0
$\begingroup$

We have two slot machines, $B_1$ and $B_2$. We've played the first machine $n_1$ times and gotten the rewards $R_1^1, \dots, R_1^{n_1}$ and played the second machine $n_2$ times and gotten the rewards $R_2^1, \dots, R_2^{n_2}$.

Based on the data $x = \{ R_1^1, \dots, R_1^{n_1},R_2^1, \dots, R_2^{n_2} \}$ we want to decide if we should play $B_1$ or $B_2$ next. The action space is $\aleph ={1,2}$ and our parameter is $Y=(R_1, R_2)$, the next reward for each machine. We consider two loss functions, $L_1(Y,a) = -R_a$ and $L_2(Y,a) = 1_{R_a = 0}$. Find the formal Bayes rule for each loss function.

Attempted solution for $L_1$:

The expected loss for action $a$ is given by the integral of the loss function weighted by the posterior distribution:

$$E[L_1(Y,a)∣x]=\int L_1(y,a)P_{Y|X}(y|x) dy $$

To minimize this expected loss, we choose the action $a$ that minimizes the integral. The Bayes rule is then given by:

$$\hat{a}(x)=arg⁡min⁡_a∫(−R_a)⋅P(Y=y∣x) $$

For $L_2$:

The expected loss for action $a$ is given by the sum of the probabilities of the loss occurring:

$$E[L_2(Y,a)∣x]=P(R_a=0∣x)$$

To minimize this expected loss, we choose the action aa that minimizes the probability of $R_a = 0$. The Bayes rule is then given by:

$$\hat{a}(x)=arg⁡min⁡_a P(R_a=0∣x) $$

Am I out in the woods or on to something?

Best regards

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.