bayesian decision making - comparing expected loss

The problem is like this:

Suppose that I am considering which country should I invest on, country A and country B, based on their GDP growth rate $$\alpha$$.

There are two possible choices for each country:

$$d_{1A}$$ : Invest in country A and $$d_{2A}$$ : Do not invest in country A;

$$d_{1B}$$ : Invest in country B and $$d_{2B}$$ : Do not invest in country B.

The loss functions for both countries are the same: $$L(d_1,\alpha)=\left\{\begin{matrix} 0 & \alpha > 0.002\\ 4 & \alpha \leqslant 0.002 \end{matrix}\right.$$ and $$L(d_2,\alpha)=\left\{\begin{matrix} 1 & \alpha > 0.002\\ 0 & \alpha \leqslant 0.002 \end{matrix}\right.$$

Assume that I can only invest on the country, the one will increase its GDP growth more rapidly.

The $$\alpha$$ of country A is from posterior $$\pi_A(\alpha|x)$$;The $$\alpha$$ of country B is from posterior $$\pi_B(\alpha|x)$$. $$\pi_A(\alpha|x)$$ and $$\pi_B(\alpha|x)$$ are known.

And I also calculate the values of the four following expected loss:

$$E\left [L(d_{1A},\alpha) \right ] = \int_{-\infty }^{0.002}{4\cdot \pi_{A}(\alpha|x)}$$ and all other $$E\left [L(d_{2A},\alpha) \right ]$$,$$E\left [L(d_{1B},\alpha) \right ]$$,$$E\left [L(d_{2B},\alpha) \right ]$$.

The issue is that how could I use these four "loss" to make a decision that which country is worth investing on? Because $$\alpha_A$$ and $$\alpha_B$$ come from different posteriors, so I cannot compare their loss directly.

• In what sense do $\alpha_A$ and $\alpha_B$ come from 'different posteriors' and why is that a problem? – Juho Kokkala Jan 18 at 6:31
• I am sorry I couldn't make it clear. The datasets of country A and country B are different, and the priors for $\alpha_A$ and $\alpha_B$ are different too. Their posteriors ($\alpha_A | x_A$ and $\alpha_B | x_B$) are produced by Gibbs sampler and are different. That's what I meant 'different posteriors'. – üpæn9l1 Jan 18 at 10:42
• And I calculate that $E[L(d_{1A},\alpha)]=0.0004$,$E[L(d_{2A},\alpha)]=1-0.0004$, and $E[L(d_{1B},\alpha)]=0.054$,$E[L(d_{2B},\alpha)]=1-0.054$. If I just focus on one country at a time, I can tell that I should invest on country A because its $E[L(d_{1A},\alpha)]=0.0004$ is smaller than $E[L(d_{2A},\alpha)]=1-0.0004$; similarly, I can tell I also should invest on country B for the same reason. But the issue is that I can only invest on only one country. How could I make the decision? – üpæn9l1 Jan 18 at 10:49