Probing Classifiers › Selectivity and What It Really Measures

The selectivity framework addresses an important technical concern. Suppose your probe achieves 90% accuracy but the control probe also achieves 80% accuracy. Does the 10% selectivity gap represent "real" syntactic knowledge? Or does it just reflect that the real labels happen to cluster in the representation space slightly better than random labels due to some incidental correlation?

The honest answer is that selectivity is a relative measure, not an absolute one. It tells you how much better the probe does on the real task compared to random labels, controlling for the probe's capacity to memorize. A high selectivity score provides positive evidence. A low selectivity score is more ambiguous: it could mean the information is absent, or it could mean the probe can memorize both real and random labels equally well.

This has led researchers to use selectivity alongside other measures. Cross-linguistic generalization tests whether a probe trained on English generalizes to French syntactic labels, which is less likely to be exploiting idiosyncratic English corpus statistics. Counterfactual analysis checks whether removing the relevant syntactic information from the input drops probe accuracy while removing irrelevant features does not. MDL probing (Voita and Titov 2020) measures compression gain, giving a cleaner signal than accuracy alone.

The MDL framework is worth understanding in more depth. The codelength of a probe is the number of bits it takes to transmit the labels given the representations under the probe's model. A probe that assigns probability $p_\theta(y_i \mid \mathbf{h}_i)$ to the correct label for each example uses $-\log_2 p_\theta(y_i \mid \mathbf{h}_i)$ bits for that example. The total codelength is:

$$L_\theta = -\sum_{i=1}^{N} \log_2 p_\theta(y_i \mid \mathbf{h}_i)$$

The compression ratio compares this to the uniform code (which assigns equal probability to all labels):

$$\text{compression} = \frac{N \log_2 K}{L_\theta}$$

where:

• $K$ is the number of classes
• $N$ is the number of examples
• A compression ratio greater than 1 means the representations help; a ratio close to 1 means they do not

This ratio is invariant to the number of examples and classes, making it directly comparable across tasks and datasets.