5.1 Ingredients of a Decision problem

1. 1.

   A set $𝒟$ of decisions which are available to the decision maker;

2. 2.

   A loss function $\mathrm{\Theta }\times 𝒟$, where $L(\theta ,d)$ is the loss associated with decision $d\in 𝒟$.

3. 3.

   A parameter space, $\theta \in \mathrm{\Theta }$, the distribution of states of nature, $\pi (\theta )$;

4. 4.

   An observation space, $y\in 𝒴$, the distribution of the observations $f(y\mid \theta$);

5. 5.

   A reward space, $ℛ$. We assume there is an ordering on these rewards. Our utility or loss function can be written as a function of the reward and our belief in what the reward is: $L(r(\theta ),d)$ where $r\in ℛ$

• •

  No decision can be taken without potential losses to the client.

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  The loss is a negative utility.

  $$L(\theta ,d)=-U(\theta ,d)$$

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  The loss function is denoted by $L(\theta ,d)$ and represents the payoff by a decision maker (statistician) if he takes the decision $d\in 𝒟$, and the real state of nature is $\theta \in \mathrm{\Theta }$.

• •

  The loss function satisfies the following properties, $L(d,d)=0$ and $L(d,\theta )$ is non-decreasing function of $|d-\theta |$

Often the decision maker must make an estimate. It can be a forecast or it can be a bid. The decision maker can be penalised for discrepancies through the use of a loss function. We now look at some common loss functions and calculate the best decision and the expected posterior loss having made the best decision.

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  Squared error loss (SLE)

  $$L(d,\theta )={(d-\theta )}^2$$

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  Absolute error loss

  $$L(d,\theta )=\left|d-\theta \right|$$

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  Asymmetric linear loss

  $L(d,\theta )$

  $=$

• •

  Hit or miss loss

  $L(d,\theta )$

  $=$