Haplotype assembly and viral quasispecies reconstruction are challenging tasks concerned with analysis of genomic mixtures using sequencing data.

In the standard MLE setting [see, e.g., Murphy, 2012, Ch. 9] we are interested in learning the These two definitions are, however, essentially equivalent. Eq. (15) is a smooth objective that can be optimized with a (stochastic) gradient descent procedure. This section contains the proofs of the results relative to the perturb and map section (Section 3.2) and The proposition now follows from arguments made in Papandreou and Y uille [2011] Its moment generating function has the form E[exp(tX)] = Γ(1 τt). As mentioned in Johnson and Balakrishnan [p. Parts of the proof are inspired by a post on stackexchange Xi'an [2016].Theorem 1.

"E-step" (Steps 1 and 2 in Algorithm 1) and a weighted maximum likelihood estimation (MLE) "M-step" (Step 3; see [ ( t 1) (t 1) One may use these in a number of different ways. The following observation is due to Chebyshev's inequality. One can use Proposition S2.1 to construct a confidence interval on, for example, the expected squared Note that 1) the bound in Proposition S2.1 is CbAS naturally controls the importance weight variance. Design procedures that leverage a trust region can naturally bound the variance of the importance weights. We used CbAS as follows.

In practice, however, recursive updating often leads to poor trade-off solutions across tasks because approximate inference is necessary for most models of interest.

It is often convenient to model high-dimensional datasets as probability distributions.

We will often use variational optimization. When the variational family is not made explicit it means that the optimization is over all functions with the correct domain and codomain, e.g.

Work done while at Harvard University.

Undirected graphical models are compact representations of joint probability distributions over random variables. To solve inference tasks of interest, graphical models of arbitrary topology can be trained using empirical risk minimization.