Estimation Theory

P1.statistics-advanced.02 · Audience: it-ml · Prerequisites: Advanced Probability

Interview-depth estimation for the data-scientist track: method of moments vs maximum likelihood, MLE's properties and its identity with cross-entropy minimisation, and the sufficiency / Fisher-information machinery that underlies estimator efficiency. Back to the foundations: N4 — Loss Functions and the Learning Objective, and Statistical Inference.

Step 1 / 4 — Method of Moments vs Maximum Likelihood (s45)

Both estimate parameters θ from data; they differ in what they match.

Method of moments (MoM). Equate the first k population moments (functions of θ) to the corresponding sample moments, then solve:

\mathbb{E}[X^r](\theta) = \frac{1}{n}\sum_{i=1}^n X_i^{\,r}, \qquad r=1,\dots,k

Simple, closed-form, and needs no full likelihood — a good initialiser. But it is generally less efficient, can produce estimates outside the parameter space (e.g. a negative variance), and ignores the distribution's shape beyond the matched moments.

Maximum likelihood (MLE). Choose θ that makes the observed data most probable:

\hat{\theta}_{\text{MLE}} = \arg\max_{\theta}\; \ell(\theta), \qquad \ell(\theta) = \sum_{i=1}^n \log p(x_i \mid \theta)

Uses the whole distributional model, so it is generally more efficient — at the cost of an optimisation and correct model specification.

Rule of thumb. MoM for a quick/robust starting point; MLE when you trust the model and want efficiency. They coincide for some models (e.g. the Gaussian mean).

📚 Go Further

Rigorous references for estimation theory.

Type	Resource
Book	Casella & Berger, Statistical Inference — sufficiency, MLE, Cramér–Rao
Book	Wasserman, All of Statistics — Fisher information, method of moments
In-app	N4 — Loss Functions, for cross-entropy as the training objective
In-app	Bayesian Inference, for MAP as regularised MLE

✅ Key takeaways

The method of moments matches sample moments to population moments — fast, closed-form, and a robust initialiser, but generally less efficient and possibly outside the parameter space.
Maximum likelihood chooses the θ that makes the data most probable; it is consistent, asymptotically normal, asymptotically efficient (attains Cramér–Rao), and invariant — though possibly biased in finite samples.
Maximising log-likelihood is identical to minimising cross-entropy / KL divergence to the empirical distribution: cross-entropy and squared-error losses are MLE under categorical and Gaussian models, and MAP is MLE plus a regulariser.
A sufficient statistic T(X) captures everything the sample says about θ (Fisher–Neyman factorisation); Rao–Blackwell says conditioning on it never hurts.
Fisher information is the expected curvature of the log-likelihood; via the Cramér–Rao bound Var(θ̂) ≥ 1/(n·I(θ)) it sets the best achievable precision for any unbiased estimator.

Next: Bayesian Inference: priors, posteriors, conjugacy, and credible intervals.

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