# Variable selection vs Model selection

So I understand that variable selection is a part of model selection. But what exactly does model selection consist of? Is it more than the following:

1) choose a distribution for your model

2) choose explanatory variables, ?

I ask this because I am reading an article Burnham & Anderson: AIC vs BIC where they talk about AIC and BIC in model selection. Reading this article I realize I have been thinking of 'model selection' as 'variable selection' (ref. comments Does BIC try to find a true model?)

An excerpt from the article where they talk about 12 models with increasing degrees of "generality" and these models show "tapering effects" (Figure 1) when KL-Information is plotted against the 12 models:

DIFFERENT PHILOSOPHIES AND TARGET MODELS ... Despite that the target of BIC is a more general model than the target model for AIC, the model most often selected here by BIC will be less general than Model 7 unless n is very large. It might be Model 5 or 6. It is known (from numerous papers and simulations in the literature) that in the tapering-effects context (Figure 1), AIC performs better than BIC. If this is the context of one’s real data analysis, then AIC should be used.

How can BIC ever choose a model more complex than AIC in model selection I do not understand! What specifically is "model selection" and when specifically does BIC choose a more "general" model than AIC?

If we are talking about variable selection, then BIC must surely always choose the model with lowest amount of variables, correct? The $2ln(N)k$ term in BIC will always penalize added variables more than the $2k$ term in AIC. But is this not unreasonable when "the target of BIC is a more general model than the target model for AIC"?

EDIT:

From a discussion in the comments in Is there any reason to prefer the AIC or BIC over the other? we see a small discussion between @Michael Chernick and @user13273 in the comments, leading me to believe that this is something that is not that trivial:

I think it is more appropriate to call this discussion as "feature" selection or "covariate" selection. To me, model selection is much broader involving specification of the distribution of errors, form of link function, and the form of covariates. When we talk about AIC/BIC, we are typically in the situation where all aspects of model building are fixed, except the selection of covariates. – user13273 Aug 13 '12 at 21:17

Deciding the specific covariates to include in a model does commonly go by the term model selection and there are a number of books with model selection in the title that are primarily deciding what model covariates/parameters to include in the model. – Michael Chernick Aug 24 '12 at 14:44

• Good question! At least part of the resolution is to distinguish between the "target" of BIC in the terminology of this paper - the true model, which it'll pick with a very large sample size - & the model it happens to pick with a particular sample size. There's no contradiction then, when considering a nested sequence of models with an increasing no. parameters, in saying that the target of BIC is the model with 9 parameters, even though at a moderate sample size the BIC picks the model with 4 parameters, & the AIC the one with 6. – Scortchi Apr 7 '16 at 9:37
• @Scortchi: Good example, but is not the concept of a target model not totally redundant when we are talking about nested models? If the context is a set of nested models (then we are talking about variable selection): BIC might have a more complex target model, but will never choose a more complex model than AIC. In any other context (we are talking about model selection) (with large sample size) the paper claims that BIC will pick a more complex ("general") target model than AIC. How this happens specifically, is still not clear to me. – Erosennin Apr 7 '16 at 10:55
• @Erosennin did you ever manage to find an answer to this general question of yours? – zipzapboing May 27 '18 at 1:41

Next, they'd run a bunch of different specifications with many different variable combinations such as OLS model: $$y_i=\sum_{j_m} X_{ij_m}\beta_{j_m}+\varepsilon_i,$$ where $j_m$ denotes variable $j$ in a model $m$. They'd pick the best model out of all models $m$ manually or in an automated routines. So, these people would call the latter stage model selection.
In the context of the paper you cited, this is all irrelevant. The paper uses BIC or AIC to select between different model specifications. It doesn't matter whether you had the variable selection as a separate step in this case. All that matters is which variables are in any particular model specification $m$, then you look at their BIC/AIC to pick the best. They account for sample sizes and number of variables.