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Maximum likelihood estimation is a method of statistical estimation: within a specified statistical model, choose the statistical parameter values that maximize the likelihood function of the observed data. For a parameter set $\Theta$ and observations $x$, its exact requirement isThe equality requires an attained, finite maximum; a supremum approached only at an excluded boundary is not enough. The resulting rule is a maximum-likelihood estimator. The likelihood function uses a probability mass function for discrete data and a probability density function for continuous data. See MIT 18.05, Reading 10b, sections 3 and 3.2.

The direction of the comparison is the interesting part: hold the observations fixed and vary the statistical parameter. Ten independent Bernoulli trials with eight successes and two failures give the maximum-likelihood estimate $\hat{p}=0.8$, as derived in log-likelihood. This chooses the member of the statistical model that assigns the greatest mass to that outcome. It does not assert an $80\%$ posterior probability that this parameter value is true. Bayesian statistics combines the same likelihood function with a prior distribution to obtain a posterior distribution; maximum a posteriori estimation maximizes that posterior mass or density instead.

The method turns many fitting tasks into one maximization problem. In the normal distribution model with known variance, it recovers the sample mean and ordinary least squares, while binomial regression and Poisson regression use different sampling assumptions to build their own likelihood functions. The log-likelihood turns independent contributions into a sum, exposing score equations and observed information. A peak's location gives an estimate; its shape records how much the observed data distinguish nearby parameter values.

Existence, uniqueness, bias, and statistical consistency are separate questions. Normal mean and variance maximum-likelihood estimators include a variance estimate with bias, likelihood supremum at an excluded boundary gives nonexistence, and the Neyman-Scott incidental parameter problem gives inconsistency. The usual asymptotic normality of a maximum likelihood estimator requires assumptions: a consistent estimator at an interior true parameter in a fixed finite-dimensional model with identifiability, sufficient smoothness and control of log-likelihood derivatives, and finite nonsingular Fisher information. A common parameter-independent support of a probability distribution is part of a standard sufficient framework; see Rigollet, MIT 18.650, Maximum Likelihood Estimation, slides 22-23. This is a description of the regular setting, not a complete theorem's hypotheses. The exact endpoint limit for a shifted exponential distribution shows why changing support can produce a different rate and a nonnormal limit. These properties do not follow just from maximizing a function.

In his 1922 paper in the Royal Society's Philosophical Transactions A, R. A. Fisher explicitly distinguished likelihood from probability and described choosing the parameter values with greatest likelihood. Section 6 also discusses comparing likelihoods after giving the maximum value one, the convention used in the figures here. See Fisher, On the Mathematical Foundations of Theoretical Statistics, section 6, pp. 326-328, DOI 10.1098/rsta.1922.0009.

With access to the uncontrolled physical gate, use single-qubit process tomography. Prepare the positive eigenstate of each of $X,Y,Z$, apply $V$, and on independent copies measure each of $X,Y,Z$. Arbitrary single-qubit rotation gates let one prepare these inputs and rotate each measurement into the computational basis. The nine measured means areThey determine the real rotation matrix $R$ of the Bloch vector, and hence the action of $V$ on every qubit state. To see precisely what this identifies, suppose another unitary $U$ has the same action. Then $V^{†}U$ commutes with every Pauli matrix. Commuting with $Z$ makes it diagonal; commuting with $X$ makes its diagonal entries equal. Therefore $U=e^{i\gamma }V$.

Consequently tomography determines $V$, and its eigenvalues, up to a common phase. More explicitly, write a representative as $V=e^{i\gamma }(\cos aI-i\sin a\mathbf{n}\cdot \mathbf{\sigma })$. Its eigenvalues are $e^{i(\gamma -a)}$ and $e^{i(\gamma +a)}$, and its Bloch vector action is rotation through $2a$ about $\mathbf{n}$. The measured rotation identifies the unordered relative eigenphase information; for example $\mathrm{Tr}R=1+2\cos (2a)$. Once a representative with a specified overall phase is supplied, diagonalizing that representative gives the corresponding two eigenvalues.

The common phase itself cannot be recovered with the stated uncontrolled access. Indeed, $(e^{i\gamma }V)\rho (e^{i\gamma }V{)}^{†}=V\rho V^{†}$ for every input, including inputs entangled with ancillas. Replacing $V$ by $e^{i\gamma }V$ therefore leaves every such preparation-and-measurement experiment unchanged, whereas it multiplies both eigenvalues by $e^{i\gamma }$. For example, $I$ and $e^{i\gamma }I$ are indistinguishable physical gates under these operations. This proves that the request for absolute eigenvalue phases needs an additional phase reference or controlled-gate assumption: the global phase of an uncontrolled quantum gate is unobservable. The controlled powers in the earlier parts supply that extra resource, since $\mathrm{diag}(I,e^{i\gamma }V)$ changes a relative phase between the two control branches.

The practical advantages are that no unknown eigenstate preparation, controlled powers, entangling gates or inverse quantum Fourier transform are needed: a fixed set of single-qubit preparations and measurements suffices. The disadvantages are statistical estimation and the phase ambiguity just proved. Estimating the bounded Pauli measurement means to additive accuracy $ϵ$ at fixed confidence requires order ${ϵ}^{-2}$ independent repetitions, giving the usual sampling cost for this direct procedure. The earlier quantum phase estimation circuit uses $N$ phase qubits and high controlled powers to resolve scale $2^{-N}$ in one coherent experiment with a constant success bound; constructing those powers from controlled uses of $V$ costs $2^N-1$ gate calls, so its exponential-looking precision is not obtained with only $N$ uses of $V$. Repetitions can raise confidence. Finally, full process tomography grows exponentially with the number of system qubits, whereas the eigenstate-based phase procedure can estimate a selected eigenphase without reconstructing the entire operator.

Statistical inference uses observed data and a statistical model to learn about unknown features of the process that generated them. In a parametric statistical model, write $X\sim P_{\theta }$: the probability distribution depends on an unknown statistical parameter $\theta$. An estimand specifies the target, such as a mean or a difference of means. A statistic is a function of the observations; an estimator is a statistic used to estimate the target. Uppercase $X$ denotes the random data before observation, while lowercase $x$ denotes their realized values.

Three questions lead to different outputs. Statistical estimation asks which target value the observations suggest. A confidence interval asks for a range produced by a procedure with calibrated coverage probability. A hypothesis test asks whether the observations depart sufficiently from a specified null hypothesis to enter its rejection region. These outputs complement one another: an estimated effect size describes magnitude, while a standard error describes the sampling distribution of an estimator. A p-value describes a tail probability under the null hypothesis.

The appeal of statistical inference is that it turns variable observations into reproducible uncertainty statements. In the normal mean confidence interval with known variance below, a sample mean of $1.2$ has standard error $0.5$: both the estimated shift and its precision matter. The calibration depends on the statistical model, including independence and the specified probability distribution. Increasing sample size improves precision under those assumptions; it does not repair a systematically misleading sampling procedure. Bayesian inference provides another framework, using a prior distribution and a likelihood function to obtain a posterior distribution.