Ok. I was wondering if this was a “build it and they will come” sort of situation. I guess you need a solution before problems arrive :p

]]>This is the spatial equivalent of a hidden Markov model, so I can think of potential applications, even though they are not developed in this paper…

]]>Why is this an important question in application?

Does the underlying model have a scientific meaning, or is it a nuisance parameter?

In the continuous case, the choice of smoothness (which is roughly equivalent to the choice of neighbourhood structure) has implications for the frequentest properties of the posterior (in particular, in a slightly un-realistic asymptotic regime, if you specify your model to be too smooth, you will have zero frequentist coverage). Is this similar here?

]]>Merci. Although it sounds too much of a “learner’s reply” to me (and not enough Bayesian to be satisfactory to me), I see your point.

]]>I would answer to the last question by saying that we do not want to estimate the distribution of the index model with that procedure, we just want to pick up the right model even if there is a bias on the ABC predictor. Indeed, we might artificially increase the posterior probability of the best index model when selecting the vector of summary statistics from the data; but no ABC procedure comparing data sets through non sufficient statistics can claim approximating the true posterior p(m|x), see Robert et al. (PNAS, 2011). The sole guarantee that remains is that ABC procedures will pick the correct model if we provide enough data (Marin et al., JRRS-B, 2014). Hence we believe that our predictor, which “uses the data twice”, do not drop any important, numerical results from the ABC machinery. And thus, we did not seek a way to compensate the bias introduced by the selection of statistics from the data.

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