In the course of daily life, anyone can stumble upon something surprising. For example, one comes to work to find a new potted plant has been put in the corridor, or suddenly notices a key that had long been missing in a handbag. Aside from making us keenly aware of our hearts thumping in our chests, the right surprise can make us giddy with delight. Perhaps even more surprising is that mathematicians are attempting to calculate degrees of “surprise” with mathematical algorithms. In turn, psychologists hope to use these to use these algorithms to collect data on how we perceive everyday life scenarios: tracking our perceptual process as we go through the day to day scenes of life, they believe, will help them understand the cognitive mechanisms in information processing.

 

Using event probability to infer surprise value

 

The mathematicians above certainly are not the first to devise measures of surprise; in fact, the problem has intrigued statisticians for over half a century. In 1948, the American scientist and mathematician Warren Weaver proposed a “Surprise Index” by which to compare the probability of one outcome with the probability of all other outcomes (thus giving one a good sense of how surprising that outcome would be). That refers to all other event to the probabilities of all other events. In an experiment with n possible outcomes and p₁, p₂……, p_n probabilities, Weaver defines the Surprise Index as:

SI =  (E(p))/p_i  = (∑_(i=1)^n*〖p^2〗_i )/p_i

SI measures the ratio between expected average probability E(p) and the probability of a particular outcome p_i. As the value of E(p) becomes relatively higher and the value of p_i becomes relatively higher, the particular outcome is more surprising.  

 

In his 1969 discussion on entrepreneurial decision-making, the British economist G.L.S. Shackle proposed a theory of “potential surprise” based on subjective expectation rather than objective probability. In his formulation, events that seem less likely are more surprising: flipping “heads” 10 times in a row would feel surprising because it is meaningful, even though it is statistically as unlikely as flipping any specific sequence. Shackle defined the potential surprise as the individual’s confidence level, that is, the individual’s confidence level to one assumption corresponds to the potential surprise level of the assumption.

 

Shackle’s theory posits that just one can state probability of an event B as P(B), so too can one state a potential surprise value of event B, which Shackle notated as y^B. The theory can also be applied to conditional events, whereby one event may or may not affect the outcome of another (the potential surprise of the second event being conditional on whether the first is true, if it is related). The psychologist John E. Fisk explains that this “the equivalent concept in probability theory is P (A∣B).” Shackle expressed the potential surprise of event A as y_0^(  A) y, for which changes are divided into two situations: (1) if event A and event B are unrelated or independent events, then the potential surprise for event A will not be affected by the occurrence or nonoccurrence of B, so y_0^(  A)=y^A whether or not; (2) however, if event A and event B are related, then the occurrence of event B will reduce our potential surprise for event A, so y_0^(  A)<y^Ay.

 

Surprise originates from uncertainty

 

“The concept of surprise is central to sensory processing, adaptation and learning, attention, and decision making,” information theorist Laurent Itti writes at the University of Southern Californias iLab’s homepage for the Formal Bayesian Theory of Surprise. In 2005, Itti and UC Irvine Professor Pierre Baldi worked together to write and test a mathematical theory to quantify surprise using Bayseian statistics. In Bayesian statistical inference, a prior probability distribution expresses one’s uncertainty about a quantity before more information or data on the quantity becomes available, while a posterior probability is the probability assigned after relevant information is taken into account. Itti and Baldi’s theory of surprise compares this difference between differences between observers’ posterior and prior beliefs. They stipulate that surprise 1) can only exist when there is uncertainty, “which can arise from intrinsic stochasticity (involving an element of randomness), missing information or limited computing resources” 2) is relative and subjective—surprise depends heavily on the observer’s expectations. The degree of surprise for the same stimulus or information varies widely between observers and even for the same observer at different times.

“In probability and decision theory it can be shown that, under a small set of axioms, the only consistent way for modeling and reasoning about uncertainty is provided by the Bayesian theory of probability,” Itti explains. “Furthermore, in the Bayesian framework, probabilities correspond to subjective degrees of beliefs in hypotheses or models which are updated, as data is acquired.” This updating process is the crux of their approach to uncertainty, and through it surprise: Bayes’ theorem is the fundamental tool by which prior belief distributions are transformed into posterior belief distributions. Represented in equation form, the prior probability distribution P (M)—the background beliefs if the observer— is transferred into the posterior distribution P (M | D) as new data D is observed:

 

P(M∣D) =(P(D∣M))/(P(D)) P(M)  

In this framework, new data D is not surprising if it leaves the observer’s beliefs unchanged; it is surprising if the posterior distribution differs substantially from the prior. Itti and Baldi formally quantify the surprise by measuring the distance between P(M) into P(M∣D):

 

S(D, M) = d[P(M), P(M∣D)]

 

Where d measures the distance between the two probability distributions

 

“Surprising” theory benefits the information processing of the scene perceptual process

 

Sense and perception studies have shown that humans recognize a scenario or stimulus in 100 milliseconds or less. When people observe these scenarios or stimuli, what information about them are they actually taking in and how do they verify this information? Researchers have proposed a number of methods to determine what information is important and what role it plays in scenario perception processing. In 1967 Norman Mackworth and A. J. Morandi tested which areas of an image viewers were most likely to fixate on. The cognitive scientists divided two simple color photographs into 64 square sections, which viewers rated in terms of informativeness based on how easy it would be to recognize a particular section on another occasion. Then a second group of viewers chose which picture they preferred more, during which their “discreet fixations” were measured in two-second intervals. The study revealed that the second group virtually only looked at the regions labeled as more informative by the first group, even after the intial 2 seconds.  James Brockmole and John Henderson’s recent research on searching for specific information and scene novelty—put simply, how much changing a picture affects a viewer’s ability to find something specific in the picture and the process by which the viewer finds it—has shown that attention deployment to the specific scenes is very narrowly focused in familiar scenes, and much more broadly focused in novel scenes. When searching for a stimulus in a scene, the viewer processes more information about the scene as the probability of finding that stimulus decreases.

This research, though, is based on subjective expectations and does not present a formula or method by which to quantify these subjective judgments. Without such a method, it would be impossible to represent them precisely. Itti and Formal Bayesian Theory of Surprise provide the necessary theoretical and methodological support. Using methods similar to those of the experiments described above, Itti and Baldi have confirmed that surprising stimuli direct a viewer’s attention when browsing images. In related experiments, Einhaeuser and Mundhenk have shown that surprising stimuli captures a viewer’s attention and impedes his or her ability to find target stimuli.

 

For studies on scene/scenario perception, the Surprise Theory can be applied to help explain the perception of information about a scene or scenario and the cognitive mechanisms involved in procession that information; related research and exploration of surprise theory will also help advance data decomposition and computational modeling of information perception and information perception in social scenarios.

 

 

Kang Tinghu and Zhang Feng are from the School of Psychology at Northwest Normal University.

 

The Chinese version appeared in Chinese Social Sciences Today, No. 455, May 27, 2013

 

                                                                                                                          Translated by Zhang Mengying

 Revised by Charles Horne