Difference between revisions of "Open Problems:84"
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| − | Given a sequence of samples $\mathbf{s}=(s_1,\dots,s_n)\in\mathbb{N}^n$, the ''sequence maximum likelihood'' estimator is the probability distribution over $\mathbb{N}$ which maximizes the probability of this sequence, i.e. | + | 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 $\mathbb{N}$ which 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}) | 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)\,. | =\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 $(s_1,\dots,s_n)$. | + | 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'' estimator is the estimator which, given $\mathbf{s}$, only considers the ''profile'' $\Phi(\mathbf{s})$ defined as the multi-set of counts: e.g., | + | 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\} | \Phi((c,a,b,b,c,d)) = \Phi((b,a, d,c,b,c)) = \{1,2,2,1\} | ||
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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'})\,. | 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 by relabeling of the domain), as shown in {{cite|AcharyaDOS-17}}. However, from a computational point of view, it is unclear whether one can compute it efficiently. | + | The PML is particularly well-suited to dealing with symmetric properties and functionals of distributions (i.e., those invariant by relabeling of the domain), as shown in {{cite|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. |
Is there a polynomial- (or even strongly subexponential)-time algorithm to compute the PML? To (additively) approximate it? | Is there a polynomial- (or even strongly subexponential)-time algorithm to compute the PML? To (additively) approximate it? | ||
Revision as of 21:03, 25 October 2017
| Suggested by | Alon Orlitsky |
|---|---|
| Source | FOCS 2017 |
| Short link | https://sublinear.info/84 |
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 $\mathbb{N}$ which 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 by 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.
Is there a polynomial- (or even strongly subexponential)-time algorithm to compute the PML? To (additively) approximate it?
