# Classification

## Computer says no Distinguishing whether a thing was generated from distribution A or B. This is a learning-theory, loss-minimisation framing; We might also consider probability distributions of categories.

Mostly this page is a list of classification target losses to assess classifiers.

## Classification loss zoo

Surprisingly subtle. ROC, AUC, precision/recall, confusion…

One of the less abstruse summaries of these is the scikit-learn classifier loss page, which includes both formulae and verbal descriptions. The Pirates guide to various scores provides an easy introduction.

### Matthews correlation coefficient

Due to Matthews (1975). This is the first choice for seamlessly handling multi-label problems within a single algorithm since its behaviour is reasonable for 2 class or multi class, balanced or unbalanced, and it’s computationally cheap. Unless you have a vastly different importance for your classes, this is a good default.

However, it is not differentiable with respect to classification certainties, so you can’t use it as, e.g., a target loss in neural nets; Therefore you use surrogate measures which are differentiable and use this to track your progress.

#### 2-class case

Take your $$2 \times 2$$. confusion matrix of true positive, false positives etc.

${\text{MCC}}={\frac {TP\times TN-FP\times FN}{{\sqrt {(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}}}$

$|{\text{MCC}}|={\sqrt {{\frac {\chi ^{2}}{n}}}}$

#### Multiclass case

Take your $$K \times K$$ confusion matrix $$C$$, then

${\text{MCC}}={\frac {\sum _{k}\sum _{l}\sum _{m}C_{kk}C_{lm}-C_{kl}C_{mk}}{{\sqrt {\sum _{k}(\sum _{l}C_{kl})(\sum _{k'|k'\neq k}\sum _{l'}C_{k'l'})}}{\sqrt {\sum _{k}(\sum _{l}C_{lk})(\sum _{k'|k'\neq k}\sum _{l'}C_{l'k'})}}}}}$

### ROC/AUC

Receiver Operator Characteristic/Area Under Curve. Supposedly dates back to radar operators in WWII. The graph of the false versus true positive rate as the criterion changes. Matthews (1975) talk about the AUC for radiology; Supposedly Spackman (1989) introduced it to machine learning, but I haven’t read the article in question. Allows us to trade off importance of false positive/false negatives.

### Cross entropy

I’d better write down form for this, since most ML toolkits are curiously shy about it.

Let $$x$$ be the estimated probability and $$z$$ be the supervised class label. Then the binary cross entropy loss is

$\ell(x,z) = -z\log(x) - (1-z)\log(1-x)$

If $$y=\operatorname{logit}(x)$$ is not a probability but a logit, then the numerically stable version is

$\ell(y,z) = \max\{y,0\} - y + \log(1+\exp(-|x|))$

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See Pólya-Gamma.

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## Hierarchical Multi-class classifiers

Read et al. (2021) discusses how to create multi-class classifiers by stacking layers of binary classifiers and using each as a feature input to the next, which is an elegant solution IMO.

## Philosophical connection to semantics

Since semantics is what humans call classifiers.

## Connection to legibility

I do think there is something interesting happening with legibility. States need to classify, apparently. Adversarial classification is my point of entry into this.

## Relative distributions

Why characterise a difference in distributions by a summary statistic? Just have an object which is a relative distribution.

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