# Problem 84: Efficient Profile Maximum Likelihood Computation

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Given a sequence of samples $\mathbf{s}=(s_1,\dots,s_n)\in\mathbb{N}^n$, the sequence maximum likelihood (SML) estimator is the probability distribution over $\mathbb{N}$ that maximizes the probability of this sequence, i.e., $$p^{\rm SML} \stackrel{\rm def}{=} \arg\!\!\!\max_{p\in\Delta(\mathbb{N})} p^{\otimes n}(\mathbf{s}) =\arg\!\!\!\max_{p\in\Delta(\mathbb{N})} \prod_{i=1}^n p(s_i)\,.$$ It is not hard to show that the SML corresponds to the empirical distribution obtained from $(s_1,\dots,s_n)$, which can be computed in linear time.
In contrast, the profile maximum likelihood (PML) estimator is the estimator which, given $\mathbf{s}$, only considers the profile $\Phi(\mathbf{s})$ defined as the multi-set of counts: e.g., $$\Phi((c,a,b,b,c,d)) = \Phi((b,a, d,c,b,c)) = \{1,2,2,1\}$$ and maximizes the likelihood of getting the profile $\Phi(\mathbf{s})$: $$p^{\rm PML} \stackrel{\rm def}{=} \arg\!\!\!\max_{p\in\Delta(\mathbb{N})} \sum_{\mathbf{s'}: \Phi(\mathbf{s'})=\Phi(\mathbf{s})} p^{\otimes n}(\mathbf{s'})\,.$$ The PML is particularly well-suited to dealing with symmetric properties and functionals of distributions (i.e., those invariant to relabeling of the domain), as shown in [AcharyaDOS-17]. In particular, in the sublinear sample regime, it provably outperforms the SML. However, from a computational point of view, it is unclear whether one can compute it efficiently.