A question on X validated about EM steps for a truncated Normal mixture led me to ponder whether or not a more ambitious completion [more ambitious than the standard component allocation] was appropriate. Namely, if the mixture is truncated to the interval (a,b), with an observed sample x of size n, this sample could be augmented into an untrucated sample y by latent samples over the complement of (a,b), with random sizes corresponding to the probabilities of falling within (-∞,a), (a,b), and (b,∞). In other words, y is made of three parts, including x, with sizes N¹, n, N³, respectively, the vector (N¹, n, N³) being a trinomial M(N⁺,p) random variable and N⁺ an extra unknown in the model. Assuming a (pseudo-) conjugate prior, an approximate Gibbs sampler can be run (by ignoring the dependence of p on the mixture parameters!). I did not go as far as implementing the idea for the mixture, but had a quick try for a simple truncated Normal. And did not spot any explosive behaviour in N⁺, which is what I was worried about. Of course, this is mostly anecdotal since the completion does not bring a significant improvement in coding or convergence (the plots corresponds to 10⁴ simulations, for a sample of size n=400).