What is the difference between an accuracy measure and an error metric?

Question

The two concepts are distinct in measure theory. Nonetheless, moving out from measure theory, the two terms are often used interchangeably. To most forecasters, especially forecast practitioners, they both refer to functions of forecast errors, e.g., MBE, MAE, or RMSE.

I am hoping somebody could elaborate on the potential harms and pitfalls when the two terms are mixed up.

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A measure $\mu$ on a set $X$ is a mapping $\mu:\mathcal{A}\rightarrow[0,\infty]$ defined on a $\sigma$-algebra $\mathcal{A}$ that satisfies non-negativity, null empty set, and $\sigma$-additivity, that is $\mu(A) \ge 0 \,\, \forall \,\, A \in \mathcal{A}$, $\mu( \emptyset)=0$, and $\mu( \sqcup_{j \in \mathbb{N}} A_j) = \sum_{j \in \mathbb{N}} \mu(A_j)$, where symbol $\sqcup$ denotes disjoint union. On the other hand, a metric is a distance measure $d:X\times X \rightarrow [0,\infty]$ that satisfies definiteness, symmetry, and triangle inequality, that is $d(x,y) = 0$ iff $x = y$, $d(x,y) = d(y,x)$, and $d(x,y)\le d(x,z) + d(z,y)$, $\forall$ $x,y,z\in X$.

I am not so sure when to use the word "measure," and when to use "metric."

Answer 1

I do not think that there is a lot of potential confusion stemming from the difference between "errors" and "accuracy", as long as people explain what measure they are using. If they note that they are using the MAPE, they can discuss "lower error" and "higher accuracy", and everyone will know which way the MAPE moved. Similarly if they prefer working with "accuracy" and define it as 1-MAPE. As you write, both terms are used in the forecasting community, and I haven't noted any particular confusion in the 13+ years I have been active there.

Difference between forecasting accuracy and forecasting error? may be useful, or possibly earlier threads in which I have pontificated about forecast error or accuracy, or even a couple of popular articles I have written on forecast accuracy measurement, e.g., Kolassa & Schütz (2007) and Kolassa (2008) and Kolassa (2009) and Kolassa & Martin (2011), also Kolassa (2016) and a forthcoming commentary on the M4 competition.

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"Accuracy measure" uses the word "measure" in its everyday sense, not in the mathematical measure theoretical way. Similarly, "error metric" uses "metric" in its everyday sense, not in the sense of a metric on a mathematical set.