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### Divide and Recombine for Large and Complex Data: Model Likelihood Functions using MCMC

In Divide & Recombine (D&R), big data are divided into subsets, each analytic method is applied to subsets, and the outputs are recombined. This enables deep analysis and practical computational performance. An innovate D\&R procedure is proposed to compute likelihood functions of data-model (DM) parameters for big data. The likelihood-model (LM) is a parametric probability density function of the DM parameters. The density parameters are estimated by fitting the density to MCMC draws from each subset DM likelihood function, and then the fitted densities are recombined. The procedure is illustrated using normal and skew-normal LMs for the logistic regression DM.

### r/MachineLearning - [D] How is the log marginal likelihood of generative models reported?

Many papers on generative models report the log-marginal likelihood in order to quantitatively compare different generative models. Since the log-marginal likelihood is intractable, the Importance Weighted Autoencoder (IWAE)'s bound is commonly reported instead. I don't understand how the bound is computed. I assume that the IWAE is first trained on the dataset and then some synthetic samples from the model in question are used to compute the marginal LL bound. However, I am not entirely sure about the procedure.

### Maximum Conditional Likelihood via Bound Maximization and the CEM Algorithm

Advantages in feature selection, robustness andlimited resource allocation have been studied. Ultimately, tasks such as regression and classification reduce to the evaluation of a conditional density. However, popularity of maximumjoint likelihood and EM techniques remains strong in part due to their elegance and convergence properties. Thus, many conditional problems are solved by first estimating joint models then conditioning them.

### A local approach to estimation in discrete loglinear models

We consider two connected aspects of maximum likelihood estimation of the parameter for high-dimensional discrete graphical models: the existence of the maximum likelihood estimate (mle) and its computation. When the data is sparse, there are many zeros in the contingency table and the maximum likelihood estimate of the parameter may not exist. Fienberg and Rinaldo (2012) have shown that the mle does not exists iff the data vector belongs to a face of the so-called marginal cone spanned by the rows of the design matrix of the model. Identifying these faces in high-dimension is challenging. In this paper, we take a local approach : we show that one such face, albeit possibly not the smallest one, can be identified by looking at a collection of marginal graphical models generated by induced subgraphs \$G_i,i=1,\ldots,k\$ of \$G\$. This is our first contribution. Our second contribution concerns the composite maximum likelihood estimate. When the dimension of the problem is large, estimating the parameters of a given graphical model through maximum likelihood is challenging, if not impossible. The traditional approach to this problem has been local with the use of composite likelihood based on local conditional likelihoods. A more recent development is to have the components of the composite likelihood be marginal likelihoods centred around each \$v\$. We first show that the estimates obtained by consensus through local conditional and marginal likelihoods are identical. We then study the asymptotic properties of the composite maximum likelihood estimate when both the dimension of the model and the sample size \$N\$ go to infinity.