# Can precision and recall of a DNN trained on human-labeled data be higher than precision and recall of the humans who labeled the data?

I was discussing Deep Learning with an academic statistician, who was criticizing the field as "lacking scientific rigor, overhyped and delivering results which are way worse than claimed". In particular, we were discussing whether a DNN classifier could be more accurate than the human experts who labeled the dataset on which it had been trained. The statistician dismissed this occurrence as "obviously impossible", and dismissed any such claims from the Machine Learning community as a clear indication that "you guys don't know what you're talking about [sic]". I replied that I couldn't see any mathematical argument for why a ML classifier couldn't in theory match the Bayes rate$$^*$$, or, put it differently, why a classifier trained on a training set $$T$$ should always be less accurate, on a test set $$E$$, than the human(s) who labeled $$T$$. All humans (experts are no exception) can and do make mistakes in labeling data, and it's entirely possible that the classifier could learn to "ignore" label noise. However, he countered that "given noisy labels, we [statisticians] know how to estimate the optimal error rate using inter-annotator error estimation [this quote may not be verbatim...I'm not 100% sure what he was referring to, but I think I got the main idea], and once this is done properly, there's no way a ML classifier could ever beat our best estimate of human accuracy".

Who is right? I think there are actually two different issues here.

1. Sometimes, the human experts who labeled the data and the humans we compare against are different. Consider this Nature paper which claims superior precision and recall (with respect to human experts) for malignant cancer detection in breast cancer mammography:

Compared to the first reader, the AI system demonstrated a statistically significant improvement in absolute specificity of 1.2% (95% confidence interval (CI) 0.29%, 2.1%; $$P$$=0.0096 for superiority) and an improvement in absolute sensitivity of 2.7% (95% CI −3%, 8.5%; $$P$$= 0.004 for non-inferiority at a pre-specified 5% margin; [..]

Compared to the typical reader, the AI system demonstrated statistically significant improvements in absolute specificity of 5.7%(95% CI 2.6%, 8.6%; $$P$$<0.001) and in absolute sensitivity of 9.4% (95% CI 4.5%, 13.9%; $$P$$<0.001; [..])

Reading the rest of the paper, it seems that the mammographies had not been labeled using the readers' (radiologists) opinions, but using the outcomes of biopsies, performed later in time. Thus, the classifier was trained on "non-noisy" labels (assuming the error rate of biopsies is order of magnitudes than the error rate of mammography-based screening). In this case, it's entirely possible that one could train a model able to beat the human reader's precision and recall.

2. "Human experts' accuracy" (or "human experts' precision and recall", for what it matters) is an ill-defined concept. Different experts have different precision and recall, and given the average size of most ML datasets, it's pretty obvious that usually single human annotates the whole dataset. Thus, in order to define a single performance to compare against the classifer's precision and recall, we need to "average" the performances of different humans in some way. I don't know how this averaging is performed, but let's assume (a BIG assumption) that the precision/recall points corresponding to different humans fall on a concave curve (ROC curves are often concave, however this is not an actual ROC curve, but a collection of points from different curves, it may well be not concave...). Then, I think averaging these points would result in a point which is below the curve (not sure how to prove it).

What do you think? Who is right?

$$^*$$: of course, I'm not expecting a ML classifier to actually match the Bayes error rate in practice, but then again, I'm not expecting a human to match it either!

• I remember discussing AI with a finance professor who remarked that he had seen it used in medical diagnosis. The response of the doctors was often, “ I hadn’t thought of that, but that’s probably right.” (On that note, while this question is interesting and certainly has a statistical component (+1), it might be a better fit on the AI stack.)
– Dave
Jan 4, 2020 at 16:55
• Hi @Dave, you state: "The response of the doctors was often, “ I hadn’t thought of that, but that’s probably right.” The response to what? To whether AI could or could not ever beat human experts (doctors)? Jan 4, 2020 at 18:08
• The diagnosis and/or treatment plan, something like, “AI says the patient has disease X with the best treatment being option Y. I knew it was disease X, but I was thinking to treat with Z. Looking at the patient’s family history more, though, yeah, treatment Y is the better option.”
– Dave
Jan 4, 2020 at 18:11
• @Dave ah, that's impressive! I didn't know about that. Jan 4, 2020 at 18:34
• Just keep in mind that this is at the anecdotal level haha (though he most certainly told me that)
– Dave
Jan 4, 2020 at 19:19

Regarding "human level labeling" versus "human level precision/recall" -- in addition to the examples you mentioned, some possible differences are

1. The data is labeled redundantly by 3 or 5 different labelers, and the majority label is used, if there's disagreement.

2. The data is "prelabeled" with some model, and labelers just need to find mistakes and correct them. This would make a big difference if it's hard to correctly annotate, but easy to confirm the correctness of an annotation.

The other, orthogonal part of the question would be whether noisy labels puts some bound on the optimal error rate. I don't know any mathematical justification for this one way or the other, but for a thought experiment, if I learned of a new animal species, say a "catdog", and was shown images of catdogs and non-catdogs with some small amount of label error, it seems very likely i'd be able to figure out which labels were incorrect and then afterwards achieve nearly optimal accuracy on the task (assuming catdogs are quite visually distinct from cats, dogs, and other animals).

While I'm sure this is subject to some no-free-lunch type theorem (ranging over all possible classification problems, I surely can't always separate out what is label noise and what's actually part of the class), it seems for most well-behaved problems this isn't an issue.

• Apologies for the late feedback. If you can add something on the perfect labels case (no noise) I think the answer would be perfect. Jan 9, 2020 at 14:35

a DNN classifier could be more accurate than the human experts who labeled the dataset on which it had been trained

The bias-variance decomposition says this observation could be explained by exactly two, non-mutually exclusive properties (other than chance): the model has lower bias, or it has lower variance. Let's set up a toy problem to illustrate that this mechanism in particular—

All humans (experts are no exception) can and do make mistakes in labeling data, and it's entirely possible that the classifier could learn to "ignore" label noise

—is the lower variance one.

Say the ground-truth (not human labels), $$Y$$, were generated like this:

$$$$Y = \beta X + \epsilon$$$$

where $$\epsilon$$ has mean $$0$$ and variance $$\sigma_\epsilon^2$$.

Say the human labels $$Y'$$ were generated like this:

$$$$Y' = \beta' X + \epsilon'$$$$

where $$\epsilon'$$ has mean $$0$$ and variance $$\sigma_{\epsilon'}^2$$. You'll see that there are two independent sources of error for the human: $$\beta' \neq \beta$$, and $$\sigma_{\epsilon'}^2 > 0$$. The former error implies the human labels are systematically incorrect, and the latter error means there's label noise.

A linear model is trained on $$\mathcal{T} \sim f(x, y')$$ via least squares. Call its outputs/predictions $$\hat{Y}$$. Here are the bias-variance decompositions for $$Y'$$ and $$\hat{Y}$$ averaged across out-of-sample/independent inputs $$x$$, as well as training sets $$\mathcal{T}$$ (so not conditional on a specific training set):

\begin{align} \text{MSE}(Y, Y') &= \text{E}_{x, \mathcal{T}} \big[ \text{E}_{\epsilon}((Y - Y')^2 \: | \: X=x, \mathcal{T} ) \big] \\ &= (\beta - \beta')^2 + \sigma_{\epsilon}^2 + \sigma_{\epsilon'}^2. \\ \text{MSE}(Y, \hat{Y}) &= \underbrace{(\beta - \beta')^2}_{\text{Bias}^2} + \underbrace{\sigma_{\epsilon}^2}_{\text{Noise}} + \underbrace{\frac{\sigma_{\epsilon'}^2}{n}(1)}_{\text{Variance}}. \end{align}

(Full derivation for the expected variance of $$\hat{Y}$$: slide 57 of this deck.)

(And here's a simulation which checks the math above. After clicking the link, click "Open with Google Colaboratory" to run it online.)

B/c the least-squares estimator is unbiased and the human is linear, the biases of the human and the model are the same. But looking at the variances, there's an easy win: theoretically, only $$n = 2$$ examples are needed from the human...to beat the human! Just enough to draw a line. The intuition to take away from this exercise is that if a model's bias is not high (likely the case for many DNNs) and human labels are quite noisy, a model can become more accurate than humans by training it on a lot of data.

The bias approach to beating the human is not so illuminating to me. It's rare for an ML practitioner to deliberately choose a model (or manipulate their data) b/c they think it has lower bias than whatever model the human expert used in their head. At that point such a practitioner is something of a domain expert, or is closely collaborating w/ one. But still worth mentioning that reducing bias is another way to beat human labels.

P.S. I'm not exactly sure why precision and recall in particular are relevant. It seems reasonable that having more data could theoretically cause any reasonable score to increase.