What is Meant by a Statistical Decision Problem?

Question

I am reading Mathematical Statistics - A Decision Theoretic Approach by Ferguson (Academic Press 1967).

A game in the book is defined as a triple $(\Theta, \mathcal A, L)$, where $L$ is a function from $\Theta\times \mathcal A\to \mathbb R$. The set $\Theta$ is to be thought of as possible "states of nature" (spooky language), $\mathcal A$ is to be thought of as the set of possible actions one can take, and $L$ is the loss one incurs when an action is taken given a state of nature.

This is formally understand. But then the author, on pg 7, defined a statistical decision problem as:

A statistical decision problem is a game $(\Theta, \mathcal A, L)$ coupled with an experiment involving a random observable $X$ whose distribution $P_\theta$ depends on the state $\theta\in \Theta$ chosen by nature.

It is not clear to me what the author intends to convey. By an "experiment" I suppose the author means a probability space, and by a random observable I suppose he means a random variable.

But what does it mean to say that the distribution of $X$ depends on the state $\theta$? Does he mean that we actually have a whose set of random variables parameterized by $\Theta$?

Please feel free to add any intuition and elaboration of these concepts. I am a complete beginner here.

Also, if you can suggest a more recent reference for reading this material then that is more than welcome.

Answer 1

Here is how I think of it, and to make it more practical, I provide the standard example of a binary decision that is both forced (you can't decide to wait and see what else you can learn) and factual (you are simply trying to decide truth, rather than trying to decide on some action based upon what you think truth is):

"State of nature"="Truth variable" - This is a random variable, the value of which you can never really know. For example, H, with two states:

H1 = there is a running car behind the door

H0 = there is not a running car behind the door

"Actions"="Decision variable" - This is a random variable representing your decision. For example, D, again with two states:

D1 = I decide there is a running car behind the door

D0 = I decide there is not a running car behind the door

"Experiment"="Observation" - Some data you can obtain (arising from the "experiment", if you choose to think of it that way) that might help inform your decision. The data is cast as a particular exemplar of a random variable, with density conditioned upon the truth variable. For example,

x (the random variable) = the noise level measured in front of the door, with

p(x|H1) = density of x if car is running behind door

P(x|H0) = density of x if no car is running behind door

X (the exemplar) = the actual measurement of the noise level in front of the door

"Loss"="Objective function" - This describes what constitutes a better result and what constitutes a worse result. This choice is arbitrary, but for all problems, one possible objective is a random consequence variable C, again with two states:

C1 = good decision (here if D=H)

C0 = bad decision (here if D~=H)

Hope this helps.

Answer 2

An experiment is just a random variable $X$ whose distribution $P_\theta$ depends on the state.

You might be able to think of this as a collection of random variables parameterised by $\theta$, but perhaps it would be better to just think of it as just a single random variable whose distribution is potentially different in different states of the world. This interpretation fits better with the intuition I'm about to give below.

What are these definitions meant to formalise?

The idea is that your payoffs $L$ depends jointly on the action you choose in $\mathcal A$ and the state of the world in $\Theta$, and you wish to choose an action to maximise your payoffs (minimise losses). However, the problem is that you don't actually know the state $\theta \in \Theta$. If you did, the problem would be (relatively) trivial. Instead, the only information you have (other than the structure of the game) is the observation of the random variable $X$. Since the distribution of $X$ varies with $\theta$, observing $X$ gives you partial information about the state $\theta$.

Here's a simple example.

Suppose you are at the office, and you're about to head out for lunch, perhaps late into the lunch break. (You're a bit of a workaholic.) However, it may or may not be raining outside. (These are the states of the world.) Thus, you may or may not bring your umbrella. (These are the actions.) You don't like getting wet in your officewear, but you'd also hate to needlessly bring an umbrella. (This describes your loss function.) Unfortunately, your office has no windows, and the Internet is on the fritz, so you can't check the weather online. However, you can look around at your colleagues who have come back from their lunch breaks: some of them may be wet from the rain outside or have umbrellas that are. This gives you partial information about whether it is in fact still raining outside, and you use this information to decide whether to bring your umbrella or not.