Bayesian Analysis of Exponential Random Graphs - Estimation of Parameters and Model Selection

Many probability models for graphs and directed graphs have beenproposed and the aim has usually been to reduce the probability of a graph tosome function that does not take the entire (graph-) structure into account, e.g.the number of edges (Bernoulli graph), dyadic properties in directed graphs(p1 Holland and Leinhardt, 1981), subgraph counts (Markov Graphs Frank andStrauss, 1986), etc. Many of these models give you analytically tractable formsfor inference about parameters while assuming dependencies that are not al-ways realistic in social science applications, whereas others make up for theirincreased realism with computational complexity. The Markov Graph of Frankand Strauss (1986), was later developed by Wasserman and Pattison (1996) intothe so called p* model, an exponential model for graphs that comprise arbitrarystatistics of graphs and attributes. In this paper we propose a procedure formaking Bayesian inference in the exponential graph framework. The aim is toobtain a joint posterior distribution of the parameters in the model, which cap-tures the uncertainty about our parameter values given the observed data. Asecond objective is to assess how much support different parameterizations ofthe model are given by data. Typically in Bayesian statistics, the expression forthe posterior distribution is not analytically tractable because of the normaliz-ing constant involving a complicated integral or sum. In the case of exponentialrandom graphs we have an additional difficulty, namely that for the p* modelthe likelihood is only known up to a constant of proportionality (with respect todata). When the likelihood is easily evaluated, the first problem is easily han-dled by means of Markov chain Monte Carlo (MCMC) methods. Here, usingthis fact, the posterior is obtained from a two-step algorithm, which samplesfrom both the sample space and the parameter space. For calculating the mar-ginal likelihood function needed for model comparison, we employ a methodsuggested by Chib and Jeliazkov (2001). This involves estimating the posteriordensity evaluated in a suitably chosen point, something which is accomplishedusing only the key components of the MCMC algorithm, taking averages overthe posterior distribution and candidate proposal distribution.