4 Review

4.1 Big picture: How to make optimal choices under statistical uncertainty?

  • A decision-maker (a.k.a. actor) chooses one option from a menu of possible actions.

  • Payoffs from that choice depend on the action chosen, and on the true value of the state of nature.

    • Payoffs may equivalently be represented as losses — negative payoffs relative to some baseline.

  • The true value of the state of nature is uncertain. Hence the decision-maker confronts a problem of decision-making under uncertainty.

    • Each possible action thus maps to a lottery over uncertain payoffs.

4.2 Uncertainty

  • Typically, the decision-maker will posses some information about the likelihood of different states of nature.

  • This information can be represented formally by treating the state of nature as a random variable drawn from a known distribution.

4.3 Optimal choice

  • Given a complete enumeration of
    • the menu of possible actions,
    • the set of possible states of nature,
    • a probability distribution over the set of states of nature, representing the likelihood of states, and – a payoff function that gives the payoffs (or losses) from each possible combination of action \(\times\) state, in some units of measure.
    then each possible action maps to a lottery over payoffs (losses).

4.3.1 Optimization: Defining objectives

  • Finally, given a lottery over payoffs, the actor chooses an optimal action guided by a decision-making principle:

    • Example: Maximize expected payoffs.
    • Example: Maximize expected utility of payoffs.
    • Example: ‘Minimax’: Choose the action to minimize possible loss, irrespective of probabilities.

  • The decision-making principle encodes the actor’s attitude towards risk – the willingness to accept losses in some uncertain states of the world, in exchange for acheiving gains in other states of the world.

    • The study of attitudes towards risk and loss is huge topic in economics, finance, and psychology. We will cover it only glancingingly.
    • For your applications, just at minimum be aware that the actor’s optimum choices may not be driven by goal to maximize expected gains (= minimize expected losses).

4.4 Integrating predictive and inferential tools into the decision calculus

In general, your predictive model serves to reduce uncertainty over future values of the states of nature.

They sharpen the probability distribution over states of nature.

(In terms of probability theory: they involve a change of measure.)

The actor can now choose an optimal action based not on her prior (or naive) beliefs, but on her posterior beliefs, conditioned on the current data.

4.5 Payoff functions: Define in terms of \(x\) (state) or \(\theta\) (parameter value)?

Depending on your appplication, it may make sense to model payoffs (or losses) as either a function of observed realized state \(x \in \mathbb{X}\), e.g., \(x\) denotes realized temperature:

\[L_0 = L_0(a;x)\]

or in terms of the value of an unobserved parameter, e.g., \(\theta\) denotes mean temperature:

\[L_1 = L_1(a; \theta)\]

4.5.1 Parameter-dependent payoffs as reduced form of state-dependent payoffs

In many cases, you can represent the \(\theta\) formulation as the reduced form of the \(x\) formulation, e.g.

\[L_1(a;\theta) = E[L_0(a;x) | \theta]\]