(which is a subset of the nodes of the graphs such that every node is connected by an edge to every other node in the subset).See Cressie (1993)or Lauritzen (1996)for detailed treatments. From an historical point of view,Hammersley (1974)explains why the Hammersley- Clifford theorem was never published as such,but only through Besag (1974).The reason is that Clifford and Hammersley were dissatisfied with the positivity constraint:The joint density could be recovered from the full conditionals only when the support of the joint was made of the product of the supports of the full conditionals(with obvious counter-examples, as in Robert and Casella 2004).While they strived to make the theorem independent of any positivity condition,their graduate student published Moussouris(1974),a counter-example that put a full stop to their endeavors. While Julian Besag can certainly be credited to some extent of the(re-)discovery of the Gibbs sampler (as in Besag 1974),Besag (1975)has the curious and anticlimactic following comment: The simulation procedure is to consider the sites cyclically and,at each stage, to amend or leave unaltered the particular site value in question,according to a probability distribution whose elements depend upon the current value at neigh- boring sites (...)However,the technique is unlikely to be particularly helpful in many other than binary situations and the Markov chain itself has no practical interpretation. So,while stating the basic version of the Gibbs sampler on a graph with discrete variables, Besag dismisses it as unpractical. On the other hand,Hammersley,together with Handscomb,wrote a textbook on Monte Carlo methods,(the first?)(Hammersley and Handscomb 1964).There they cover such topics as "Crude Monte Carlo"(which is(3));importance sampling;control variates;and "Conditional Monte Carlo",which looks surprisingly like a missing-data completion ap- proach.Of course,they do not cover the Hammersley-Clifford theorem but,in contrast to Besag (1974),they state in the Preface We are convinced nevertheless that Monte Carlo methods will one day reach an impressive maturity. Well said! 7(which is a subset of the nodes of the graphs such that every node is connected by an edge to every other node in the subset). See Cressie (1993) or Lauritzen (1996) for detailed treatments. From an historical point of view, Hammersley (1974) explains why the Hammersley￾Clifford theorem was never published as such, but only through Besag (1974). The reason is that Clifford and Hammersley were dissatisfied with the positivity constraint: The joint density could be recovered from the full conditionals only when the support of the joint was made of the product of the supports of the full conditionals (with obvious counter-examples, as in Robert and Casella 2004). While they strived to make the theorem independent of any positivity condition, their graduate student published Moussouris (1974), a counter-example that put a full stop to their endeavors. While Julian Besag can certainly be credited to some extent of the (re-)discovery of the Gibbs sampler (as in Besag 1974), Besag (1975) has the curious and anticlimactic following comment: The simulation procedure is to consider the sites cyclically and, at each stage, to amend or leave unaltered the particular site value in question, according to a probability distribution whose elements depend upon the current value at neigh￾boring sites (...) However, the technique is unlikely to be particularly helpful in many other than binary situations and the Markov chain itself has no practical interpretation. So, while stating the basic version of the Gibbs sampler on a graph with discrete variables, Besag dismisses it as unpractical. On the other hand, Hammersley, together with Handscomb, wrote a textbook on Monte Carlo methods, (the first?) (Hammersley and Handscomb 1964). There they cover such topics as “Crude Monte Carlo“ (which is (3)); importance sampling; control variates; and “Conditional Monte Carlo”, which looks surprisingly like a missing-data completion ap￾proach. Of course, they do not cover the Hammersley-Clifford theorem but, in contrast to Besag (1974), they state in the Preface We are convinced nevertheless that Monte Carlo methods will one day reach an impressive maturity. Well said! 7