By rejecting the notion of information systems as being a series of independent “facts”, this chapter proposes a different normative and descriptive account of dia- gnostic decision-making which is supported by both theoretical and empirical work. In this, the emphasis is placed on the knowledge derived from the relationships between ordinal data. However, this approach is not epistemological. These rela- tionships are shown to have significant numerical worth, and it is their discovery which drives information search.
The central assertions of the pseudodiagnosticity paradigm that only paired data have diagnostic worth, that the selection of data pairs is normatively correct, and that the tendency to select non-paired data may only be explained as a failure of logic have been challenged not only through computational modeling but, also, through the development of the Relational Information Theory model. Thus, it has been clearly demonstrated that there exists an alternative strategy that is not only more effective than the selection of paired data but is also normatively correct. That there is also evidence to support the idea that people actively follow this strategy in information search strongly suggests that data selection is neither illogical nor the result of confirmation bias.
The structural problems with the standard two hypotheses and two diagnostic criteria pseudodiagnosticity exercises have been highlighted, with the presence of an-
chor information in D1|H1 being shown to strongly influence cell selection patterns.
It is only by extending the exercises to include at least three hypotheses that these influences appear to be largely negated, with clear selection patterns and strategies starting to emerge.
Chapter 3
A conjecture for the quantum
calculation of likelihood ratios
3.1
Introduction
One problem with the pseudodiagnosticity paradigm, discussed in Chapter 2, is its reliance upon categorical answers to questions framed as Bayesian problems. While this experimental structure affords the decision-maker an opportunity to use non-Bayesian techniques to provide reasonable answers to questions, the axiomatic difficulties associated with the na¨ıve Bayes’ classifier raise questions about how es- timations of probability should be calculated if required. In particular, the reliance of Bayes’ theorem upon the multiplication of marginal probabilities, in the absence of statistical information such as estimates of covariate overlap, is problematic if the conditional independence of the posterior data is not guaranteed.
Such issues have led some researchers to attempt to reconceptualise psychology and decision-making theory using quantum mechanics - an approach with intuitive merit given that both disciplines apply statistical axioms to analyse and interpret probabilistic systems. For instance, Busemeyer and Bruza (2012) have considered the effects of state transitions within a quantum, Hilbert space to describe how the order in which relevant information is considered may affect product choice. Equally, Khrennikov (2009) has used a quantum mechanical approach to explain psychological phenomena such as the violation of the law of total probability. How-
ever, within psychological theory there has been no attempt to consider the topic of Bayesian rationality within a quantum framework.
A promising starting point for the application of quantum mechanics to Bayesian
rationality is the L¨uders (1951) postulate, which allows for the estimation of particle
correlations through state projection. However, the L¨uders’ postulate fails under
nonlocal conditions, such as the Einstein, Podolsky, and Rosen paradox (Graft,
2017). Thus, at best, the L¨uders’ postulate is a special case expression designed
solely for use with sub-atomic particle ensembles (Graft, 2017), rather than the more common individual measurements associated with decision-making theory.
An alternative theoretical approach is that of Caves et al. (2002a) who have taken a radically subjectivist view of Bayes’ theorem by applying de Finetti’s epi- stemologically driven view of statistics (see de Finetti, 1974). In doing so, Caves et al. have argued that their “quantum Bayesian” statistical systems are best in- terpreted by methods in which the Bayesian likelihood ratio is seen to be both external to the system and subjectively imposed on it by the observer (Timpson, 2008). However, from a decision-making perspective, the Caves et al. approach is problematic. Bayes’ theorem and, in particular, the na¨ıve Bayes’ classifier have been used extensively to interpret information systems and develop normative decision- making models (Oaksford and Chater, 2007). While subjectivity may play a role in a descriptive model of human decision-making, its use in normative analysis could suggest the presence of a cognitive “homonculus” with the power to influence de- cision outcomes. Yet at a human scale, for instance, an observer’s belief as to the chances of a fair coin landing either “heads” or “tails” has no known effect. Rather, within normative decision-making theory, the “heads:tails” likelihood ratio of 0.5:0.5 is only meaningful when considered as a property of the coin’s own internal statist- ical system rather than as some ephemeral and arbitrary qualia.
Despite the evident progress made in the application of quantum mechanics to decision-making theory, the lack of an orthodox Copenhagen-based theoretical coun- terpoint to Caves et al. has impeded the development of new, non-subjective, and
normative decision-making models in psychology. It is this knowledge gap which this chapter aims to fill.