# #StackBounty: #bayesian #decision-theory Two class bayesian decision theory

### Bounty: 50

I’m new to decision theory, but in the many “intro to bayesian decision theory” tutorials, the two-category classification example is usually given. It boils down to deciding action 1 if it’s risk is lower, $$R(a_1|textbf{x}) < R(a_2|textbf{x})$$ and action 2 otherwise (with $$lambda_{ij}$$ representing losses).

where

$$begin{equation} R(a_1|textbf{x}) = lambda_{11}P(w_1|textbf{x}) + lambda_{12}P(w_2|textbf{x}) end{equation}$$

$$begin{equation} R(a_2|textbf{x}) = lambda_{21}P(w_1|textbf{x}) + lambda_{22}P(w_2|textbf{x}) end{equation}$$

My question is: Can I use a non-parametric classifier in this bayesian framework for decisons? For example, a decision tree can give a probabilistic result which I would call $$P(w_i|textbf{x})$$. But, I’m dealing with rare-event classification, so I would like to include prior probabilities of the classes, $$P(w_1)$$ and $$P(w_2)$$.

Then there are couple number of ways to represent the decision rule, but neither seems to fit what I’m trying to do.

1) $$lambda_{11}P(w_1|textbf{x}) + lambda_{11}P(w_2|textbf{x}) < lambda_{21}P(w_1|textbf{x}) + lambda_{22}P(w_2|textbf{x})$$

This way doesn’t include the prior probabilities

2) $$(lambda_{21} – lambda_{11})P(textbf{x}|w_1)P(w_1) > (lambda_{12} – lambda_{22})P(textbf{x}|w_2)P(w_2)$$

This way requires class conditional distributions for the features.

Am I missing something or is this simply not possible?

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