Direct comparison

AIC vs. BIC for Model Selection

Q: Can I compare AIC or BIC values across models fit to different datasets, or with different sample sizes?

No. Both criteria are only meaningful for models fit to the exact same data and the same response variable — including the same set of observations if any rows were dropped for missingness in one model but not another. An AIC or BIC value has no standalone interpretation; only the difference between models fit to identical data is informative.

Q: If AIC and BIC pick different models, which one is "right"?

Neither is simply more correct — they're answering different questions. AIC is estimating which model will predict new data best; BIC is estimating which model is most likely to be the true, parsimonious specification. When they disagree, it's usually because BIC's heavier penalty has dropped a parameter that improves prediction slightly without being essential to the underlying structure. Report both and state explicitly which objective (prediction vs. parsimony) the paper is prioritizing, rather than silently picking whichever result is more convenient.

Q: Do AIC and BIC replace the need to check model assumptions?

No. Both criteria compare relative fit among the models you specified; neither one tells you whether any of those models are well-specified in the first place (correct distributional family, no omitted confounders, correctly modeled dependence structure). A model with the lowest AIC or BIC among a badly misspecified candidate set is still a badly misspecified model.

AIC's complexity penalty is fixed; BIC's grows with sample size. Compare the formulas, what each optimizes for, and when to use which.

Written and maintained by CASRAI Editorial Board

Last updated 31 August 2026

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In summary

Both criteria start from the same maximized log-likelihood and subtract a complexity penalty, and both require the models being compared to share the same response variable and be fit to the same data. The difference is the penalty term: AIC charges a flat 2 per parameter regardless of sample size, while BIC charges k·ln(n) per parameter — a penalty that grows with n and exceeds AIC's once a dataset has more than about 7-8 observations (ln(n) > 2 once n > e² ≈ 7.39). That single difference means AIC is asymptotically efficient at minimizing out-of-sample prediction error but is not consistent (it does not reliably converge on the true model as sample size grows, and tends to favor slightly more complex models even with unlimited data), while BIC is consistent under standard regularity conditions (it does converge on the true model, if that true model is among the candidates, as n → ∞) but is not asymptotically efficient for prediction. Practically: reach for AIC when the goal is picking the model that will predict best, and for BIC when the goal is identifying the most defensible, parsimonious specification — they will often agree on a small candidate set, and disagree more often as either the number of candidate models or the sample size grows large.

How do AIC, BIC compare side by side?

The table below compares AIC, BIC across 10 procurement-relevant dimensions, from full name / origin through requires a real likelihood.

Side-by-side comparison

Dimension	AIC	BIC
Full name / origin	Akaike Information Criterion — Hirotugu Akaike, presented 1971, published 1974.	Bayesian Information Criterion (also Schwarz Information Criterion, SIC) — Gideon Schwarz, 1978.
Formula	AIC = 2k − 2ln(L̂)	BIC = k·ln(n) − 2ln(L̂)
Penalty per parameter	Fixed at 2, regardless of sample size.	ln(n), which grows with the sample size — stricter than AIC's penalty once n exceeds roughly 7-8.
What it approximates	An estimate of relative out-of-sample predictive accuracy (Kullback-Leibler divergence from the unknown true model).	A large-sample approximation to the Bayes factor — roughly, the posterior probability the model is the correct one, under specific prior assumptions.
Large-sample statistical property	Asymptotically efficient — minimizes mean prediction error as n grows, but is not guaranteed to select the true model even with infinite data.	Consistent under standard regularity conditions — selects the true model as n → ∞, if that model is among the candidates — but is not asymptotically efficient for prediction.
Behavior with a small candidate set vs. many candidates	Tends to keep more parameters as choices multiply; overfitting risk rises with the number of candidate models compared.	Its heavier, sample-size-scaled penalty resists overfitting more as candidate models multiply, at the cost of sometimes underfitting in smaller samples.
Small-sample correction	AICc = AIC + (2k² + 2k)/(n−k−1) is the standard correction; commonly recommended once the observations-to-parameters ratio is small (a widely used rule of thumb is n/k < 40). AICc converges to AIC as n → ∞.	No equivalently standardized small-sample correction is in common use — BIC's own ln(n) term already scales with sample size by construction.
Comparing more than two models	Rank by AIC and read the gap: relative likelihood of model i vs. the best model is exp((AIC_min − AIC_i)/2); models within about 2 of the minimum are considered to have substantial support.	Ranked the same way, but differences are more often read as rough evidence strength on a Bayes-factor-style scale (e.g. a difference under 2 is weak evidence, over 10 is decisive) rather than a relative-likelihood calculation.
Requires nested models?	No — any models fit to the same response and data can be compared, nested or not.	No, same as AIC — neither criterion requires nesting, unlike a likelihood ratio test.
Requires a real likelihood	Yes — not valid for quasi-likelihood methods (e.g. quasi-Poisson, GEE), which have no proper likelihood to subtract 2ln(L̂) from.	Yes, same requirement — for quasi-likelihood models, a QIC-style substitute is used instead of either AIC or BIC.

Common questions

Common questions about AIC vs BIC

Can I compare AIC or BIC values across models fit to different datasets, or with different sample sizes?

No. Both criteria are only meaningful for models fit to the exact same data and the same response variable — including the same set of observations if any rows were dropped for missingness in one model but not another. An AIC or BIC value has no standalone interpretation; only the difference between models fit to identical data is informative.

If AIC and BIC pick different models, which one is "right"?

Neither is simply more correct — they're answering different questions. AIC is estimating which model will predict new data best; BIC is estimating which model is most likely to be the true, parsimonious specification. When they disagree, it's usually because BIC's heavier penalty has dropped a parameter that improves prediction slightly without being essential to the underlying structure. Report both and state explicitly which objective (prediction vs. parsimony) the paper is prioritizing, rather than silently picking whichever result is more convenient.

Do AIC and BIC replace the need to check model assumptions?

No. Both criteria compare relative fit among the models you specified; neither one tells you whether any of those models are well-specified in the first place (correct distributional family, no omitted confounders, correctly modeled dependence structure). A model with the lowest AIC or BIC among a badly misspecified candidate set is still a badly misspecified model.

Going deeper