# Uncertainty around utility maximizer in decision analysis

Hi all,

Following the Decision Analysis section from Stan Users Guide, suppose I have:

d^* = \textrm{arg max}_d \ \mathbb{E}[U(x) \mid d].

where the expectation is taken with respect to the posterior distribution of outcomes x conditional on decisions d, p(x\vert d).

If the decisions are continuous (eg, how much money to invest), I wonder if it is possible to derive a posterior distribution for the maximizer d^*. For instance, for each posterior draw s, I compute {d^*}^{(s)} = \textrm{arg max}_d \ \big[U(x) \mid d \big]^{(s)} (ie., the â€śoptimal decision for each posterior drawâ€ť).

My underlying motivation is to run decision analysis for different settings/datasets, which seem to yield slightly different optimal decisions. So I wanted to have an idea of how much uncertainty there is around those â€śoptimizersâ€ť.

Does this make any sense?

Thank you!

You can definitely get the posterior distribution of a utility maximum.
It goes like this:

1. Fit your model to data and get posterior inferences for your parameter vector theta. Or if you have no data model, just use your prior for theta as your posterior.
2. Get a bunch of posterior simulation draws for theta.
3. For each simulation draw, work out your utility function and find the d that maximizes it.
4. VoliĂ : you now have a bunch of posterior simulations of the optimal decision, d.

You might be able to do step 3 within the generated quantities block of the Stan model. I donâ€™t know what optimizers are available within Stan.

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Awesome! Thank you very much, Professor!