Corection for UQLab Bayesian Inference Manuell: section on affine invariant ensemble algorithm


I would like to suggest corrections for the UQLab user maunal Bayesian Inference for model calibration and inverse problems, Section 1.3.4 Affine invariant ensemble algorithm.
In this section it is claimed that the Affine invariant ensemble algorithm" runs an ensemble of C Markov chains", but this is only correct without “Markow”. It seems that the formulation “ensemble Markov chain” in Goodman and Weare (2010) created some confusion. Recalling the content of page 67 in Goodman and Weare (2010) with the notions in the UQLab manual, e.g. using that the parameters in the UQLab manual belong to the set \mathcal{D}_{\mathbf{X}} being a subset \mathbb{R}^M, one gets:

  • It holds that the ensemble as in Goodman and Weare which I will denote as Goodman-Weare-ensamble transforms to \left( \mathcal{D}_{\mathbf{X}} \right)^C instead to \left(\mathbb{R}^M\right)^C
  • If one want to deal with a Markov chain, one has to consider \left( \left(\mathbf{x}_j^1\right)_{j=1}^C, \, \left(\mathbf{x}_j^2 \right)_{j=1}^C, \,\left(\mathbf{x}_j^3\right)_{j=1}^C, \cdots \right) being a Markow chain on the Goodman-Weare-ensamble \left(\mathcal{D}_{\mathbf{X}} \right)^C .
  • If one is considering for integer j between 1 and C then it holds that \left(\mathbf{x}_j^1, \mathbf{x}_j^2, \mathbf{x}_j^3, \cdots\right) is not a Markow chai, see also Goodman and Weare (2010), text below equation (5). Considering some values y_1,y_2 \in \mathcal{D}_{\mathbf{X}} we see that the probability we get for x_j^3 if we request that x_j^2 = y_2 depends on the probabilities we have for x_i^2 with i \neq j and i \in \{1,\dots,C\}, but the corresponding probabilities arising if we request that x_i^1 =y_1 will be different to those probabilities arsing if we do not fix the value of x_i^1.

Best regards