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classifiers_as_tests_v2.tex
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classifiers_as_tests_v2.tex
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\pdfoutput=1
%% Author: PGL Porta Mana
%% Created: 2022-04-14T17:18:22+0200
%% Last-Updated: 2022-05-30T00:06:18+0200
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Probabilities for classifier outputs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand*{\propertitle}{A probability transducer\\for machine-learning classifiers}%\\ {\Large}}
% title uses LARGE; set Large for smaller
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\newcommand*{\headtitle}{Probability transducer for classifiers}
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%\hspace*{\stretch{0}}%
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{\protect\centering K. Dyrland \href{https://orcid.org/0000-0002-7674-5733}{\protect\includegraphics[scale=0.16]{orcid_32x32.png}}\\\scriptsize\epost{\email{kjetil.dyrland}{gmail.com}}}%
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\\\footnotesize Dept of Computer science, Electrical Engineering and Mathematical Sciences\\Western Norway University of Applied Sciences, Bergen, Norway
\\\textsuperscript{\ensuremath{\dagger}}\amp\ Mohn Medical Imaging and Visualization Centre, Bergen, Norway
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\noindent \mynotep{}
The present work sells probabilities for \ml\ algorithms at a bargain price. It shows that these probabilities are just what those algorithms need in order to increase their sales.
% \\\noindent\emph{\footnotesize Note: Dear Reader
% \amp\ Peer, this manuscript is being peer-reviewed by you. Thank you.}
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% {\footnotesize PACS: ***}\qquad%
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\section{The inadequacy of common classification approaches}
\label{sec:goal_class}
As the potential of using \ml\ algorithms in important fields such as medicine or drug discovery increases \autocites{lundervoldetal2019,chenetal2018,green2019}, the \ml\ community ought to keep in mind what the actual needs and inference contexts in such fields are. We must avoid trying (intentionally or unintentionally) to convince such fields to change their needs, or to ignore their own contexts just to fit \ml\ solutions that are available and fashionable at the moment. Rather, we must make sure the that solutions fit needs \amp\ context, and amend them if they do not.
The \ml\ mindset and approach to problems such as classification in such new important fields is often still inadequate in many respects. It reflects simpler needs and contexts of many inference problems successfully tackled by machine learning earlier on.
A stereotypical \enquote{cat vs dog} image classification, for instance, has four very important differences from a \enquote{disease \tI\ vs disease \tII} medical classification, or from an \enquote{active vs inactive} drug classification:
\begin{enumerate}[label=(\roman*),wide]
\item\label{item:gain_loss}Nobody presumably dies or loses large amounts of money if a cat image is misclassified as dog or vice versa. But a person can die if a disease is misdiagnosed; huge capitals can be lost if an ultimately ineffective drug candidate is pursued. The \emph{gains and losses} -- or generally speaking the \emph{utilities} -- of correct and incorrect classifications in the former problem and in the two latter problems are vastly different.
\item\label{item:decisions_classes} To what purpose do we try to guess whether an image's subject is a cat or a dog? For example because we must decide whether to put it in the folder \enquote{cats} or in the folder \enquote{dogs}. To what purpose do we try to guess a patient's disease or a compound's chemical activity? A clinician does not simply tell a patient \enquote*{You probably have such-and-such disease. Goodbye!}, but has to decide among many different kinds of treatments. The candidate drug compound may be discarded, pursued as it is, modified, and so on. \emph{The ultimate goal of a classification is always some kind of decision}, not just a class guess. In the cat-vs-dog problem there is a natural one-one correspondence between classes and decisions. But in the medical or drug-discovery problems \emph{the set of classes and the set of decisions are very different}, and have even different numbers of elements.
\item\label{item:optimal_truth} If there is a 70\% probability that an image's subject is a cat, then it is natural to put it in the folder \enquote{cats} rather than \enquote{dogs} (if the decision is only between these two folders). If there is a 70\% probability that a patient has a particular health condition, it may nonetheless be better to dismiss the patient -- that is, to behave as if there was no condition. This is the optimal decision, for example, when the only available treatment for the condition would severely harm the patient if the condition were not present. Such treatment would be recommended only if the probability for the condition were much higher than 70\%. Similarly, even if there is a 70\% probability that a candidate drug is active it may nonetheless be best to discard it. This is the economically most advantageous choice if pursuing a false-positive leads to large economic losses. \emph{The target of a classification is not what’s most probable, but what’s optimal.} \mynotep{add something on proper probabilities, connecting to~\ref{item:determ_statist}}
\item\label{item:determ_statist} The relation from image pixels to house-pet subject may be almost deterministic; so we are effectively looking for or extrapolating a function $\texts{pet} \mo f(\texts{pixels})$ contaminated by little noise. But the relation between medical-test scores or biochemical features on one side, and disease or drug activity on the other, is typically \emph{probabilistic}; so a function $\texts{disease} \mo f(\texts{scores})$
% scores${}\!\mapsto\!{}$disease
or $\texts{activity} \mo f(\texts{features})$
% features${}\!\mapsto\!{}$activity
does not even exist. \emph{We are assessing statistical relationships} $\P(\texts{disease} , \texts{scores})$ or $\P(\texts{activity} , \texts{features})$ instead, which include deterministic ones as special cases.
\end{enumerate}
In summary, there is place to improve classifiers so as to
\begin{enumerate*}[label=(\roman*)]
\item quantitatively take into account actual utilities,
\item separate classes from decisions,
\item target optimality rather than \enquote{truth}, %-- which cannot be targeted because it is unknown.
\item output and use proper probabilities.
\end{enumerate*}
% \begin{enumerate*}[label=(\roman*)]
% \item neglect actual gains and losses or treat them only qualitatively,
% \item neglect proper probabilities and often demotes density regression (statistical inference) to functional regression (interpolation),
% \item confuse classes and decisions,
% \item mistake truth for optimality.
% \end{enumerate*}
\medskip
In artificial intelligence and machine learning it is known how to address all these issues in principle -- the theoretical framework is for example beautifully presented in the first 18 chapters or so of Russell \amp\ Norvig's \parencites*{russelletal1995_r2022} text \autocites[see also]{selfetal1987,cheeseman1988,cheeseman1995_r2018,pearl1988,mackay1995_r2005}.
Issues~\ref{item:gain_loss}--\ref{item:optimal_truth} are simply solved by adopting the standpoint of \emph{Decision Theory}, which we briefly review in \sect~\ref{sec:utility_classification} below and discuss at length in a companion work \autocites{dyrlandetal2022}. In short: The set of decisions and the set of classes pertinent to a problem are separated if necessary. A utility is associated to each decision, relative to the occurrence of each particular class; these utilities are assembled into a utility matrix: one row per decision, one column per class. This matrix is multiplied by a column vector consisting in the probability for the classes. The resulting vector of numbers contains the expected utility of each decision. Finally we select the decision having \emph{maximal expected utility}, according to the principle bearing this name. Such procedure also takes care of the class imbalance problem \autocites[\cf\ the analysis by][(they use the term \enquote{cost} instead of \enquote{utility})]{drummondetal2005}.
Clearly this procedure is computationally inexpensive and ridiculously easy to implement in any \ml\ classifier. The difficulty is that this procedure requires \emph{sensible probabilities} for the classes, which brings us to issue~\ref{item:determ_statist}, the most difficult.
Some machine-learning algorithms for classification, such as support-vector machines, output only a class label. Others, such as deep networks, output a set of real numbers that can bear some qualitative relation to the plausibilities of the classes. But these numbers cannot be reliably interpreted as proper probabilities, that is, as the degrees of belief assigned to the classes by a rational agent \autocites{mackay1992d,galetal2016}[\chaps~2, 12, 13]{russelletal1995_r2022}; or, in terms of \enquote{populations} \autocites{lindleyetal1981}%[\sect~II.4]{fisher1956_r1967}
, as the expected frequencies of the classes in the hypothetical population of units (degrees of belief and frequencies being related by de~Finetti's theorem \autocites[\chap~4]{bernardoetal1994_r2000}{dawid2013}). Algorithms that internally do perform probabilistic calculations, for instance naive-Bayes or logistic-regression classifiers \autocites[\sect~3.5, \chap~8]{murphy2012}[\sects~8.2, 4.3]{bishop2006}[\chap~10, \sect~17.4]{barber2007_r2020}, unfortunately rest on strong probabilistic assumptions, such as independence and particular shapes of distributions, that are often unrealistic (and their consistency with the specific application is rarely checked). Only particular classifiers such as Bayesian neural networks \autocites{nealetal2006}[\sect~5.7]{bishop2006} output sensible probabilities, but they are computationally very expensive. The stumbling block is the extremely high dimensionality of the feature space, which makes the calculation of the probabilities
\[
\P(\texts{class}, \texts{feature} \| \texts{training data})
\]
(a problem opaquely called \enquote{density regression} or \enquote{density estimation}\autocites{ferguson1983,thorburn1986,hjort1996,dunsonetal2007}) computationally unfeasible.
If we solved the issue of outputting proper probabilities then the remaining three issues would be easy to solve, as discussed above.
\bigskip
In the present work we propose an alternative solution to calculate proper class probabilities, which can then be used in conjunction with utilities to perform the final classification or decision.
This solution consists in a sort of \enquote{transducer} or \enquote{calibration curve} that transforms the algorithm's raw output into a probability. It has a low computational cost, can be applied to all commonly used classifiers and to simple regression algorithms, does not need any changes in algorithm architecture or in training procedures, and is grounded on first principles. The probability thus obtained can be combined with utilities to perform the final classification task.
Moreover, this transducer has three other great benefits, which come automatically with its computation. First, it allows us to calculate both the probability of class conditional on features, and \emph{the probability of features conditional on class}. In other words it allows us to use the classification algorithm in both \enquote{discriminative} and \enquote{generative} modes \autocites[\sect~21.2.3]{russelletal1995_r2022}[\sect~8.6]{murphy2012}, even if the algorithm was not designed for a generative use. Second, its computation also gives a quantification of how much the probability would change if we had further training data. Third, it can give an \emph{evaluation of the whole classifier} -- including an uncertainty about such evaluation -- that allows us to compare it with other classifiers to choose the optimal one.
In \sect~\ref{sec:transducer} we present the general idea behind the probability-transducer and its calculation. Its combination with the rule of expected-utility maximization to perform classification is discussed in \sect~\ref{sec:utility_classification}; we call this combined use the \enquote{augmentation} of a classifier.
In \sect~\ref{sec:demonstration} we demonstrate the implementation, use, and benefits of classifier augmentation in a concrete drug-discovery classification problem and dataset, with a \RF\ and a \CNN\ classifiers.
Finally, we give a summary and discussion in \sect~\ref{sec:summary_discussion}, including some teasers of further applications to be discussed in future work.
\section{An output-to-probability transducer}
\label{sec:transducer}
In the present section we explain the main idea in informal and intuitive terms, and give its mathematical essentials only. Stricter logical derivation and more mathematical details are left to appendix\mynotep{\ldots}.
\subsection{Main idea: algorithm output as a proxy for the features}
\label{sec:essential_idea}
Let us consider first the essentials behind a classification (or regression) problem. We have the following quantities:
\begin{itemize}
\item the \emph{feature} values of a set of known units,
\item the \emph{classes} of the same set of units,
\end{itemize}
which together form our \emph{training data}; and
\begin{itemize}[resume]
\item the feature value of a new unit,
\end{itemize}
where the \enquote{units} can be widgets, images, patients, drug compounds, and so on, depending on the classification problem. From these quantities we would like to infer the class of the new unit. This inference consists in probability values (we omit \enquote{new} in equations for brevity)
\begin{equation}
\label{eq:prob_essential}
\P(\texts{class of unit} \| \texts{feature of unit},\,
\texts{classes \amp\ features of training data})
\end{equation}
for each possible class.
These probabilities are obtained through the rules of the probability calculus \autocites{jaynes1994_r2003}[\chaps~12--13]{russelletal1995_r2022}{gregory2005,hailperin2011,jeffreys1939_r1983}; in this case specifically through the so-called de~Finetti theorem \autocites[\chap~4]{bernardoetal1994_r2000}{dawid2013} which connects training data and new unit.
Combined with a set of utilities, these probabilities allow us to determine an optimal, further decision to be made among a set of alternatives. Note that the inference~\eqref{eq:prob_essential} includes deterministic interpolation, \ie\ the assessment of a function $\texts{class} \mo f(\texts{feature})$, as a special case, when the probabilities are essentially $0$s and $1$s.
A trained classifier should ideally output the probabilities above when applied to the new unit. Machine-learning classifiers trade this capability for computational speed -- with an increase in the latter of several orders of magnitude \autocites[to understand this trade-off in the case of neural-network classifiers see \eg][]{mackay1992,mackay1992b,mackay1992d}[\sect~16.5 esp.~16.5.7]{murphy2012}[see also the discussion by][]{selfetal1987}. Thus their output cannot be considered a probability, but \emph{it still carries information about both class and feature variables}.
% We can introduce the known output variable provided by the classifier in the probability~\eqref{eq:prob_essential}:
% \begin{equation}
% \label{eq:prob_essential_output}
% \P(\texts{class of unit} \| \texts{output for unit},
% \texts{feature of unit}, \texts{training data}) \ .
% \end{equation}
% This probability must be numerically equal to~\eqref{eq:prob_essential} because the classifier's output cannot give more information about the class than is already present in the feature and in the training data (the classifier would be biased otherwise).
Now we acknowledge the fact that the information contained in the feature and in the training data, relevant to the class of the new unit, is simply inaccessible to us because of computational limitations. We do have access to the output for the new unit, however, which does carry relevant information. Thus want to calculate a probability such as $\P(\texts{class of unit} \| \texts{output for unit})$.
To calculate this probability, however, it is necessary to have examples of further pairs $(\texts{class of unit}, \texts{output for unit})$, of which the new unit's pair can be considered a \enquote{representative sample} \autocites[for a critical analysis of the sometimes hollow term \enquote{representative sample} see][]{kruskaletal1979,kruskaletal1979b,kruskaletal1979c,kruskaletal1980} and vice versa -- exactly for the same reason why we need training data in the first place.
For this purpose, can we use the pairs $(\texts{class of unit}, \texts{output for unit})$ of the training data? This would be very convenient, as those pairs are readily available. But answer is no. The reason is that the outputs of the training data are produced from the features \emph{and the classes} jointly; this is the very point of the training phase. There is therefore a direct informational dependence between the classes and the outputs of the training data. For the new unit, on the other hand, the classifier produces its output from the feature alone. The new unit is not a representative sample of the training data.
We need a data set where the outputs are generated by simple application of the algorithm to the feature, as it would occur in its concrete use, and the classes are known. The \emph{test data} of standard \ml\ procedures are exactly what we need. The new unit can be considered a representative sample of the test data, and vice versa. We call such data \enquote{transducer-calibration data}, owing to its specific purpose.
The probability we arrive at is therefore
\begin{equation}
\label{eq:prob_output}
\P(\texts{class of unit} \| \texts{output for unit},\,
\texts{classes \amp\ outputs for calibration data}) \ .
\end{equation}
For algorithms that yield an output much simpler than the features, such as a vector of few real components, the probability above can be exactly calculated. Thus, once we obtain the classifier's output for the new unit, we can calculate a probability for the new unit's class.
\medskip
The idea above can also be informally understood in two ways. First: the classifier's output is regarded as a proxy for the feature. Second: the classifier is regarded as something analogous to a \emph{diagnostic test}, such as any common diagnostic or prognostic test used in medicine for example. We do not take diagnostic-test results at face value -- if a flu test is \enquote{positive} we do not conclude that the patient has the flu -- but rather arrive at a probability that the patient has the flu, given some statistics about results of tests performed on \enquote{verified samples} of true-positive and true-negative patients \autocites[\chap~5]{soxetal1988_r2013}[\chap~5]{huninketal2001_r2014}[see also][]{jennyetal2018}.
The probability values~\eqref{eq:prob_output} for a fixed class and variable output give us a sort of \enquote{calibration curve} (or calibration hypersurface for multidimensional outputs) of the output-to-probability transducer for the classifier. See \fig~\ref{fig:prob_curve_RF} as an example. It must be stressed that such curve needs to be calculated only once, and it can be used for all further applications of the classifier to new units.
\medskip
What is the relation between the ideal incomputable probability~\eqref{eq:prob_essential} and the probability~\eqref{eq:prob_output} obtained by proxy? It can be shown that the two are related by convex combination or mixing:
\begin{multline}
\label{eq:relation_two_probs}
\P(\texts{class} \| \texts{output},\,
\texts{classes \amp\ outputs for calibration data}) ={}\\[\jot]
\sum_{\mathclap{\substack{\text{feature}\\\text{training data}}}}
w(\texts{feature}, \texts{training data})\times
\P(\texts{class} \| \texts{feature},\, \texts{training data})
\end{multline}
where $w$ are positive and normalized functions. This means that the proxy probability is generally less sharp, that is, farther away from $0$ and $1$, than the ideal one. This is an obvious consequence of the loss of information about the feature value. Such mixing, on the other hand, only leads to more conservative probabilities, not to over-confident ones.
\mynotep{add note about the fact that the goodness of the results still greatly depends on the goodness of the classifier.}
\mynotew{add that \emph{no new test set is needed}! The calibration set also works as \enquote{test set} thanks to the probability calculus}
\subsection{Calculation of the probabilities}
\label{sec:calculation_transducer}
Let us denote by $c$ the class value of a new unit, by $y$ the output of the classifier for the new unit, and by $D \defd \set{c_{i}, y_{i}}$ the classes and classifier outputs for the transducer-calibration data.
It is more convenient to focus on the \emph{joint} probability of class and output given the data,
\begin{equation}
\label{eq:prob_output}
\p(c, y \| D) \ ,
\end{equation}
rather than on the conditional probability of the class given the output and calibration data,~\eqref{eq:prob_output}.
The joint probability is calculated using standard non-parametric Bayesian methods \autocites[for introductions and reviews see \eg][]{walker2013,muelleretal2004b,hjort1996}. \enquote{Non-parametric} in this case means that we do not make any assumptions about the shape of the probability curve as a function of $c,y$ (contrast this with logistic regression, for instance), or about special independence between the variables (contrast this with naive-Bayes). The only assumption made -- and we believe it is quite realistic -- is that the curve must have some minimal degree of smoothness. This assumption allows for much leeway, however: \fig\mynotep{\ldots} for instance shows that the probability curve can still have very sharp bends, as long as they are not cusps.
Non-parametric methods differ from one another in the kind of \enquote{coordinate system} they select on the infinite-dimensional space of all possible probability curves, that is, in the way they represent a general positive normalized function.
We choose the representation discussed by Dunson \amp\ Bhattacharya\autocites{dunsonetal2011}[see also the special case presented by][]{rasmussen1999}. The end result of interest in the present section is that the probability density $p(c,y \| D)$, with $c$ discrete and $y$ continuous and possibly multi-dimensional, is expressed as a sum
\begin{equation}
\label{eq:representation_P}
p(c, y \| D) = \sum_{k} q_{k}\ A(c \| \alpha_{k})\ B(y \| \beta_{k})
\end{equation}
of a finite but large number of terms. Each term is the product of a positive weight $q_{k}$, a probability distribution $A(c \| \alpha_{k})$ for $c$ depending on parameters $\alpha_{k}$, and a probability density $B(c \| \beta_{k})$ for $y$ depending on parameters $\beta_{k}$. The parameter values can be different from term to term, as indicated by the index ${}_{k}$. The weights $\set{q_{k}}$ are normalized.
This mathematical representation can approximate (under some norm) any smooth probability density in $c$ and $y$. It has the advantages of being automatically positive and normalized, and of readily producing the marginal distributions for $c$ and for $y$:
\begin{equation}
\label{eq:marginals}
p(c) = \sum_{k} q_{k}\ A(c \| \alpha_{k})\ ,
\qquad
p(y) = \sum_{k} q_{k}\ B(y \| \beta_{k}) \ ,
\end{equation}
from which also the conditional distributions are easily obtained:
\begin{subequations} \label{eq:conditionals}
\begin{align}
\label{eq:conditional_c_given_y}
p(c \| y) &= \sum_{k} \frac{q_{k}\ B(y \| \beta_{k})
}{\sum_{l} q_{l}\ B(y \| \beta_{l})}\
A(c \| \alpha_{k})\ \ ,
\\
\label{eq:conditional_y_given_c}
p(y \| c) &= \sum_{k} \frac{q_{k}\ A(c \| \alpha_{k})
}{\sum_{l} q_{l}\ A(c \| \alpha_{l})}\
B(y \| \beta_{k}) \ .
\end{align}
\end{subequations}
These will be important in the following discussion.
The parametric distributions $A(c \| \alpha)$ and $B(y \| \beta)$ are chosen by us according to convenience. For the first we use a simple categorical distribution with parameters $\alpha$. In a binary-classification case, where the class variable $c$ takes on conventional values $\set{0,1}$, it is
\begin{equation}
\label{eq:distr_class}
A(c \| \alpha) = c\ \alpha + (1-c)\ (1-\alpha) \equiv