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A computationally and cognitively plausible model of supervised and unsupervised learning

09 Jun 2013-pp 145-156
TL;DR: In this paper, a new model of learning based on empirical psychological results on association learning is proposed, and two forms of this model are developed, the Informatron as a chance-corrected Perceptron, and AdaBook as an chancecorrected AdaBoost procedure.
Abstract: Both empirical and mathematical demonstrations of the importance of chance-corrected measures are discussed, and a new model of learning is proposed based on empirical psychological results on association learning. Two forms of this model are developed, the Informatron as a chance-corrected Perceptron, and AdaBook as a chance-corrected AdaBoost procedure. Computational results presented show chance correction facilitates learning.

Summary (3 min read)

1 Introduction*

  • The issue of chance correction has been discussed for many decades in the context of statistics, psychology and machine learning, with multiple measures being shown to have desirable properties, including various definitions of Kappa or Correlation, and the psychologically validated ΔP measures.
  • The authors discuss the relationships between these measures, showing that they form part of a single family of measures, and that using an appropriate measure can positively impact learning.

1.1 What’s in a “word”?

  • The typical task is a word association model, but other tasks may focus on syllables or rimes or orthography.
  • The “word” is not a well-defined unit psychologically or linguistically, and is arguably now a backformed concept from modern orthology.
  • Thus the authors use “word” for want of a better word, and the scare quotes should be imagined to be there at all times, although they are frequently omitted for readability!.
  • An extended abstract based on an earlier version has been submitted for presentation to the Cognitive Science Society (in accordance with their policy of being of “limited circulation”).

1.2 What’s in a “measure”?

  • A primary focus of this paper is the inadequacy of currently used measures such as Accuracy, True Positive Rate, Precision, F-score, etc.
  • Alternate chance-corrected measures have been advocated in multiple areas of cognitive, computational and physical science, and in particular in Psychology in the specific context of association learning [1-3], where ΔP is considered “the normative measure of contingency”.
  • In parallel, discontent with misleading measures of accuracy was building in Statistics [4,5], Computational Linguistics [6] and Machine Learning [7] and extended to the broader Cognitive Science community [8].
  • Reversions to older methods such as Kappa and Correlation (and ROC AUC, AUK, etc.) were proposed and in this paper the authors explore learning models that directly optimize such measures.

2 Informedness, Correlation & DeltaP

  • The concept of chance-corrected accuracy measures has been reinvented several times in several contexts, with some of the most important being Kappa variants [4,5].
  • Empirically ΔP’ is stronger than ΔP in these experiments, and TP and TP’ are much weaker, with TP failing to achieve a significant result for either Children or Adults in their experiments.
  • The authors further define True and False Positives and Negatives based on whether the prediction P or N was accurate or not (counts, TP, TN, FP, FN; probs tp, tn, fp, fn; rates tpr=tp/rp, tnr=tn/rn, fpr=fp/rn).
  • Whilst the above systematic notation is convenient for derivations and proofs, these probabilities are known by a number of different names and the authors will use some of these terms (and shortened forms) for clarity of equations and discussions.
  • Also Recall and Sensitivity are synonyms for true positive rate (tpr), whilst Inverse Recall and Specificity correspond to true negative rate (tnr).

3 Association Learning & Neural Networks

  • The authors have seen that chance-corrected ΔP measures are better models both from a statistical point of view (giving rise to probabilities of an informed prediction or marked predictor) and also from an empirical psychology perspective (reflecting human association strength more accurately).
  • They also have the advantage over correlation of being usable separately to provide directionality or together to provide the same information as correlation.
  • This raises the question of whether their statistical and neural learning models reflect appropriate statistics.
  • The statistical models traditionally directly maximize accuracy or minimize error, without chance correction, and many neural network and convext boosting models can shown to be equivalent to such statistical models, as the authors show in this section and the next.
  • The authors question is whether these can be generalized with a bioplausble chance-correcting model.

3.2 A family of neural update rules

  • Voting, bagging, boosting and stacking ensembles may also be construed to obey (9) for appropriate choices of f and g.
  • In (10) the authors see the original Hebb update rule in three forms.
  • For sparse (word to word) association learning, Wjk simply corresponds to unnormalized cjk contingency table entries of (8), being normalized counts cjk = Cjk/N = Wjk/N.
  • Often authors of neuroplausible models have the rider that cells may correspond to a cluster of neurons rather than one.

3.3 The Informatron

  • To model chance-correction, the authors require a matrix that reflects Informedness gains (in “dollars”) rather than counts (10) or errors (11-13).
  • The authors now show an equation corresponding to (7-8) clarifying the role of the shadow neurons: Zik = f(Σj g(Xij) Wjk / Sik) with Sik = Yek – Dik (17) Furthermore it doesn’t explicitly give multiclass Informedness but that is a straightforward higher level embedding, and it doesn’t model features or kernels, which is an obvious lower level recursion.
  • These XOR and EQV circuits correspond to their (p≠r) resp. (p=r), allowing comparison of prediction and reality in their model.

4 Fusion and Boosting

  • The authors also noted earlier that both MLPs and Boosting can also be modelled by (9), and in particular AdaBoost [22] assumes a weak learner g and uses that to learn a strong learner in a very similar way to the Perceptron algorithms they have been considering.
  • The standard algorithms define that as Error <0.5, or Accuracy >0.5, where Error is the sum of fp and fn, and Accuracy is the sum of tp and tn (Table 1), and Accuracy +.
  • A technique to fix this is simply to calculate GiniK = (K+1)/2, where Gini (being originally designed for ROC AUC) can be applied to any chance-corrected measure K where 0 marks the chance level, mapping this chance level to ½.
  • To complete the discussion of AdaBoost, it suffices to note that the different trained classifiers result from training the same weak learner on different weightings (or resamplings) of the available training set, with weights given by the odds Acc / Err.
  • The authors have now introduced a neural model that directly implements ΔP or ΔP’ (which is purely a matter of direction and both directions are modelled in Fig. 1).

5 Results & Conclusions

  • The most commonly used training algorithm today is SVM, closely followed by AdaBoost, which is actually usually better than SVM when SVM is boosted rather than the default Decision Stump (which is basically the best Perceptron possible based on a single input variable).
  • To test their boosting algorithm, which the authors call AdaBook because of its Bookmaker corrected accuracy), they used standard UCI Machine Learning datasets relating to English letters (recognizing visually, acoustically or by pen motion).
  • These were selected consistent with their language focus.
  • Thus the authors have shown that the use of chance-corrected measures, ΔP rather than TP or TPR, etc. is not only found empirically in Psychological Association experiments, but leads to improved learning in Machine Learning experiments.

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Content maybe subject to copyright    Report

Archived at the Flinders Academic Commons:
http://dspace.flinders.edu.au/dspace/
This is the authors’ version of an article published in
Lecture Notes in Computer Science. The original publication
is available by subscription at:
http://link.springer.com/
doi: 10.1007/978-3-642-38786-9_17
Please cite this article as:
Powers, D.M. (2013). A computationally and cognitively
plausible model of supervised and unsupervised
learning. In D Liu et. al (Ed.), Advances in Brain Inspired
Cognitive Systems: Vol 7888, 6th International
Conference, BICS 2013, Beijing, China, June 9-11, 2013.
Proceedings (pp. 145-156) Berlin: Springer Berlin
Heidelberg.
Copyright (2013) Springer-Verlag. All right
s reserved. Please
note that any alterations made during the publishing
process may not appear in this version. The final publication
is available at link.springer.com”.
Archived at the Flinders Academic Commons: http://dspace.flinders.edu.au/dspace/

A computationally and cognitively plausible model
of supervised and unsupervised learning
David M. W. Powers
1,2
1
CSEM Centre for Knowledge & Interaction Technology, Flinders University,
Adelaide, South Australia
2
Beijing Municipal Lab for Multimedia & Intelligent Software, BJUT
Beijing, China
powers@acm.org
Abstract. Both empirical and mathematical demonstrations of the importance
of chance-corrected measures are discussed, and a new model of learning is
proposed based on empirical psychological results on association learning. Two
forms of this model are developed, the Informatron as a chance-corrected
Perceptron, and AdaBook as a chance-corrected AdaBoost procedure.
Computational results presented show chance correction facilitates learning.
Keywords: Chance-corrected evaluation, Kappa, Perceptron, AdaBoost
1 Introduction
*
The issue of chance correction has been discussed for many decades in the context of
statistics, psychology and machine learning, with multiple measures being shown to
have desirable properties, including various definitions of Kappa or Correlation, and
the psychologically validated ΔP measures. In this paper, we discuss the relationships
between these measures, showing that they form part of a single family of measures,
and that using an appropriate measure can positively impact learning.
1.1 What’s in a “word”?
In the Informatron model we present, we will be aiming to model results in human
association and language processing. The typical task is a word association model,
but other tasks may focus on syllables or rimes or orthography. The “word” is not a
well-defined unit psychologically or linguistically, and is arguably now a backformed
concept from modern orthology. Thus we use “word” for want of a better word, and the
scare quotes should be imagined to be there at all times, although they are frequently
omitted for readability! (Consider “into” vs “out of”, “bring around” vs “umbringen”.)
*
An extended abstract based on an earlier version has been submitted for presentation to the
Cognitive Science Society (in accordance with their policy of being of “limited circulation”).
Archived at the Flinders Academic Commons: http://dspace.flinders.edu.au/dspace/

1.2 What’s in a “measure”?
A primary focus of this paper is the inadequacy of currently used measures such as
Accuracy, True Positive Rate, Precision, F-score, etc. Alternate chance-corrected
measures have been advocated in multiple areas of cognitive, computational and
physical science, and in particular in Psychology in the specific context of
(unsupervised) association learning [1-3], where ΔP is considered “the normative
measure of contingency”.
In parallel, discontent with misleading measures of accuracy was building in
Statistics [4,5], Computational Linguistics [6] and Machine Learning [7] and
extended to the broader Cognitive Science community [8]. Reversions to older
methods such as Kappa and Correlation (and ROC AUC, AUK, etc.) were proposed
and in this paper we explore learning models that directly optimize such measures.
2 Informedness, Correlation & DeltaP
The concept of chance-corrected accuracy measures has been reinvented several times
in several contexts, with some of the most important being Kappa variants [4,5].
This is an ad hoc approach that subtracts from accuracy (Ac) an estimate of the
chance-level accuracy (EAc) and renormalizes to the form of a probability
Κ=(AcEAc)/(1EAc). But different forms of chance estimate, different forms of
normalization, and different generalizations to multiple classes or raters/predictors,
lead to a whole family of Kappa measures of which ΔP turns out to be one, and ΔP’
another [9]. The geometric mean of these two unidirectional measures is correlation,
which is thus a measure of mean association over both directions of an AB relation
between events. Perruchet and Pereman [3] focus on an A, B word sequence and
define ΔP as a chance-corrected version of TP = P(B|A), corresponding to Precision
(proportion of events A that predict B correctly), whilst ΔP’ corrects TP’ = P(A|B)
which is better known as TPR, Sensitivity or Recall, meaning the proportion of events
B that are predicted by A on the assumption that forward prediction AB is
normative. They argue for comparing TP with a baseline of how often event B occurs
when not preceded by A so that ΔP = P(B|A) P(B|¬A) and ΔP= P(A|B) P(A|¬B).
Empirically ΔP’ is stronger than ΔP in these experiments, and TP and TP’ are
much weaker, with TP failing to achieve a significant result for either Children or
Adults in their experiments. Why should the reverse direction be stronger? One
reason may be that an occurrence in the past is more definite for the speaker and has
been more deeply processed for the hearer. Furthermore, often a following segment
may help disambiguate a preceding one. Thus in computational work at both word
level and phoneme/grapheme level, the preceding two units and the succeeding three
units, seem to be optimal in association-based syntax and morphology learning
models [10,11], and two-side context has also proven important in semantic models
[12]. However, Flach [7] and Powers [8] independently derived ΔP’-equivalent
measures, not ΔP, as a skew/chance independent measure for AB predictions as the
information value relates to (and should be conditioned on) the prevalence of B not A.
Archived at the Flinders Academic Commons: http://dspace.flinders.edu.au/dspace/

In view of these Machine Learning proofs we turn there to introduce and motivate
definitions in a statistical notation that conflicts with that quoted above from the
Psychology literature. We use systematic acronyms [7,8] in upper case for counts,
lower case for rates or probabilities. In dichotomous Machine Learning [7] we
assume that we have for each instance a Real class label which is either Positive or
Negative (counts, RP or RN, rates rp=RP/N and rn=RN/N where we have N instances
labelled). We assume that our classifier, or in Association Learning the predictor,
specifies one Predicted class label as being the most likely for each instance (counts,
PP or PN, probs pp and pn). We further define True and False Positives and Negatives
based on whether the prediction P or N was accurate or not (counts, TP, TN, FP, FN;
probs tp, tn, fp, fn; rates tpr=tp/rp, tnr=tn/rn, fpr=fp/rn).
Table 1: Prob notation for dichotomous contingency matrix.
+R
R
+P
tp
fp
P
fn
tn
rp
rn
1
Whilst the above systematic notation is convenient for derivations and proofs,
these probabilities (probs) are known by a number of different names and we will use
some of these terms (and shortened forms) for clarity of equations and discussions.
The probs rp and rn are also known as Prevalence (Prev) and Inverse Prevalence
(IPrev), whilst pp and bn are Bias and Inverse Bias (IBias) resp. Also Recall and
Sensitivity are synonyms for true positive rate (tpr), whilst Inverse Recall and
Specificity correspond to true negative rate (tnr). The term rate is used when we are
talking about the rate of finding or recalling the real item or label, that is the
proportion of the real items with the label that are recalled. When we are talking
about the accuracy of a prediction in the sense of how many of our predictions are
accurate we use the term accuracy, with Precision (Prec) or true positive accuracy
being tpa=tp/pp, and Inverse Precision or true negative accuracy being tna=tn/pn, and
our (perverse) prediction accuracy for false positives being fpa=fp/pp. We also use
fpa and fna correspondingly for the perverse accuracies predicting the wrong (false)
class. Names for other probs [13] won’t be needed.
The chance-corrected measure ΔP’ turns out to be the dichotomous case of
Informedness, the probability that a prediction is informed with respect to the real
variable (rather than chance). This was proven based on considerations of odds-
setting in horse-racing, and is well known as a mechanism for debiasing multiple
choice exams [8,13]. It has also been derived as skew-insensitive Weighted Relative
Accuracy (siWRAcc) based on consideration of ROC curves [7]. As previously
shown in another notation, it is given by:
ΔP’ = tprfpr = tpr+tnr1 = Sensitivity + Specificity 1 (1)
Archived at the Flinders Academic Commons: http://dspace.flinders.edu.au/dspace/

The inverse concept is Markedness, the probability that the predicting variable is
actually marked by the real variable (rather than occuring independently or randomly).
This reduces to ΔP in the dichotomous case:
ΔP = tpafpa = tpa+tna1 = Prec + IPrec 1 (2)
As noted earlier, the geometric mean of ΔP and ΔP’ is Matthews Correlation
(Perruchet & Pereman, 2004), and kappas and correlations all correspond to different
normalizations of the determinant of the contingency matrix [13]. It is noted that ΔP’ is
recall-like, based on the rate we recall or predict each class, whilst ΔP is precision-like,
based on the accuracy of our predictions of each label.
The Kappa interpretation of ΔP and ΔP’ in terms of correction for Prevalence and
Bias [9,13] is apparent from the following equations (noting that Prev<1 is assumed,
and Bias<1 is thus a requirement of informed prediction, and E(Acc)<1 for any
standard Kappa model):
Kappa = (AccuracyE(Acc)) / (1E(Acc)) (3)
ΔP’ = (Recall Bias) / (1 Prevalence) (3)
ΔP = (PrecisionPrevalence)/(1 Bias) (4)
If we think only in terms of the positive class, and have an example with high natural
prevalence, such as water being a noun say 90% of the time, then it is possible to do
better by guessing noun all the time than by using a part of speech determining
algorithm that is only say 75% accurate [6]. Then if we are guessing our Precision
will follow Prevalence (90% of our noun predictions will be nouns) and Recall will
follow Bias (100% of our noun occurences will be recalled correctly, 0% of the others).
We can see that these chance levels are subtracted off in (3) and (4), but unlike the
usual kappas, a different chance level estimate is used in the denominator for
normalization to a probability and unlike the other kappas, we actually have a well
defined probability as the probability of an informed prediction or of a marked
predictor resp. The insight into the alternate denominator comes from consideration
of the amount of room for improvement. The gain due to Bias in (3) is relative to
the chance level set by Prevalence, as ΔP’ can increase only so much by dealing with
only one class how much is missed by this blindpositive focus of tpr or Recall on
the positive class is captured by the Inverse Prevalence, (1 Prevalence).
Informedness and Markedness in the general multiclass case, with K classes and the
corresponding one-vs-rest dichotomus statistics indexed by k, are simply
Informedness = Σ
k
Bias
k
ΔP
k
(5)
Markedness = Σ
k
Prev
k
ΔP
k
(6)
Informedness can also be characterized as an average cost over the contingency table
cells c
pr
where the cost of a particular prediction p versus the real class r is given by
the Bookmaker odds: what you win or lose is inversely determined by the prevalence
of the horse you predict (bet on) winning (p=r) or losing (pr) using a programming
convention for Boolean expressions here, (true,false)=(1,0), define Gain G
pr
to have
G
pr
= 1/(Prev
p
–D
pr
) where D
pr
= (pr) (7)
Informedness = Σ
p
Bias
p
[Σ
r
c
pr
G
pr
]
(8)
Archived at the Flinders Academic Commons: http://dspace.flinders.edu.au/dspace/

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Both empirical and mathematical demonstrations of the importance of chance-corrected measures are discussed, and a new model of learning is proposed based on empirical psychological results on association learning.