Bayes test function with a discrete prior

Question

Let X be exponential with mean θ. Consider testing H0 : θ = 1 versus H1 : θ = 2 with a single observation.

Loss function: 0-1 Loss function.

So the risk of the test function φ is R(1, φ) = E1 (φ(X)) and R(2, φ) = E2 (1 − φ(X)).

Prior distribution θ with θ({1}) = 2/3, and θ({2}) = 1/3

Find the Bayes test function for the prior θ.

Hint: Your test should minimize [R(1, φ) + 2R(2, φ)]/3.

I know how to deal with questions when there's a squared error loss and a continuous prior. But this 0-1 loss function and discrete prior.

The only book that helped me a little was the Theory of Statistics | Mark J. Schervish. but I could only find little information about bayesian hypothesis testing on pages 218-222.

Any guidance,direction would be highly appreciated.

Answer 1

Here your test function $\varphi$ is such that $\varphi(x) = 1$ indicates that you should favor $H_1$ based on observation $x$ and $\varphi(x) = 0$ if it indicates that you should favor $H_0$. The risk function you indicated then corresponds to the probability of a mistake given $H_0$ or $H_1$ : indeed, $E_{\theta = 1}(\varphi(X)) = P_{\theta = 1}(\varphi(X) = 1)$ and $E_{\theta = 2}(1 - \varphi(X)) = P_{\theta = 2}(\varphi(X) = 0)$.

You want to find $\varphi$ that minimizes the expected risk under the prior distribution, which is : $E(R) = P(\theta = 1)E_{\theta = 1}(\varphi(X)) + E_{\theta = 2}(\varphi(1 - X)) = \frac{2}{3}E_{\theta = 1}(\varphi(X)) + \frac{1}{3}E_{\theta = 2}(1 - \varphi(X))$, hence the hint.

Now, you can write the expression above as an integral and try to find $\varphi$ which minimizes it. Remember that $\varphi(x)$ can only be $0$ or $1$...