**F**or reasons too long to describe here, I recently came across a 2013 paper by Dutta and Bhattacharya (from ISI Kolkata) entitled MCMC based on deterministic transforms, which sounded a bit dubious until I realised the *deterministic* label apply to the choice of the transformation and not to the Metropolis-Hastings proposal… The core of the proposed method is to make a proposal that simultaneously considers a move and its inverse, namely from x to either x’=T(x,ε) or x”=T⁻¹(x,ε) , where ε is an independent random noise, possibly degenerated to a manifold of lesser dimension. Due to the symmetry the acceptance probability is then a ratio of the target, multiplied by the x-Jacobian of T (as in reversible jump). I tried the method on a mixture of Gamma distributions target (in red) with an Exponential scale change and the resulting sample indeed fitted said target.

The authors even make an argument in favour of a unidimensional noise, although this amounts to running an implicit Gibbs sampler. Argument based on a reduced simulation cost for ε, albeit the full dimensional transform x’=T(x,ε) still requires to be computed. And as noted in the paper this also requires checking for irreducibility. The claim for higher efficiency found therein is thus mostly unsubstantiated…

*“The detailed balance requirement also demands that, given x, the regions covered by the forward and the backward transformations are disjoint.”*

The above statement is also surprising in that the generic detailed balance condition does not impose such a restriction.

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