?

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    $\begingroup$ I think to call it the main application is likely too strong, but it is a possible application. $\endgroup$
    – Richard Hardy
    Commented Dec 18, 2021 at 7:56

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Generalized estimating equations might be considered an example of this idea. The joint likelihood of correlated data is specified through marginal distributions and a correlation structure. If you were to verify the operating characteristics of confidence intervals for such a model you could simulate correlated normal observations, apply the the probability integral transformation to turn these into uniform observations, and apply the probability integral transformation once again to turn these into another multivariate distribution of interest.

If you are interested in a feature of the distribution of $X_1$ (e.g., the mean, a percentile, etc.) given values of $X_2$ through $X_5$, these other variables can be included as covariates in a univariate model for $X_1$. When it comes to inference one can apply a link function to the estimator so that it is approximately normally distributed and use the delta method to estimate the standard error and invert a Wald test to produce a confidence interval.

A similar solution can be approximated through simulation, but it is computationally intensive.

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  • $\begingroup$ @ Geoffrey Johnson : Thank you so much for your answer! Suppose you already fit a multivariate copula to your data and want to find out the most probable value of "X1" given "X2 = x2...X5 = x5" . I thought the only way to do this is to treat the multivariate conditional distribution P(X1 | X2 = x2...X5 = x5) as a function and use MCMC to generate many random samples from P(X1 | X2 = x2...X5 = x5) . Then, you would take the average of the many random samples and consider this average as the most most probable value , i.e. E( P(X1 | X2 = x2...X5 = x5) ), where "E" is the expected value. Thanks! $\endgroup$
    – stats_noob
    Commented Dec 21, 2021 at 3:53
  • $\begingroup$ That is certainly a viable numerical approach for estimating the unknown fixed true population mean, $E[X_1|X_2=x_2,...,X_5=x_5]$, given the observed data. Alternatively, you can construct an estimator for $E[X_1|X_2=x_2,...,X_5=x_5]$ given the observed data, e.g. the maximum likelihood estimator. This value should be in close agreement with your numerical approach. $\endgroup$
    – Geoffrey Johnson
    Commented Dec 21, 2021 at 13:17
  • $\begingroup$ To account for the uncertainty around having estimated this population quantity, a Wald, score, or likelihood ratio test can be inverted to form a confidence interval (a set of plausible hypotheses in the parameter space determined by the observed data). You could rely on the normal or chi-square approximation to the sampling distribution of the test statistic, or you could utilize numerical methods for this as well. $\endgroup$
    – Geoffrey Johnson
    Commented Dec 21, 2021 at 13:18

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