In the development of the python confidence interval library, for the analytic confidence intervals of some metrics I’ve been relying on results from the remarkable Confidence interval for micro-averaged F1 and macro-averaged F1 scores paper by Kanae Takahashi, Kouji Yamamoto, Aya Kuchiba and Tatsuki Koyama.

In the paper they derive confidence intervals for Micro F1 and Macro F1 (and by extension to Micro Precision/Recall, since they are equal to Micro F1).

However there are a few common variants that the paper didn’t address:

The next sections derive the confidence intervals for these missing metrics in the spirit of the paper above, using the delta method. Some of the sections are a bit more verbose than in the paper that (elegently) combines some steps together, however I found it helpful to break it down a bit more.

These were implemented in the python confidence interval library.

The computation flow for confidence intervals with the delta method

This is the computation flow we’re going to go through:

  1. Approximate the (normal) distribution of the confusion matrix probabilities \(p_{ij}\).
  2. Express the metrics as functions of \(p_{ij}\): \(metric(p_{ij})\).
  3. Use the delta method approximate the (normal) distrubution of \(metric(p_{ij})\).
  4. Plug in our estimate of \(p_{ij}\) based on the observed data, and get the variance of \(metric(p_{ij})\)
  5. Once we have the variance, we can get the confidence interval lower and upper bounds using the gaussian distribution.

The confusion matrix multinomial distribution

\(C_{ij}\) is the confusion matrix: the number of predictions with the ground truth category i, that were actually predicted as j. Note that here wee keep the scikit-learn notation, intead of the notation in the paper that’s transposed. We have an actual observed confusion matrix \(\hat{C_{ij}}\), but we assumed it was sampled from a distribution \(C_{ij}\).

The core assumption here is that \(C_{ij}\) has a multinomial distribution with parameters \(p_{ij}\).

\[E(C_{ij}) = n p_{ij}\] \[Cov(C_{ij}, C_{ij}) = Var(C_{ij}) = np_{ij}(1-p_{ij})\]

And when ij != kl:

\[Cov(C_{ij}, C_{kl}) = -np_{ij}p_{kl}\]

By combining the two cases above, the covariance matrix of the multinomial distribution can be written as: \(Cov(C_{ij}, C_{kl}) = n * [diag(p)_{ij} - (pp^T)_{ij}]\)

We don’t know what \(p_{ij}\) actually is. But our best guess for it, the maximum likelihood estimator, is:

\(\hat{p_{ij}} = \frac{C_{ij}}{n}\).

n = \(\sum_{ij}C_{ij}\), is the total number of predictions.

The central limit theorem applied on the confusion matrix

\(C_{ij}\) can be seen as the sum of n individual trial binary variables \(X_{ij}\), where \(X_{ij}=1\) with probability \(p_{ij}\).

\[\hat{p_{ij}} = \frac{\sum_{k=1}^{N}{X_{ijk}}}{n} = \frac{\hat{C_{ij}}}{n}\]

From the central limit theorem, since \(\hat{p_{ij}}\) is the average of many variables, we know that it has a normal distribution. We also know it’s mean and covariance, since \(\hat{p_{ij}} = \frac{\hat{C_{ij}}}{n}\) and we know from above what the distribution of \(C_{ij}\) is.

\[E[\hat{p}] = \frac{E[C]}{n}, Cov(\hat{p}) = \frac{Cov(C)}{n^2}\] \[\hat{p_{ij}} \sim Normal(E[p], \frac {diag(p) - (pp^T)]}{n}\]

Expressing the metrics in terms of the confusion matrix

  • Binary F1: \(metric(p) = F1_{binary} = \frac {2p_{11} }{2p_{11} + p_{01} + p_{10}} = \frac {2p_{11} }{d}\)

  • Macro Recall: \(metric(p) = R = \frac{1}{r}\sum_{i=1}^{r} \frac{p_{ii}}{\sum_j{p_{ij}}}\)

  • Macro Prcecision: \(metric(p) = P = \frac{1}{r}\sum_{i=1}^{r} \frac{p_{ii}}{\sum_j{p_{ji}}}\)

\(r\) = number of categories

If we plug in our estimate for p, we get the (point estimation of the) metric.

The multi-variate delta method

The various metrics above are functions \(metric(\hat{p})\). We know from above that \(\hat{p}\) has approximately a normal distribution. The multi-variate delta method gives us a recepie to get the distribution of the \(metric(\hat{p})\):

\[metric(\hat{p}) \sim Normal(metric(p), \frac{\partial metric(p)^T}{\partial p} Cov(p) \frac{\partial metric(p)}{\partial p})\]

We also know from above that \(Cov(p) = \frac {diag(p) - pp^T}{n}\)

Now the only thing missing is to compute those derivatives!

Computing the derivative for binary F1

\[f1 = \frac {2p_{11} }{2p_{11} + p_{01} + p_{10}} = \frac {2p_{11} }{d}\]

\(\frac{\partial f1}{\partial p_{10}} = \frac{\partial f1}{\partial p_{01}} = -2\frac {p_{11}} {d^2} = -\frac {f1} {d}\) \(\frac{\partial f1}{\partial p_{11}} = \frac 2 {d} - \frac {4 p_{11}} {d^2} = \frac {2(1-f1)} {d}\)

The code can be found here.

Computing the derivative for Macro Recall

\(metric(p) = R = \frac{1}{r}\sum_{i=1}^{r} \frac{p_{ii}}{\sum_j{p_{ij}}} =\frac{1}{r}\sum_{i=1}^{r} \frac{p_{ii}}{d_i} = \frac{1}{r}\sum_{i=1}^{r}R_i\)

\(\frac{\partial R}{\partial p_{ii}} =\frac{1}{r} [\frac{1}{di} - \frac{p_{ii}}{di^2}] = \frac{1}{r} \frac{1-R_i}{d_i}\) \(\frac{\partial R}{\partial p_{ij}} = -\frac{1}{r} \frac{p_{ii}}{d_i^2} = -\frac{1}{r} \frac{R_i}{d_i}\)

In terms of computation, the non diagonal row elements will all be the same expression of the \(R_i\) of that row.

The code can be found here

Computing the derivative for Macro Precision

\(metric(p) = P = \frac{1}{r}\sum_{i=1}^{r} \frac{p_{ii}}{\sum_j{p_{ji}}} =\frac{1}{r}\sum_{i=1}^{r} \frac{p_{ii}}{d_i} = \frac{1}{r}\sum_{i=1}^{r}P_i\)

\[\frac{\partial P}{\partial p_{ii}} =\frac{1}{r} [\frac{1}{di} - \frac{p_{ii}}{di^2}] = \frac{1}{r} \frac{1-P_i}{d_i}\] \[\frac{\partial p}{\partial p_{ji}} = -\frac{1}{r} \frac{p_{ii}}{d_i^2} = -\frac{1}{r} \frac{P_i}{d_i}\]

Note how for precision we derive for \(p_{ji}\) instead of \(p_{ij}\). In terms of computation, the non diagonal columns elements will all be the same expression of the \(P_i\) of that column.

The code can be found here