I am given a single scaled Cauchy: $f_{X|θ}(s)=\frac{2}{τ}⋅\frac{θ}{θ^2+s^2}$ and the hypotheses $H_0: θ=1.4$ versus $H_1:θ=2.3$. If a single data point X is drawn then how can I find the Bayes Factor for $H_0$ versus $H_1$ given $X=4.1$?
I'm having trouble figuring how to solve the problem. I know that it should be getting a ratio $(\frac{H_0}{H_1})$, but am unsure about where the $X=4.1$ fits into this.
In case of both simple hypothesis, bayes factor matches with LR thus
$$\frac{\mathbb{P}[\mathbf{x}|\theta_0]}{\mathbb{P}[\mathbf{x}|\theta_1]}=\frac{\frac{1.4}{1.4^2+4.1^2}}{\frac{2.3}{2.3^2+4.1^2}}\approx0.72$$
If you have no particular prior information about the parameter, it is more likely that $\theta=2.3$ instead of $1.4$