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).
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While answering a question related with the EM algorithm on X validated, I realised a global (or generic) feature of the (objective) E function, namely that![E(\theta'|\theta)=\mathbb E_{\theta}[\log\,f_{X,Z}(x^\text{obs},Z|\theta')|X=x^\text{obs}]](https://s0.wp.com/latex.php?latex=E%28%5Ctheta%27%7C%5Ctheta%29%3D%5Cmathbb+E_%7B%5Ctheta%7D%5B%5Clog%5C%2Cf_%7BX%2CZ%7D%28x%5E%5Ctext%7Bobs%7D%2CZ%7C%5Ctheta%27%29%7CX%3Dx%5E%5Ctext%7Bobs%7D%5D&bg=000000&fg=B0B0B0&s=0&c=20201002)
![\log\,f_X(x^\text{obs};\theta')+\mathbb E_{\theta}[\log\,f_{Z|X}(Z|x^\text{obs},\theta')|X=x^\text{obs}]](https://s0.wp.com/latex.php?latex=%5Clog%5C%2Cf_X%28x%5E%5Ctext%7Bobs%7D%3B%5Ctheta%27%29%2B%5Cmathbb+E_%7B%5Ctheta%7D%5B%5Clog%5C%2Cf_%7BZ%7CX%7D%28Z%7Cx%5E%5Ctext%7Bobs%7D%2C%5Ctheta%27%29%7CX%3Dx%5E%5Ctext%7Bobs%7D%5D&bg=000000&fg=B0B0B0&s=0&c=20201002)
![\log\,f_X(x^\text{obs};\theta')=\log\,\mathbb E_{\theta}\left[\frac{f_{X,Z}(x^\text{obs},Z;\theta')}{f_{Z|X}(Z|x^\text{obs},\theta)}\big|X=x^\text{obs}\right]](https://s0.wp.com/latex.php?latex=%5Clog%5C%2Cf_X%28x%5E%5Ctext%7Bobs%7D%3B%5Ctheta%27%29%3D%5Clog%5C%2C%5Cmathbb+E_%7B%5Ctheta%7D%5Cleft%5B%5Cfrac%7Bf_%7BX%2CZ%7D%28x%5E%5Ctext%7Bobs%7D%2CZ%3B%5Ctheta%27%29%7D%7Bf_%7BZ%7CX%7D%28Z%7Cx%5E%5Ctext%7Bobs%7D%2C%5Ctheta%29%7D%5Cbig%7CX%3Dx%5E%5Ctext%7Bobs%7D%5Cright%5D&bg=000000&fg=B0B0B0&s=0&c=20201002)
Xi'an's Og 