Which model should be chosen using AICc/BIC and P-value? I am using GLM for modeling the binary data. For the predictive variables I am using different number of Zernike Basis functions. To test how many Zernike basis functions I need, I am using AICc/BIC to choose the best model. For example, I use k={1,2,3,...,20} basis functions and AICc chooses the model which has k=5 basis functions, say X1,X2,X3,X4,X5. However, using GLM shows me that not all these 5 basis functions are significant, i.e., the p-value>0.05. For example the coefficients of X2 and X4 are not significant. What should I do now? In the end when I want to find one model for my data should I use the model which only has X1,X3,X5? or Should I not care about the p-value of the coefficients and include X1,X2,X3,X4,X5?
 A: The AIC, the BIC and the $p$-values all address different questions. For feature selection (variable selection, model selection), only the former two are relevant. See e.g. Rob J. Hyndman's blog posts "Statistical tests for variable selection" and "Facts and fallacies of the AIC".
AIC is best suited for forecasting purposes as it targets maximization of the likelihood of a new observation from the same data generating process (under normality of errors, maximization of likelihood is equivalent to minimization of square loss). BIC is a consistent model selector and as such may be useful in finding which model among a given set is the true model (if the true model is in the set). So depending on your goal (prediction or identification of the data generating process), AIC or BIC may be relevant for you.
In this particular application I suspect that AIC is more relevant as the models you are considering probably are not thought to contain any true model among them.
Furthermore, model averaging as suggested by @Björn might be a good (better) alternative solution.
A: For a prediction situation AIC is more relevant than BIC or p-values. However, to pick a single model on the basis of any of these and then to analyze the data as if no model selection had been done, is likely inappropriate (unless one candidate model is so vastly better than the others as to remove all model uncertainty). Model averaging is usually more appropriate, if the goal is prediction. 
