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3. In the Potts model, X1, X2, ..., Xn are binary random variables taking the values 0 and 1, with a probability depending on a parameter α. The dependence between the Xi's is defined as follows: P(X1 = 1 | α) = 1/2 and for i = 2, 3, ..., n, P(Xi = Xi-1 | X1 = x1, X2 = x2, ..., Xi-1 = xi-1, α) = P(Xi = Xi-1 | Xi-1 = xi-1, α) = exp(α) / (1 + exp(α)). Let x* = (x*1, x*2, ..., x*n) denote an observed outcome from the Potts model and let S* = ∑(i=2 to n) 1{x*i = x*i-1}.
(a) Write down P(X = x* | α) and show that S* is a sufficient statistic under the Potts model. [5 marks]
(b) Given that the prior of α is U(0, 6), describe an ABC (Approximate Bayesian Computation) algorithm for obtaining a sample from the posterior distribution π(α | x*) using S*.
(c) State why it is more efficient to use S* over x* for the acceptance criterion of a proposed value in the ABC algorithm and quantify the gain in efficiency for the Potts model.
(d) What are the disadvantages of using a summary statistic in the ABC algorithm?
(e) Describe how the ABC algorithm can be made more efficient (with an increased acceptance rate) at the cost of sampling from an approximate posterior distribution.