Abstract
In this chapter we describe how to formalize statistical decision problems. These involve making decisions whose utility depends on an unknown state of the world. In this setting, it is common to assume that the state of the world is a fundamental property that is not influenced by our decisions. However, we can calculate a probability distribution for the state of the world, using a prior belief and some data, where the data we obtain may depend on our decisions.
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Notes
- 1.
These distributions may be singular, that is, they may be concentrated in one point. For example, \(\sigma ^*\) is singular, if \(\sigma ^*(a)=1\) for some \(a\) and \(\sigma ^*(a')=0\) for all \(a'\ne a\).
- 2.
For that reason, policies are also sometimes called decision functions or decision rules in the literature.
- 3.
We obtain a different probability of observations under the binomial model, but the resulting posterior, and hence the policy, is the same.
- 4.
For example, through Lipschitz conditions on the policy.
- 5.
The following figures are not really accurate, as they are liberally adapted from different studies.
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Dimitrakakis, C., Ortner, R. (2022). Decision Problems. In: Decision Making Under Uncertainty and Reinforcement Learning. Intelligent Systems Reference Library, vol 223. Springer, Cham. https://doi.org/10.1007/978-3-031-07614-5_3
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DOI: https://doi.org/10.1007/978-3-031-07614-5_3
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