Last updated: 2019-04-04
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The goal here is to introduce some basic ideas from decision theory, and particularly the notions of loss, decision rule, and integrated risk, in the context of a simple prediction problem.
To understand this vignette you will need to be familiar with the concept of probability distributions and expectations.
Consider the problem of predicting an outcome \(Y\) on the basis of inputs (or “features” or “predictors”) \(X\). Typically \(Y\) might be a one-dimensional outcome (discrete or continuous) and \(X\) a multi-dimensional input. If \(Y\) is discrete then this is often referred to as a “classification problem”; if \(Y\) is continuous then this is often referred to as a “regression problem”.
As a concrete example, here we attempted to classify an elephant tusk as being from a forest or savanna elephant (\(Y\)) based on its genetic data (\(X\)).
To make a rational decision about what value \(\hat{Y}\) to predict for \(Y\) we must specify how “bad” different types of errors are.
That is, we must specify, for each possible value of \(Y\), and each possible prediction \(\hat{Y}\), a (real) value \(L(\hat{Y},Y)\) that measures how “wrong” the prediction is. Big values of \(L\) indicate worse predictions. \(L\) is called the “loss function”.
For example, if \(Y\) is continuous and real-valued then some simple common loss functions are:
If \(Y\) is discrete then a simple common loss function is 0-1 loss, which is 0 if the prediction is correct and 1 otherwise: \(L(\hat{Y},Y) = I(\hat{Y} \neq Y)\) where \(I\) denotes the indicator function.
In this context a decision rule is simply a way of predicting \(Y\) from \(X\). That is it is a function \(\hat{Y}(X)\), which for any given \(X\) produces a predicted value \(\hat{Y}\) for \(Y\).
Now consider applying the decision rule \(\hat{Y}(X)\) to a series of \((X,Y)\) pairs coming from some probability distribution \(p(X,Y)\). A natural way to measure how good (or bad) the decision rule is, is to compute the expected loss (sometimes referred to as the Integrated Risk, and here denoted \(r\)):
\[r(\hat{Y}) := \int \int L(\hat{Y}(X),Y) p(X,Y) \,dX \,dY.\]
So what decision rule \(\hat{Y}\) is “optimal” in terms of minimizing the expected loss \(r\)? It is easy to show that the following decision rule minimizes \(r\):
Optimal Decision Rule: For each \(X\) choose the prediction \(\hat{Y}_\text{opt}(X)\) that minimizes the conditional expected loss: \[\hat{Y}_\text{opt}(X) = \arg \min_a \int L(a, Y) \, p(Y | X) dY.\]
The proof is straightforward. Since \(p(X,Y)= p(Y|X) p(X)\), we can rewrite the expected loss for any decision rule \(\hat{Y}\) as: \[r(\hat{Y}) = \int \biggl[ \int L(\hat{Y}(X),Y) p(Y|X) \, dY \biggr] p(X) \, dX\] Note that, by definition, \(\hat{Y}_\text{opt}(X)\) minimizes the term inside \([]\). Thus, for any decision rule \(\hat{Y}\) we have that the risk is no better than \(\hat{Y}_\text{opt}\): \[r(\hat{Y}) \geq \int \biggl[ \int L(\hat{Y}_\text{opt}(X),Y) p(Y|X) \, dY \biggr] p(X) \, dX = r(\hat{Y}_\text{opt}).\]
As defined above, finding \(\hat{Y}_\text{opt}(X)\) requires solving an optimization problem. For the standard loss functions given above, solving this optimization problem is easy provided that the conditional distribution \(p(Y|X)\) is easy to work with.
For example, using squared loss, the optimization problem becomes \[\hat{Y}_\text{opt}(X) = \arg \min_a f(a)\] where \[f(a) = \int (a-Y)^2 \, p(Y | X) \, dY.\] We can optimize this by differentiating with respect to \(a\) and setting the derivative to 0. Differentiating \(f(a)\) gives \[f'(a) = 2 \int (a-Y) p(Y|X) \, dY = 2(a - E(Y|X)).\] which is zero at \(a=E(Y|X)\).
Thus, for squared loss, the optimal decision rule is to predict \(Y\) using its conditional mean given \(X\): \(\hat{Y}_\text{opt}(X) = E(Y | X)\).
Although not quite as easy to show, under absolute loss the optimal decision rule is to set \(\hat{Y}\) to the median of the conditional distribution \(Y|X\).
Under 0-1 loss, with \(Y\) discrete,
the optimal decision rule is to set \(\hat{Y}\) to the mode of the conditional distribution \(p(Y|X)\). That is \[\hat{Y}_\text{opt}(X) = \arg \max_a p(Y=a |X).\] Showing this is left as an Exercise.
The conditional distribution \(Y|X\) is sometimes be referred to as the “posterior” distribution of \(Y\) given data \(X\), and computing this distribution is sometimes referred to as “performing Bayesian inference for \(Y\)”.
Thus, the above result (“Optimal Decision Rule” section) can be thought of as characterizing the optimality of Bayesian inference in terms of a “frequentist” measure (\(r\)) which measures long-run performance across many samples \((X,Y)\) from \(p(X,Y)\). For example, predicting \(Y\) by its posterior mean, \(E(Y|X)\), is optimal in terms of expected squared loss (with expectation taken across \(p(X,Y)\)).
Because of this connection with Bayesian inference, the optimal value \(r(\hat{Y}_\text{opt})\) is sometimes referred to as the “Bayes risk”, and \(\hat{Y}_\text{opt}\) is referred to as a “Bayes decision rule”.
Note that the optimal decision rule depends on the distribution \(p(Y,X)\) – or, more specifically, on the conditional distribution \(p(Y|X)\). Typically one does not know this distribution exactly, and so one cannot implement the optimal decision rule. (An exception is in artificial simulation experiments, where the “true” distribution \(p(Y,X)\) is known; in these cases the optimal rule can be computed, and may provide a useful yardstick against which other rules can be compared.)
One way (but not the only way) to proceed in practice is to perform Bayesian inference for \(Y\) anyway, by simply positing (assuming) some “prior” distribution \(p(Y)\), and a “likelihood” \(p(X|Y)\), and using these to compute a posterior distribution \(p(Y|X)\). The result above shows that inference based on this posterior will be optimal, on average, across large numbers of samples of \((X,Y)\) drawn from \(p(X,Y)= p(Y)p(X|Y)\). But, of course, the result does not guarantee optimality, on average, across samples from some other distribution, \(q(X,Y)\) say. One might summarize this as “Bayesian inference is optimal, on average, if both the prior distribution and likelihood are `correct’”.
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