What would be an example of a really simple model with an intractable likelihood? Approximate Bayesian computation is a really cool technique for fitting basically any stochastic model, intended for models where the likelihood is intractable (say, you can sample from the model if you fix the parameters but you cannot numerically, algorithmically or analytically calculate the likelihood). When introducing approximate Bayesian computation (ABC) to an audience it is nice to use some example model that is really simple but still somewhat interesting and that has an intractable likelihood.
What would be a good example of a really simple model that still has an intractable likelihood?
 A: One example I came through a few weeks ago and quite like for its simplicity is the following one: given an original normal dataset
$$
x_1,\ldots,x_n\stackrel{\text{iid}}{\sim}\text{N}(\theta,\sigma^2)\,,
$$
the reported data is (alas!) made of the two-dimensional summary
$$
S(x_1,\ldots,x_n)=(\text{med}(x_1,\ldots,x_n),\text{mad}(x_1,\ldots,x_n))\,,
$$
which is not sufficient and which does not have a closed form joint density.
A: Two distributions that are used a lot in the literature are:


*

*The g-and-k distribution. This is defined by its quantile function (inverse cdf) but has an intractable density. Rayner and MacGillivray (2002) is a good overview of these, and one of many ABC papers which use it as a toy example is Drovandi and Pettitt (2011).

*Alpha stable distributions. These are defined by their characteristic function but have an intractable density except for a couple of special cases. This has applications in finance and is often used in sequential ABC papers, for example Yildirim et al (2013).

