Difference Between AIC and BIC (With Table)

The AIC and BIC are commonly used in model criteria. Model selection criteria are rules that are used to choose a statistical model from a list of candidate models based on observable data. Though both concepts refer to model selection, they are not similar. There are several differences between the two techniques of model selection.

AIC vs BIC

The main difference between AIC (Akaike’s Information Criteria) and BIC (Bayesian Information Criteria) is that they are both utilized for model selection. However, they are designed for certain applications and can provide distinct outcomes. AIC has unlimited and somewhat large dimensions when compared to BIC.

AIC or the Akaike Information Criteria is a scoring and model selection method. AIC was developed under the idea that your true model requires an unlimited number of parameters and aims to minimize information loss by approximating it with a given finite-dimensional model.

BIC or the Bayesian Information Criteria is an evaluation of the purpose of the possibility, based on whether or not the model is correct, using a specific Bayesian framework. BIC was developed as a large-sample approximation to Bayesian selection from a given set of finite-dimensional models.

Comparison Table Between AIC and BIC

Parameters Of ComparisonAICBIC
Definition AIC is an evaluation of a continuous and corresponding interval among the undefined, correct, and justified likelihood of the information.BIC is an appropriate evaluation of the purpose of the possibility following the model under a certain Bayesian structure.
Developed byHirotsugu AkaikeGideon E. Schwarz 
Published in19741978
FormulaAIC = 2k – 2ln(L^)BIC = k ln(n) – 2ln(L^)
Model-selection parameterAIC seeks to choose a model that minimizes the Kullback–Leibler divergence between the training and predicted values.BIC would like to select a model that maximizes the posterior probability of the data given the model.
Assumptions.  Most optimistic assumptionsLess optimal assumptions
 Penalty The penalty for AIC is 2kThe penalty for BIC is ln (n) k.

 What is AIC?

The Akaike information criteria, abbreviated AIC, was developed by Japanese statistician Hirotugu Akaike. It was initially referred to as “an information criteria.” Akaike presented it for the first time in English at a conference in 1971, and the proceedings of the conference were published in 1973. In 1974, the first official paper was published. Because the approach is based on the idea of entropy in information theory, Akaike called it the “entropy maximization principle.”

The goal of AIC is to find the model that best describes the variation in the dependent variable with the fewest number of independent factors. As a result, choosing a simpler model over a complicated one is advantageous.

According to AIC, the best-fit model is the one that explains the most variance with the fewest number of independent variables.AIC is currently the base of a paradigm for statistical foundations and is commonly utilized for statistical inference.

When assessing the amount of information lost by a model, AIC considers the trade-off between the model’s quality of fit and its simplicity. To put it another way, AIC addresses both the danger of overfitting and the risk of underfitting.

What is BIC?

The Bayesian information criteria, abbreviated BIC, is also known as the Schwarz information criterion. Gideon E. Schwarz developed the criteria as an asymptotic approximation to a transformation of a candidate model’s Bayesian posteriorprobability.

In 1978, Gideon E. Schwarz created and published the theory. Its popularity arises from its computational simplicity and efficient performance in a wide range of modeling frameworks, including Bayesian applications where prior distributions can be difficult to obtain.

In Bayesian statistics, the Bayesian Information Criteria is used to pick between two or more possible alternative models. It is feasible to raise the probability of a model by adding parameters, however, doing so may result in overfitting. The BIC overcomes this problem by providing a penalty term for the model’s parameter count. In comparison to AIC, BIC has a longer penalty term.

BIC seeks to select a model that maximizes the posterior probability of the data given the model. When choosing amongst many models, models with lower BIC values are often favoured. A lower BIC does not always imply that one model is superior to another. The BIC is only a heuristic since it contains approximations. Differences in BIC, in particular, should never be interpreted as modified Bayes factors.

Main Differences Between AIC and BIC

  1. AIC stands for Akaike’s Information Criteria, whereas BIC stands for Bayesian Information Criteria.
  2. Hirotsugu Akaike published Akaike’s Information Criteria in 1974, while Gideon E. Schwarz published Bayesian Information Criteria in 1978.
  3. The AIC may be thought of as a metric for determining the quality of fit of any estimated statistical model, whereas the BIC is a model selection method among a class of parametric models with varying numbers of parameters.
  4. In Bayesian Information Criteria, the penalty for additional parameters is greater than in Akaike’s Information Criteria.
  5. Under the assumption that the “true model” is not in the candidate set, AIC is asymptotically best for picking the model with the least mean squared error, while BIC is not asymptotically optimal.
  6. The overall goal of Akaike’s Information Criteria is to identify an unknown model with high dimensional realism, On the other hand, the Bayesian Information Criteria, identifies only True models.
  7. The Bayesian Information Criteria is generally consistent, while Akaike Information Criteria is not.

Conclusion

Depending on their unique purposes and a separate set of asymptotic assumptions, AIC and BIC are both almost correct. The AIC typically risks selecting a model that is too large, whereas the BIC criterion frequently risks selecting a model that is too small.

In modeling, there is always the danger of either under-fitting for small n or over-fitting for high n. In circumstances where n is small, criteria with lower under-fitting rates, such as AIC, frequently appear better, but in cases where n is big, more parsimonious criteria, such as BIC, often appear better.

It is important to note that AIC or BIC cannot tell how well a given model explains your data; it would only tell if it achieves a better balance between model complexity and specificity than other models.

References

  1. https://journals.sagepub.com/doi/abs/10.1177/0049124103262065
  2. https://www.jstor.org/stable/43495189