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Is there any way to use BIC in model selection for gam? And if so then how to extract bic?

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As long as the different observations $\{y_i, \mathbf{x}_i\}_{1 \leq i \leq T}$ that you use to fit your GAM are i.i.d. (and assuming that the number of observations $T$ is reasonably large) it is perfectly possible to compute its BIC.

Let's assume that your model has the following form:

$$ g(\mathbf{E}(y)) = \alpha + f_{\theta_1}^{(1)}(x^{(1)}) + f_{\theta_2}^{(2)}(x^{(2)})+ f_{\theta_3}^{(3)}(x^{(3)}) + \dots $$

where $\mathbf{x} = [x^{(1)},\dots,x^{(N)}]$, and where $\theta_1,\dots,\theta_N$ are the free parameters of the functions you are fitting on your data. Then the total number of free parameters in your GAM will be

$$ 1+ \sum_{j=1}^N |\theta_j| $$

i.e. the number of free parameters in each function, plus one (for $\alpha$).

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