Inverse theory is a field of mathematics that aims to solve problems in which we wish to use data to constrain a model when we only know how to predict data given the model. There are many examples of such Inverse Problems, and most of my research has either developed or used inverse theory. Consequently my related publications are most of those on my main web page. However, I have a separate page describing my contributions to the development of inverse theory.
An interesting and related area of research is in the field of Elicitation Theory. If experts have either knowledge that is tacitly held in their minds or skills that were acquired through practise, in order to use that knowledge to quantitative effect it is often desirable to quantify the knowledge. All human knowledge is subjective and uncertain, so one might try to accumulate such knowledge into probability distributions. However, simply asking people to describe their knowledge in terms of numbers or probabilities is known to provide highly biased answers under many conditions, even when asking experts in the field of interest. The theory of statistics devoted to the study of how best to ask people questions in order to obtain the least biased answers is called Elicitation Theory.
In Curtis and Wood (2004), we apply experimental design theory of Curtis et al., (2004) to design exactly the optimal next question that should be asked of an expert such that the answer is most likely to fill the most significant gap in accumulated, quantified knowledge. The method has the feature that it is applied in real time during the interrogation, and accounts for all of the information derived from previous questions. This strategy should surely be optimal, although our specific method can without doubt be improved.
Another related and interesting phenomenon that has been the subject of much recent study and debate in Economics literature is that of 'herding' in humans - the observation that under some conditions humans flock after each other rather than taking independent decisions or forming and acting on their own beliefs. It has been shown that this can be perfectly rational behaviour: in situations in which I believe that you have better information than me, it may be rational for me to follow your example. What is more, in some situations the very belief that is propagated through a herd can become self-verifying: if we all believe that a stock in a free market is going to increase in price, it is rational for us all to buy it, thus increasing the price of the stock. In such situations, Elicitation Theory can play a crucial role in analysing motivations and actions of the herd.
What is not so clear is why Scientific communities exhibit herding behaviour. In Science there is (by assumption) an underlying, immutable reality that can not be changed by the herding of beliefs about its nature. Nevertheless, scientific communities 'herd' around current paradigms and are even observed to reject observed data because it does not fit a paradigm rather than challenge the paradigm. In a recent paper, Baddeley et al., (2004) discuss the potential parallels between Scientific and Economic herding. I continue to investigate this field because the recent advances in mathematics of herding in Social Scientific literature will surely shed new light on the underlying mechanisms or motivations behind Scientific herding behaviour, even though such motivations may be quite different from those assumed in the Social Sciences.Related Papers in refereed journals: