#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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