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# Hedged Maximum Likelihood Estimation 

Robin Blume-Kohout Perimeter Institute for Theoretical Physics ∗ 

This paper proposes and analyzes a new method for quantum state estimation, called hedged maximum likelihood (HMLE). HMLE is a quantum version of Lidstone’s Law, also known as the “add β ” rule. A straightforward modification of maximum likelihood estimation (MLE), it can be used as a plugin replacement for MLE. The HMLE estimate is a strictly positive density matrix, slightly less likely than the ML estimate, but with much better behavior for predictive tasks. Single- qubit numerics indicate that HMLE beats MLE, according to several metrics, for nearly all “true” states. For nearly-pure states, MLE does slightly better, but neither method is optimal. 

Quantum state estimation is a basic task in quantum information science [1], simple to describe but hard to do right. The estimator gets N independently and identi- cally prepared (i.i.d.) quantum systems, performs mea- surements on them, analyzes the data, and reports a single-system density matrix ˆ ρ . The goal is to report the most “accurate” answer possible. There is room for sub- stantial debate over what this means, which the present paper will avoid by adopting three common assumptions: (i) we are concerned with N copies of an unknown “true” state ρ ; (ii) the goal is to get ˆ ρ as close as possible to ρ , according to some metric d ( ρ, ˆ ρ ); and (iii) measurement outcomes are intrinsically random, and we are concerned with average (over measurement outcomes, not over ρ ) performance. I will not consider how to choose a mea- surement, seeking instead a protocol that works well for all measurements. 

Maximum likelihood estimation (MLE) [2–5], the most common protocol, tends to report rank-deficient esti- mates with zero eigenvalues [6]. Those eigenvalues repre- sent probabilities. Assigning a zero probability indicates extraordinary confidence – confidence that the data do not support. For predictive purposes, this “zero eigen- value problem” can be be disastrous in practice. 

This paper suggests an alternative, hedged maximum likelihood . HMLE is a simple modification of MLE that can be used as a plug-in substitute for it. The mod- ification consists, in its entirety, of the following rule. Replace the standard likelihood function L ( ρ ) = Pr(observed data | ρ ) with the product of L ( ρ ) and a “hedging function” 

where det( · ) is the determinant, and β ≈ 1 2 is a positive constant chosen at the estimator’s dis- cretion. The rest of this Letter is devoted to explaining, deriving, and analyzing this procedure. 

Background: HMLE is motivated by a rule for esti- mating classical probabilities called Lidstone’s Law [7, 8] – or, more colloquially, “add β ”. Suppose we have ob- served N samples from an unknown i.i.d. distribution p = { p$_{1}$ . . . p$_{K}$ } , and have seen n$_{k}$ “ k ”s. What prob- abilities ˆ p should we assign for the next sample? The 

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likelihood, L ( p ) = ∏ k p n$_{k}$ k , is maximized by the natural and obvious estimate 

This can be disastrous in practice. Suppose some letter k has not yet been observed, so n$_{k}$ = 0, and MLE assigns ˆ p$_{k}$ = 0. This is fine if p$_{k}$ really is zero, but it’s equally plausible that p$_{k}$ is positive but small. If it is, the conse- quences of this error depend on what the estimate is used for. They are catastrophic when the estimate is used for predictive tasks, such as data compression or gambling [9, 10]. 

Compression and gambling define operational interpre- tations of ˆ p , and identify relative entropy as a measure of error: 

A gambler maximizes his bankroll’s expected growth rate by gambling a fraction ˆ p$_{k}$ of it on outcome “ k ”, and a compressor gets optimal performance by replacing “ k ” with a codeword of length − log ˆ p$_{k}$ . If the true proba- bilities are p and the estimate is ˆ p , then the gambler’s wealth grows as $( n ) = $(0) e n (const − H ( p ) − D ( p || ˆ p $^{))}$, where H ( p ) ≡ − ∑ k p$_{k}$ log p$_{k}$ is the entropy of p . Similarly, the length of the compressor’s compressed string grows as L = n [ H ( p ) + D ( p || ˆ p )]. In both cases, H ( p ) is the un- avoidable cost of p ’s randomness, while D ( p || ˆ p ) is the ad- ditional cost of estimating it incorrectly. Setting ˆ p$_{k}$ = 0 thus implies extreme strategies for gambling (bet the en- tire bankroll against “ k ”) and data compression (map “ k ” to an infinitely long codeword). Either way, if the next letter is “ k ”, the consequences are disastrous. 

“Add β ” avoids these catastrophes by hedging against as-yet-unseen possibilities. It assigns probabilities 

The lowest probability that can be assigned is β N + Kβ ≈ β $_{N}$. Like Eq. 2, this rule has a statistical derivation. It is the Bayes estimator (i.e., it minimizes expected cost) for 

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