Bayesian Decision Theory

Introduction

Earlier, we introduced Bayesian inference and derived posterior distributions for unknown parameters. Markov chains model sequential data, and Markov Chain Monte Carlo (MCMC) methods use the Markov property to sample from complex posterior distributions.

So far, our development of Bayesian statistics has focused on inference: given observed data, how should we update our beliefs about unknown quantities? But inference alone does not tell us what to do. In many applications we must ultimately choose an action based on uncertain information. Medical diagnosis, spam filtering, and autonomous driving are typical examples. Bayesian decision theory provides the principled framework for making such optimal decisions under uncertainty.

The setup is as follows. An agent must choose an action \(a\) from a set of possible actions \(\mathcal{A}\). The consequence of this action depends on the unknown state of nature \(h \in \mathcal{H}\), which the agent cannot observe directly. To quantify the cost of choosing an action \(a\) when the true state is \(h\), we introduce a loss function \(l(h, a)\).

The key idea of Bayesian decision theory is to combine the loss function with the posterior distribution \(p(h \mid \boldsymbol{x})\) obtained from observed evidence \(\boldsymbol{x}\) (or a dataset \(\mathcal{D}\)). For any action \(a\), the posterior expected loss (or posterior risk) is defined as follows.

When \(\mathcal{H}\) is continuous, the sum is replaced by an integral against the posterior density, as in the continuous-parameter setting below. A rational agent should select the action that minimizes this expected loss. This leads to the central concept of this page.

Equivalently, by defining the utility function \(U(h, a) = -l(h, a)\), which measures the desirability of each action in each state, the Bayes estimator can be written as \[ \pi^*(\boldsymbol{x}) = \arg\max_{a \in \mathcal{A}} \, \mathbb{E}_{p(h \mid \boldsymbol{x})}[U(h, a)]. \] This utility-maximization formulation is natural in economics, game theory, and reinforcement learning, where the focus is on maximizing expected rewards rather than minimizing losses.

The power of this framework lies in its generality. By choosing different loss functions, we recover many familiar estimators and decision rules as special cases. We will see three such recoveries in this page. For continuous parameters, squared-error loss yields the posterior mean and absolute-error loss yields the posterior median. For discrete labels, zero-one loss yields the maximum a posteriori (MAP) estimate, and this is the decision-theoretic justification for choosing the most probable class at inference time. After establishing these special cases, we connect the Bayesian formulation above to its frequentist counterpart and show that, once a prior is fixed, averaging the frequentist risk over that prior and minimizing posterior expected loss lead to the same optimal rule under mild regularity.

Continuous Parameters: Squared and Absolute Error Loss

We first specialize the framework to the setting where the unknown state of nature is a continuous real-valued parameter. To match standard usage in parametric statistics, we write \(\theta \in \Theta \subseteq \mathbb{R}\), with \(\Theta\) an interval, in place of \(h \in \mathcal{H}\). The action space \(\mathcal{A}\) is also taken to be \(\Theta\), so that an action \(a\) is a candidate value for \(\theta\). The interval assumption guarantees that the posterior mean and posterior median found below are themselves available actions. This is the situation of point estimation revisited from a decision-theoretic standpoint. We treat the scalar case (\(\Theta \subseteq \mathbb{R}\)) for clarity of exposition.

The vector extension (\(\boldsymbol{\theta} \in \Theta \subseteq \mathbb{R}^k\) with squared-norm loss \(\|\boldsymbol{\theta} - \boldsymbol{a}\|^2\)) follows the same logic.

Two loss functions are particularly natural in this setting: squared-error loss, which penalizes large deviations heavily, and absolute-error loss, which is more robust to outliers. Each leads to a familiar Bayesian point estimator as the corresponding optimal action.

Squared-Error Loss and the Posterior Mean

Under this loss, the posterior expected loss admits a clean closed-form minimizer: the posterior mean.

The posterior mean is therefore the Bayes estimator under squared-error loss, often denoted \(\hat{\theta}_{\text{MMSE}}\) (minimum mean squared error). The same algebra as in the bias-variance decomposition \(\operatorname{MSE}(\hat{\theta}) = \operatorname{Var}(\hat{\theta}) + \operatorname{Bias}(\hat{\theta})^2\) of the mean squared error, applied to the posterior instead of the sampling distribution, gives \[ \rho(a \mid \boldsymbol{x}) = \operatorname{Var}_{p(\theta \mid \boldsymbol{x})}[\theta] + (\mathbb{E}_{p(\theta \mid \boldsymbol{x})}[\theta] - a)^2, \] which makes the minimality of the posterior mean visible at a glance.

Absolute-Error Loss and the Posterior Median

Squared-error loss penalizes large deviations more aggressively than small ones. A deviation of 2 incurs four times the loss of a deviation of 1. Absolute-error loss is linear in the magnitude of the deviation and is therefore more robust when the posterior has heavy tails. The optimal action under this loss is the posterior median.

Frequentist Risk, Bayes Risk, and Minimax

So far we have framed decision theory from the Bayesian perspective. Given observed data, we minimize the expected loss with respect to the posterior. The classical, or frequentist, perspective inverts this conditioning. There, the parameter \(\theta\) is treated as an unknown but fixed quantity, and the data \(\boldsymbol{x}\) is treated as random. We evaluate a decision rule by averaging its loss over hypothetical data realizations drawn from the data-generating distribution \(p(\boldsymbol{x} \mid \theta)\).

These two perspectives appear to start from opposite premises, yet they are connected by a precise equivalence theorem. Establishing this equivalence is the central goal of this section. Along the way, we will introduce minimax, the third major notion of optimality, and discuss its interpretation as a worst-case guarantee.

Convention. Throughout this section, we work in the parametric setting introduced for continuous parameters above, but allow \(\theta \in \Theta\) to be either scalar- or vector-valued. The state of nature is the parameter \(\theta\) (replacing the general \(h \in \mathcal{H}\) of the introduction), and a decision rule is a measurable function \(\delta : \mathcal{X} \to \mathcal{A}\) mapping data to actions. When \(\mathcal{A} = \Theta\), the decision rule is exactly a point estimator in the sense of statistical inference. Whereas \(\pi^*(\boldsymbol{x})\) denoted the optimal decision rule under a given criterion, we now use \(\delta\) for an arbitrary candidate decision rule whose risk we wish to evaluate. Optimality will then be defined as minimization of one of several risk notions introduced below.

Frequentist Risk

Three observations clarify the role of this definition. First, the parameter \(\theta\) is held fixed while the data is averaged over, which is the opposite conditioning from the posterior expected loss \(\rho(a \mid \boldsymbol{x})\). Second, \(R(\theta, \delta)\) is a function of \(\theta\). A single decision rule produces a risk function over the entire parameter space, not a single number. Third, comparing two decision rules therefore requires comparing two functions, and one risk curve may be lower for some \(\theta\) but higher for others. This last observation motivates the notions of admissibility and minimax developed below.

Worked Example: Estimating a Gaussian Mean

To make the definition concrete, consider the canonical setting of estimating the mean of a Gaussian. Suppose data \(x_1, \ldots, x_N \overset{\text{i.i.d.}}{\sim} \mathcal{N}(\theta, \sigma^2)\) with \(\sigma^2\) known, and suppose we use the squared-error loss \(l_2(\theta, a) = (\theta - a)^2\). The corresponding risk is the mean squared error \[ R(\theta, \delta) = \mathbb{E}_{p(\boldsymbol{x} \mid \theta)}\!\left[(\theta - \delta(\boldsymbol{x}))^2\right] = \operatorname{MSE}(\delta). \]

From the decomposition of the mean squared error into variance and squared bias (where, in the language of estimators, \(\hat{\theta} = \delta(\boldsymbol{x})\)), \[ \operatorname{MSE}(\delta) = \operatorname{Var}_{p(\boldsymbol{x} \mid \theta)}[\delta(\boldsymbol{x})] + \operatorname{Bias}(\delta)^2, \] where \(\operatorname{Bias}(\delta) := \mathbb{E}_{p(\boldsymbol{x} \mid \theta)}[\delta(\boldsymbol{x})] - \theta\). We use this identity to compute the risk of three estimators.

(i) The sample mean \(\delta_1(\boldsymbol{x}) = \bar{x} = \tfrac{1}{N}\sum_n x_n\). By linearity of expectation, \(\mathbb{E}[\bar{x}] = \theta\) (unbiased), and by the variance of a sum of independent variables, \(\operatorname{Var}[\bar{x}] = \sigma^2/N\). Hence \[ R(\theta, \delta_1) = \frac{\sigma^2}{N}. \] The risk is constant in \(\theta\). The quality of the sample mean as an estimator does not depend on the true value.

(ii) The constant estimator \(\delta_2(\boldsymbol{x}) = \theta_0\) for some fixed \(\theta_0 \in \mathbb{R}\). The variance is zero, but the bias is \(\theta_0 - \theta\), giving \[ R(\theta, \delta_2) = (\theta_0 - \theta)^2. \] The risk is zero when \(\theta = \theta_0\) (a perfect guess) and grows quadratically as \(\theta\) moves away.

(iii) The shrinkage estimator \(\delta_3(\boldsymbol{x}) = w \bar{x} + (1 - w) \theta_0\), a convex combination of the sample mean and the fixed value \(\theta_0\) with shrinkage weight \(w \in (0, 1)\). At this stage we treat \(w\) as an arbitrary tuning parameter. We will see momentarily that one particular choice arises from a Bayesian construction. Computing the bias, \[ \mathbb{E}_{p(\boldsymbol{x} \mid \theta)}[\delta_3] - \theta = w\theta + (1-w)\theta_0 - \theta = (1-w)(\theta_0 - \theta), \] and the variance \(\operatorname{Var}[\delta_3] = w^2 \sigma^2 / N\), we obtain \[ R(\theta, \delta_3) = w^2 \frac{\sigma^2}{N} + (1-w)^2 (\theta_0 - \theta)^2. \] The risk interpolates between the two extremes: \(w = 1\) recovers the sample mean and \(w = 0\) recovers the constant estimator.

Bayesian origin of the shrinkage weight. The estimator \(\delta_3\) arises naturally as the posterior mean under a Gaussian prior \(\theta \sim \mathcal{N}(\theta_0, \tau^2)\) (with known data variance \(\sigma^2\)). The resulting posterior is again Gaussian, and its mean is the convex combination \(\delta_3\) with the specific weight \[ w = \frac{N\tau^2}{N\tau^2 + \sigma^2}. \] A diffuse prior (large \(\tau^2\)) gives \(w \to 1\), so the data dominates and \(\delta_3 \to \delta_1\). A sharp prior (small \(\tau^2\)) gives \(w \to 0\), so the prior dominates and \(\delta_3 \to \delta_2\). The full derivation appears in our treatment of the Normal model with known variance.

Comparing the three risk functions reveals a fundamental phenomenon. The sample mean \(\delta_1\) has the lowest risk when \(\theta\) is far from \(\theta_0\) (since the squared-bias term in \(\delta_3\) grows quadratically in \(|\theta - \theta_0|\) while the risk of \(\delta_1\) stays at \(\sigma^2/N\)). The constant \(\delta_2\) achieves zero risk only at the single point \(\theta = \theta_0\) and is uniformly worse than \(\delta_3\) outside a small neighborhood of \(\theta_0\) (the explicit threshold can be computed by setting \(R(\theta, \delta_2) = R(\theta, \delta_3)\)).

The shrinkage estimator \(\delta_3\) is optimal in an intermediate range. There \(\theta\) is close enough to \(\theta_0\) to benefit from the prior but not so close that \(\delta_2\) overtakes it. No single estimator is uniformly best, and the optimal choice depends on the unknown true \(\theta\).

Admissibility is a partial-order criterion. It rules out estimators that are uniformly worse than some alternative, but typically leaves an entire family of admissible candidates from which to choose. To select among admissible rules, one needs an additional principle. Two such principles are averaging the risk over a prior (the Bayesian solution) and worst-casing it over \(\theta\) (the minimax solution).

Bayes Risk and the Equivalence Theorem

The Bayesian approach to selecting among decision rules averages the frequentist risk against a prior distribution \(p(\theta)\), collapsing the risk function to a single number per rule.

The Bayes risk is an integral over both \(\theta\) and \(\boldsymbol{x}\): \[ r(p, \delta) = \int \int l(\theta, \delta(\boldsymbol{x})) \, p(\boldsymbol{x} \mid \theta) \, p(\theta) \, d\boldsymbol{x} \, d\theta. \] The next theorem, the central result of this section, shows that this minimization problem reduces to a pointwise one. For each fixed \(\boldsymbol{x}\), we choose the action that minimizes the posterior expected loss. This is the precise sense in which, once a prior is fixed, Bayesian and frequentist optimality coincide.

Maximum Risk and Minimax

The Bayes risk requires a prior, which a strict frequentist may consider unjustified. An alternative is to evaluate a rule by its worst-case risk over all parameter values.

The minimax criterion encodes a deeply pessimistic view. The decision-maker assumes that nature will select the worst possible \(\theta\) against the chosen rule, and seeks to minimize the damage in this adversarial scenario. In the worked example above, the maximum risks are easily computed: \(R_{\max}(\delta_1) = \sigma^2/N\) (the constant risk of the sample mean), while \(R_{\max}(\delta_2) = +\infty\) and \(R_{\max}(\delta_3) = +\infty\) (since the squared-bias terms blow up as \(|\theta - \theta_0| \to \infty\)). On the real line, \(\delta_1\) is therefore the minimax rule among these three candidates, not because it is uniformly best but because it is the only one with bounded worst-case risk.

A deep result in classical decision theory connects the two optimality criteria. Under suitable regularity conditions (compactness of \(\Theta\), continuity of the loss, and existence of a so-called least favorable prior, the prior against which the smallest attainable Bayes risk is largest), every minimax rule arises as a Bayes rule against this least favorable prior. The full statement and proof require additional machinery (saddle-point theory, complete-class theorems) and are beyond the scope of this page.

Classification (Zero-One Loss)

Perhaps the most common application of Bayesian decision theory in machine learning is classification. Given an input \(\boldsymbol{x} \in \mathcal{X}\), we wish to assign it the optimal class label. We now return to the discrete-label setting of the introduction (rather than the continuous-parameter setting of the previous two sections), where the state of nature is the unknown true class label. To apply the general framework, we specify the states, actions, and loss function for this particular setting.

Suppose that the states of nature correspond to class labels \[ \mathcal{H} = \mathcal{Y} = \{1, \ldots, C\}, \] and that the possible actions are also the class labels: \(\mathcal{A} = \mathcal{Y}\). A natural loss function in this context is the zero-one loss, which penalizes misclassification equally regardless of which classes are confused.

Notation. In this classification context, we use the starred symbol \(y^*\) to emphasize the true (but unknown) class label, and \(\hat{y}\) for the predicted label. This parallels the estimator notation \(\hat{\theta}\) for the predicted value of an unknown parameter. The asterisk plays a purely notational role here and does not change the meaning of the symbol.

Under the zero-one loss, the posterior expected loss for choosing label \(\hat{y}\) becomes \[ \rho(\hat{y} \mid \boldsymbol{x}) = p(\hat{y} \neq y^* \mid \boldsymbol{x}) = 1 - p(y^* = \hat{y} \mid \boldsymbol{x}). \] Thus, minimizing the expected loss is equivalent to maximizing the posterior probability: \[ \pi^*(\boldsymbol{x}) = \arg\max_{y \in \mathcal{Y}} \, p(y \mid \boldsymbol{x}). \]

In other words, the optimal decision under zero-one loss is to select the mode of the posterior distribution. We call this the maximum a posteriori (MAP) estimate. Zero-one loss thus supplies the decision-theoretic justification for reporting the most probable class.

The Reject Option

In some scenarios the cost of an incorrect classification may be so high that it is preferable for the system to abstain from making a decision when it is uncertain. Such scenarios include safety-critical applications such as medical diagnosis or autonomous driving. The reject option formalizes this abstention.

Under this approach, the set of available actions is expanded to include a reject action: \[ \mathcal{A} = \mathcal{Y} \cup \{0\}, \] where action \(0\) represents the reject option, that is, declining to assign a label. The loss function is then defined as \[ l(y^*, a) = \begin{cases} 0 & \text{if } y^* = a \text{ and } a \in \{1, \ldots, C\} \\\\ \lambda_r & \text{if } a = 0 \\\\ \lambda_e & \text{otherwise} \end{cases} \] where \(\lambda_r\) is the cost of the reject action and \(\lambda_e\) is the cost of a classification error.

For the reject option to be meaningful, we require \(0 \lt \lambda_r \lt \lambda_e\). Otherwise rejecting is either free (and we would always reject) or more expensive than guessing (and we would never reject).

Under this framework, instead of always choosing the label with the highest posterior probability, the optimal policy chooses a label only when the classifier is sufficiently confident: \[ a^* = \begin{cases} \hat{y}^* & \text{if } p^* \gt \lambda^* \\\\ \text{reject} & \text{otherwise} \end{cases} \] where \[ \begin{align*} \hat{y}^* &:= \arg\max_{y \in \{1, \ldots, C\}} p(y \mid \boldsymbol{x}), \\\\ p^* &:= p(\hat{y}^* \mid \boldsymbol{x}) = \max_{y \in \{1, \ldots, C\}} p(y \mid \boldsymbol{x}), \\\\ \lambda^* &:= 1 - \frac{\lambda_r}{\lambda_e}. \end{align*} \]

Confusion Matrix

The Bayes estimator tells us what the optimal decision rule is, but in practice, we need to measure how well a classifier actually performs on real data. The standard tool for this is the (class) confusion matrix, which provides a complete summary of the outcomes of classification decisions. We recall the four possible outcomes from our treatment of Type I and Type II errors in statistical inference and hypothesis testing.

For binary classification (\(y \in \{0, 1\}\)), each prediction falls into one of four categories:

• True Positives (TP): instances correctly classified as positive.
• True Negatives (TN): instances correctly classified as negative.
• False Positives (FP): instances incorrectly classified as positive (Type I error).
• False Negatives (FN): instances incorrectly classified as negative (Type II error).

Understanding the distinction between FP and FN is critical because the costs associated with each error type may differ significantly depending on the application. In safety-critical systems, a false negative (missing a dangerous condition) is typically far more costly than a false positive (raising an unnecessary alarm). For example, in medical screening, a false positive means unnecessarily alarming a healthy patient, whereas a false negative means failing to detect a disease in someone who needs treatment.

Notation. In the confusion matrix below, \(P\) and \(N\) denote the total counts of actual positive and actual negative instances respectively (with the hat versions \(\hat{P}\) and \(\hat{N}\) for the predicted totals), and \(M = P + N\) is the total number of labeled examples. These symbols are local to this evaluation context and should not be confused with the probability operator \(P(\cdot)\) of earlier pages.

In the context of Bayesian decision theory, the confusion matrix quantifies the empirical performance of the Bayes estimator (or any other decision rule) by counting how often the predicted labels match or mismatch the true labels.

In practice, a binary classifier typically outputs a probability \(p(y = 1 \mid \boldsymbol{x})\), and the final prediction depends on a decision threshold \(\tau \in [0, 1]\). For any fixed threshold \(\tau\), the decision rule is \[ \hat{y}_{\tau}(\boldsymbol{x}) = \mathbb{1}\{p(y = 1 \mid \boldsymbol{x}) \geq \tau\}. \]

Given a set of \(M\) labeled examples, we can compute the empirical counts for each cell of the confusion matrix. For example, \[ \begin{align*} \text{FP}_{\tau} &= \sum_{n=1}^{M} \mathbb{1}\{\hat{y}_{\tau}(\boldsymbol{x}_n) = 1, \, y_n = 0\},\\\\ \text{FN}_{\tau} &= \sum_{n=1}^{M} \mathbb{1}\{\hat{y}_{\tau}(\boldsymbol{x}_n) = 0, \, y_n = 1\}. \end{align*} \]

Since the confusion matrix depends on the choice of \(\tau\), different thresholds lead to different trade-offs between error types. How should we choose \(\tau\), and how can we evaluate a classifier's performance across all possible thresholds? This motivates the ROC and PR curves developed in the following sections.

Receiver Operating Characteristic Curves

Rather than evaluating a classifier at a single threshold, we can characterize its performance across all thresholds simultaneously. The receiver operating characteristic (ROC) curve achieves this by plotting two rates derived from the confusion matrix. To obtain these rates, we normalize the confusion matrix per row, which yields the conditional distribution \(p(\hat{y} \mid y)\). Since each row sums to 1, these rates describe the classifier's behavior separately within the positive and negative populations.

Notation. Each rate below is defined as an empirical ratio of counts. For large \(M\), the law of large numbers identifies it with the corresponding population conditional probability, which we write as \(p(\hat{y}_\tau = \cdot \mid y = \cdot)\) using \(\tau\) as a subscript on the prediction rather than as a conditioning event (since \(\tau\) is a fixed parameter, not a random variable).

• True positive rate (TPR) (or Sensitivity / Recall): \[ \text{TPR}_{\tau} := \frac{\text{TP}_{\tau}}{\text{TP}_{\tau} + \text{FN}_{\tau}} \approx p(\hat{y}_\tau = 1 \mid y = 1). \]
• False positive rate (FPR) (or Type I error rate / Fallout): \[ \text{FPR}_{\tau} := \frac{\text{FP}_{\tau}}{\text{FP}_{\tau} + \text{TN}_{\tau}} \approx p(\hat{y}_\tau = 1 \mid y = 0). \]
• False negative rate (FNR) (or Type II error rate / Miss rate):\[ \text{FNR}_{\tau} := \frac{\text{FN}_{\tau}}{\text{TP}_{\tau} + \text{FN}_{\tau}} \approx p(\hat{y}_\tau = 0 \mid y = 1). \]
• True negative rate (TNR) (or Specificity):\[ \text{TNR}_{\tau} := \frac{\text{TN}_{\tau}}{\text{FP}_{\tau} + \text{TN}_{\tau}} \approx p(\hat{y}_\tau = 0 \mid y = 0). \]

By plotting TPR against FPR as a parametric curve in \(\tau\), we obtain the ROC curve. The overall quality of a classifier is often summarized using the AUC (Area Under the Curve). A higher AUC indicates better discriminative ability aggregated over all threshold values, with a maximum of 1.0 for a perfect classifier.

The figure below compares two classifiers, one trained using logistic regression and the other using a random forest. Both classifiers provide predicted probabilities for the positive class, so we can vary \(\tau\) and compute the corresponding TPR and FPR.

A diagonal line is also drawn. It represents a random classifier, that is, one that assigns labels purely by chance. On this line, the TPR equals the FPR at every threshold. If a classifier's ROC curve lies on this diagonal, it means the classifier is performing no better than random guessing. In contrast, any performance above the diagonal indicates that the classifier is capturing some signal, while performance below the diagonal (rare in practice) would indicate worse-than-random behavior.

In our demonstration, the logistic regression model has been intentionally made worse and yields an AUC of 0.78, while the random forest shows superior performance with an AUC of 0.94. These results mean that, overall, the random forest is much better at distinguishing between the positive and negative classes compared to the underperforming logistic regression model.

(Data: 10,000 samples, 20 total features, 5 features are informative, 2 clusters per class, 5% label noise.)

Equal Error Rate (EER)

The ROC curve provides a comprehensive view of the trade-off between TPR and FPR, but it can be useful to summarize this trade-off with a single number. The equal error rate (EER) is the operating point where \(\text{FPR} = \text{FNR}\), the threshold at which the two error rates are symmetric. EER is a practical operating-point summary that does not assume a particular cost structure or class prior. It is determined by the two error rates alone.

Note. The EER is generally not the Bayes-optimal threshold. Under equal misclassification costs, the Bayes-optimal threshold corresponds to a posterior probability of \(1/2\) whatever the class prior, since the posterior already accounts for it, and it depends on the calibration of the classifier's score, not on the FPR/FNR symmetry.

EER is particularly useful in applications such as biometric authentication (fingerprint or face recognition), where a single interpretable operating point is desired for system reporting.

The figure below shows the EER point for our two models, that is, the point where the FPR and FNR curves intersect:

A perfect classifier would achieve an EER of 0, which corresponds to the top-left corner of the ROC curve. In practice, one may tune the decision threshold to operate at or near the EER, depending on the application's requirements.

The ROC curve and EER evaluate a classifier by conditioning on the true class (that is, how the classifier behaves within each population). However, in many practical settings we care about a different question: given that the classifier predicted positive, how likely is it to be correct? This leads us to precision-recall analysis.

Precision-Recall (PR) Curves

Here, we normalize the confusion matrix per column to obtain \(p(y \mid \hat{y})\), which conditions on the predicted label. The column-normalized confusion matrix answers a complementary question: among the instances predicted as positive (or negative), what fraction are truly positive (or negative)? This perspective is especially important when precision is critical, such as in medical diagnosis or fraud detection.

• Positive predictive value (PPV) (or Precision):\[ \text{PPV}_{\tau} := \frac{\text{TP}_{\tau}}{\text{TP}_{\tau} + \text{FP}_{\tau}} \approx p(y = 1 \mid \hat{y}_\tau = 1). \]
• False discovery rate (FDR):\[ \text{FDR}_{\tau} := \frac{\text{FP}_{\tau}}{\text{TP}_{\tau} + \text{FP}_{\tau}} \approx p(y = 0 \mid \hat{y}_\tau = 1). \]
• False omission rate (FOR):\[ \text{FOR}_{\tau} := \frac{\text{FN}_{\tau}}{\text{FN}_{\tau} + \text{TN}_{\tau}} \approx p(y = 1 \mid \hat{y}_\tau = 0). \]
• Negative predictive value (NPV):\[ \text{NPV}_{\tau} := \frac{\text{TN}_{\tau}}{\text{FN}_{\tau} + \text{TN}_{\tau}} \approx p(y = 0 \mid \hat{y}_\tau = 0). \]

Note that within each column, the rates sum to 1: \(\text{PPV} + \text{FDR} = 1\) and \(\text{FOR} + \text{NPV} = 1\).

To summarize a system's performance, we often use a precision-recall (PR) curve. It is especially informative when classes are imbalanced (that is, when the positive class is rare) or when false positives and false negatives have different costs. This curve plots precision against recall as the decision threshold \(\tau\) varies.

Let precision be \(\mathcal{P}(\tau)\) and recall be \(\mathcal{R}(\tau)\). For binary labels \(\hat{y}_n, y_n \in \{0, 1\}\), at threshold \(\tau\) these are computed as \[ \begin{align*} \mathcal{P}(\tau) &:= \frac{\sum_n y_n \hat{y}_n}{\sum_n \hat{y}_n} = \text{PPV}_\tau, \\\\ \mathcal{R}(\tau) &:= \frac{\sum_n y_n \hat{y}_n}{\sum_n y_n} = \text{TPR}_\tau, \end{align*} \] using \(\mathcal{P}, \mathcal{R}\) (calligraphic) for the precision and recall functions of \(\tau\). These are the same quantities as the \(\text{PPV}_\tau\) and \(\text{TPR}_\tau\) introduced earlier, renamed in keeping with standard machine-learning terminology and to emphasize their role as functions of \(\tau\) when plotting curves. (The calligraphic \(\mathcal{P}\) is local to this section and is unrelated to the count \(P\) of actual positives or to the probability operator.)

By plotting \(\mathcal{P}(\tau)\) against \(\mathcal{R}(\tau)\) as \(\tau\) varies, we obtain the PR curve.

A crucial difference between the ROC and PR curves is the baseline for a random classifier. While a random classifier yields an AUC of 0.5 in expectation on an ROC curve, the baseline for a PR curve is the fraction of positive samples in the dataset: \[ y_{\text{baseline}} = \frac{P}{P + N}. \]

To see why, consider a random classifier that predicts positive with a constant probability \(q\) (independent of the input). On average it achieves \(\text{TP} = qP\) and \(\text{FP} = qN\), so its precision is \(\text{TP}/(\text{TP}+\text{FP}) = qP/(qP + qN) = P/(P+N)\) regardless of \(q \in (0, 1]\). Varying \(q\) sweeps recall over \((0, 1]\) while precision stays constant, and so traces a horizontal line at this baseline. This makes the PR curve a much more informative evaluation tool for highly imbalanced datasets, as the "no-knowledge" score can be near zero while the "perfect" score remains 1.0.

However, raw precision values can be noisy as the threshold varies. To stabilize this, interpolated precision is often computed. For a given recall level \(r\), it is defined as the maximum precision observed for any recall level greater than or equal to \(r\): \[ \mathcal{P}_{\text{interp}}(r) := \max_{r' \geq r} \mathcal{P}(r'). \] One common definition of the average precision (AP) is the area under this interpolated PR curve. It provides a single-number summary that reflects the classifier's ability to maintain high precision as recall increases (that is, as the decision threshold \(\tau\) is lowered).

In our case, the logistic regression produced an AP of 0.73, while the random forest achieved a much stronger AP of 0.93. This indicates that the random forest is significantly more robust at identifying positive cases without sacrificing precision.

Note. In settings where multiple PR curves are generated (for example, one for each query in information retrieval or one per class in multi-class classification), the mean average precision (mAP) is computed as the mean of the AP scores over all curves. mAP offers an overall performance measure across multiple queries or classes.

Class Imbalance

In many real-world applications, the class distribution is far from uniform. A dataset is considered imbalanced when one class has significantly fewer examples than the other, for instance 5% positive samples and 95% negative samples. In such cases, naive metrics like accuracy can be highly misleading. A model that always predicts the majority class achieves high accuracy without correctly identifying any minority class instances. How do ROC-AUC and PR-AP behave differently under class imbalance?

The ROC-AUC metric is relatively insensitive to class imbalance at the population level, because both TPR and FPR are computed within their respective populations. TPR is a ratio within the positive samples and FPR is a ratio within the negative samples. Consequently, the class proportions do not directly affect the population ROC curve. (In practice, however, empirical AUC estimates can become noisy when one class is severely under-represented, since the minority sample provides few examples for stable estimation.)

On the other hand, the PR-AP metric is more sensitive to class imbalance. This is because precision depends on both populations: \[ \text{Precision} = \frac{\text{TP}}{\text{TP} + \text{FP}}. \] When the negative class vastly outnumbers the positive class, even a small false positive rate can produce a large absolute number of false positives, significantly reducing precision.

To demonstrate this effect, we create a dataset where 90% of the samples belong to the negative class and only 10% belong to the positive class, then train both the logistic regression and random forest models again:

ROC Curve (Imbalanced)

Precision-Recall Curve (Imbalanced)

The ROC curves remain relatively smooth and the AUC does not drop drastically, even though the dataset is highly imbalanced. The PR curves, however, reveal a distinct difference, with noticeably lower AP scores. This highlights how class imbalance makes it harder to achieve high precision and recall simultaneously. Even though the random forest outperforms logistic regression, it still struggles to detect the rare positive cases effectively.

In summary, PR curves focus on precision and recall, which directly reflect how well a model identifies the minority class. Precision, in particular, is sensitive to even a small number of false positives, and so gives a more realistic picture of performance when the positive class is scarce. Thus, PR-AP is often the preferred metric when the focus is on correctly identifying the minority class, for example in fraud detection, rare-disease screening, or anomaly detection. When both classes are of comparable practical importance, ROC-AUC remains an informative summary.