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Focused Information Criterion

Select a candidate statistical model by the estimated risk of its estimator for a declared focus parameter, allowing the preferred model to change when the inferential target changes.

Version
v4 · 2026-09-19 · History
Domain-specific #
1861
Domain group
Formal Sciences
Origin domain
Experimental Design & Statistics
Subdomain
Statistical Model Selection → Experimental Design & Statistics
Aliases
Focussed Information Criterion, FIC

Core Idea

The Focused Information Criterion (FIC) is a statistical model-selection method that asks which candidate model estimates a declared focus parameter most accurately. The focus, written mu, is an estimand meaningful under every candidate model: a regression mean at a specified covariate value, a quantile, a treatment contrast, a survival probability at a chosen time and patient profile, or another scientifically chosen functional. Candidate model S supplies an estimator mu_hat_S. FIC estimates the risk of that estimator for this particular focus and selects a candidate with minimum estimated focus risk. Change the focus and the selected model may legitimately change.

How would you explain it like I'm…

Best Guesser for One Thing

Imagine you have several maps, and you want to pick one. The best map for finding the ice cream shop might not be the best map for finding the park. The Focused Information Criterion is a way grown-ups pick a math guess-tool by first saying exactly what question they want answered, and then choosing the tool that answers that one question best.

Pick the Model for Your Question

Scientists often have several different math models that could explain their data, and they have to pick one. The Focused Information Criterion says: first decide exactly which number you care about, like how likely one kind of patient is to recover. Then, for each model, estimate how far off its guess for that one number is likely to be, and pick the model with the smallest expected error. A simple model might be steadier but a bit off; a complicated one might be closer on average but wobblier. If you change which number you care about, a different model might win, and that's okay.

Target-Specific Model Selection

The Focused Information Criterion (FIC) is a way to choose among statistical models based on how well each one estimates a specific quantity you declare in advance, called the focus parameter. The focus could be an average outcome for a particular kind of person, a survival probability at a certain time, or the effect of a treatment. For each candidate model, FIC estimates the error (risk) of that model's estimate of the focus and picks the one with the lowest estimated risk. This differs from criteria like AIC or BIC, which give each model one overall score without knowing what you'll use it for. Smaller models usually have less variance but more bias; bigger models reduce bias but add variance. Because FIC balances these for your chosen target, changing the focus can legitimately change which model is best.

 

The Focused Information Criterion (FIC) selects among candidate statistical models according to how accurately each estimates a declared focus parameter mu, an estimand defined under every candidate, such as a regression mean at given covariates, a quantile, a treatment contrast or a survival probability. Each candidate S yields an estimator mu_hat_S, and FIC estimates that estimator's risk, typically mean squared error, choosing the candidate with minimum estimated focus risk. Unlike AIC or BIC, which assign one global score without reference to the downstream estimand, FIC makes the target an input, so the selected model can change with the focus. The theory uses local misspecification: a wide model has core parameters theta and optional parameters gamma, and the true distribution sits at gamma_0 + delta/sqrt(n) around the narrow model. Under this regime, sqrt(n)(mu_hat_S - mu_true) converges to a limit whose risk splits into squared bias plus variance; omitting parameters lowers variance but adds focus-relevant bias. Since delta cannot be estimated consistently, FIC uses a bias-corrected estimate of this limiting risk. The local setup is an asymptotic device keeping bias and variance comparable, not a claim that nature changes with n. Merely paying extra attention to one coefficient, or picking by global fit and checking the focus afterward, is not FIC.

Scope of Application

FIC applies where several plausible statistical models estimate the same scientifically meaningful quantity and the analyst can derive or approximate each estimator's risk under a defensible comparison regime. Its original scope is large-sample likelihood inference over nested submodels. Subsequent formulations cover regression, generalized linear models, survival analysis, semiparametric and nonparametric settings, quantile targets, time series, high-dimensional procedures, and focused model averaging. The domain can change; the common-focus and risk-estimation obligations do not.

Clarity

FIC forces the phrase “best model” to be completed. The analyst must name the target, the loss, the candidate class, and the regime under which the score estimates risk. This turns an ambiguous global superlative into an auditable conditional claim: “among these candidates, under this local-misspecification approximation and squared-error loss, model S has the smallest estimated risk for mu.”

Manages Complexity

Model-selection complexity grows combinatorially when q optional parameters produce up to 2^q submodels. FIC gives these candidates a common focus-specific currency. Once the wide-model information, focus derivatives, and subset projections are computed, many candidate risks can be compared without separately inventing a scientific objective for every model.

Abstract Reasoning

Focus invariance check. Verify that mu(theta,gamma) denotes one quantity under every candidate. Parameter names alone do not establish this; transformations, links, and conditional-versus-marginal interpretations can change the estimand.

Local experiment. In the original theory, place the true parameter at gamma_0 + delta/sqrt(n). For each subset S, derive the limiting distribution of sqrt(n)(mu_hat_S-mu_true). The focus gradient converts omitted parameter directions into bias; the information matrix converts estimated directions into variance.

Knowledge Transfer

Within statistics, FIC transfers by preserving roles rather than formulas. A generalized linear model changes the likelihood and focus derivative; a Cox model adds a baseline hazard and censoring structure; a semiparametric model changes nuisance estimation; a quantile target changes the estimator and risk. In every case, the analyst can ask: What is the common focus? What candidate estimators exist? What is their focus risk? Which terms estimate bias and variance? Which model minimizes the justified estimate?

Relationships to Other Abstractions

Local relationship map for Focused Information CriterionParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Focused InformationCriterionDOMAINPrime abstraction: Statistical Inference — presupposesStatisticalInferencePRIMEPrime abstraction: Selection — is a kind ofSelectionPRIME

Current abstraction Focused Information Criterion Domain-specific

Parents (2) — outgoing typed relations

  • Focused Information Criterion is a kind of Selection Prime

    prime:selection — proposed strict subsumption parent. FIC takes a candidate population and a focus-risk criterion, then gives one or more models greater retention.

  • Focused Information Criterion presupposes Statistical Inference Prime

    prime:statistical_inference — proposed strict presupposition parent. Focus estimators, limiting distributions, information matrices, and model-selection risk are meaningful only inside sample-to-process reasoning with quantified.

Hierarchy paths (4) — routes to 4 parentless roots

  • Focused Information Criterion → is a kind of Selection

Neighborhood in Abstraction Space

Focused Information Criterion sits in a sparse region of the domain-specific corpus (68th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Statistical Learning & Model Failure Modes (41 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25