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Very recently, Transformation based Markov Chain Monte Carlo (TMCMC) wasproposed by Dutta and Bhattcharya (2013) as a much efficient alternative to theMetropolis-Hastings algorithm, Random Walk Metropolis (RWM) algorithm,especially in high dimensions. The main advantage of this algorithm is that itsimultaneously updates all components of a high dimensional parameter by someappropriate deterministic transformation of a single random variable, therebyreducing time complexity and enhancing the acceptance rate. The optimal scalingof the additive TMCMC approach has already been studied for the Gaussianproposal density by Dey and Bhattacharya(2013). In this paper, we discussdiffusion-based optimal scaling behavior for non-Gaussian proposal densities -in particular, uniform, Student's t and Cauchy proposals. We also considerdiffusion based optimal scaling for non-Gaussian proposals when the targetdensity is discontinuous. In the case of the Random Walk metropolis (RWM)algorithm these non-regular situations have been studied by Neal and Roberts(2011) in terms of expected squared jumping distance (ESJD), but the diffusionbased approach has not been considered. Although we could not formally proveour diffusion result for the Cauchy proposal, simulation based results led usto a conjecture that the diffusion result still holds for the Cauchy case. Wecompare our diffusion based TMCMC approach with that of ESJD based RWM approachfor the very challenging Cauchy proposal case, showing that our former approachclearly outperforms the latter.

A Brief Review of Optimal Scaling of the Main MCMC Approaches and Optimal Scaling of Additive TMCMC Under Non-Regular Cases