Is there a way to calculate the maximum likelihood from a tree regression model? When fitting a tree regressor model, I would like to calculate the AIC and BIC metrics. However I need the maximum of the likelihood function to do this.
Is there a closed form solution or some other way to calculate the likelihood function from a tree regressor? I haven't been able to find any information online, other than a closed form solution in an OLS framework.
 A: To compute the BIC or AIC for a model, the observed dataset has to have an associated conditional distribution. For instance,

*

*In a linear regression, a dataset $\mathcal{D} = \{(t_n, {\bf x}_n) \vert t_n\in\mathbb{R}, {\bf x}_n\in\mathbb{R}^M\}$ is assumed to be conditionally distributed as

$$
t_n\vert {\bf x}_n\sim\mathcal{N}({\bf w}^T{{\bf x}_n}, \sigma^2)
$$


*In a logistic regression, a dataset $\mathcal{D} = \{(t_n, {\bf x}_n) \vert t_n\in\{0,1\}, {\bf x}_n\in\mathbb{R}^M\}$ is assumed to be conditionally distributed as

$$
t_n\vert {\bf x}_n\sim\text{Blli}(\sigma({\bf w}^T{{\bf x}_n}))
$$


*In an ARCH(1) model,  a dataset $\mathcal{D} = \{t_n \vert t_n\in\mathbb{R}\}$ is assumed to be conditionally distributed as

$$
t_n\vert t_{n-1}\sim\mathcal{N}(0, \sigma(t_{n-1}))
$$
And so on...
A classical decision tree, however, does not assume a conditional distribution for the data. There is no associated likelihood function, hence BIC cannot be computed.
If you wanted to compute the BIC, you'd need to assign to your model some sort of likelihood function.
A: A regression tree is still a linear model (if you define the correct interaction terms). So in principle it is possible to calculate AIC and BIC with the OLS formula.
