The Role of Summary Statistics

Over here, it makes mention of "summary statistics".

For example, in the Bayesian Setting where we want to estimate some parameter "thetha": A randomly chosen value of the "thetha_i" is selected by randomly sampling the priors. This value of "thetha_i" is used to generate a series of realizations "y1, y2, y3...yn", collectively denoted as "y_i".

A "summary statistic" for each "Yi" generated through simulation is denoted as : S(Yi). A "distance" (e.g. euclidean distance) is evaluated between the "summary statistics" of the original data and the simulated data, e.g. D[ S(Yi), S(Y0) ] ... where Y0 is the original data. IF D[ S(Yi), S(Y0) ] > "some threshold" , THEN the choice of "thetha_i" is accepted.

We repeat this procedure many times, and keep track of all accepted values of "thetha_i". These (many) values of "thetha_i" can be used to make the posterior distribution of "thetha_i", and a final value of "thehta_i" can be selected.

Throughout this whole procedure, I am still confused about what exactly is a "summary statistic". Does anyone know what is the functional form of the "summary statistic"?

• A summary statistic is an oxymoron for statistic. It can be any transform of the sample. Sep 17 at 15:16
• @Xi'an That sounds basically right. I don't recall encountering a formal definition of "summary statistic." However, I have imagined it is a special kind of statistic with the property that it depends on all the data. That is, if there exist $i$ and $n$ for which $\tau(x_1,\ldots,x_i,\ldots,x_n)=\tau(x_1,\ldots,x_{i-1},y_i,x_{i+1},\ldots, x_n)$ for all $x_1,x_2,\ldots,x_n;\,y_i,$ then $\tau$ is not a summary statistic.
– whuber
Sep 20 at 16:41