R 626 B – Randomness and Natural Scenes 1 Running head : R 626 B – RANDOMNESS AND NATURAL SCENES Subjective randomness and natural scene statistics

Accounts of subjective randomness suggest that people consider a stimulus random when they cannot detect any regularities characterizing the structure of that stimulus. We explore the possibility that the regularities people detect are shaped by the statistics of their natural environment. We do this by testing the hypothesis that people’s perception of randomness in two-dimensional binary arrays (images with two levels of intensity) is inversely related to the probability with which the array’s pattern would be encountered in nature. We estimated natural scene probabilities for small binary arrays by tabulating the frequencies with which each pattern of cell values appears. We then conducted an experiment in which we collected human randomness judgments. The results show an inverse relationship between people’s perceived randomness of an array pattern and the probability of the pattern appearing in nature. R626B – Randomness and Natural Scenes 3 Subjective randomness and natural scene statistics People are very sensitive to deviations from their expectations about randomness. For example, the game Yahtzee involves repeatedly rolling five six-sided dice. If you were to roll all sixes six times in a row, you would probably be quite surprised. The probability of such a sequence arising by chance is 1/630. However, the low probability of such an event is not sufficient to explain its apparent non-randomness, as any other ordered sequence of the same number of dice rolls has the same probability. Consequently, recent accounts of human subjective randomness (our sense of the extent to which an event seems random) have focused on the regularities in an event. These regularities suggest a process other than chance might be at work (Griffiths & Tenenbaum, 2003, 2004; Falk & Konold, 1997; Feldman, 1996, 1997). The basic idea behind these accounts is that stimuli will appear random when they do not express any regularities. An important challenge for any account of subjective randomness based on the presence of regularities is to explain why people should be sensitive to a particular set of regularities. In the example given above, systematic runs of the same number may suggest loaded dice, or some other non-random process influencing the outcomes. However, for other kinds of stimuli, such as the oneor two-dimensional binary arrays used in many subjective randomness experiments, explanations are more difficult to come by. A common finding in these experiments is that people consider arrays in which cells take different values from their neighbors (such as the one-dimensional array 0010101101) more random than arrays in which cells take the same values as their neighbors (such as 0000011111) (Falk & Konold, 1997). This result makes it clear that people are sensitive to certain regularities, such as cells having the same values as their neighbors. However, it is difficult to explain why these regularities should be more important than others that seem a priori plausible, such as neighboring cells differing in their values. In this paper, we explore a possible explanation for the origins of the regularities that influence subjective randomness judgments for one class of stimuli: two-dimensional binary arrays. These stimuli are essentially images, with the cells in the array having the appearance of a grid of black and white pixels (see Figure 1). We might thus expect that the kinds of regularities detected by the visual system should R626B – Randomness and Natural Scenes 4 play an important role in determining their perceived randomness. A great deal of recent research suggests that the human visual cortex efficiently codes for the structure of natural scenes – scenes containing natural elements such as trees, flowers, and shrubs that represent the visual environment in which humans evolved (Olshausen & Field, 2000; Simoncelli & Olshausen, 2001). We consider the possibility that the kinds of regularities that people detect in two-dimensional binary arrays are those that are characteristic of natural scenes. Preliminary support for the idea that the statistics of natural scenes may explain subjective randomness was provided by a study conducted by Schreiber and Griffiths (2007). In this study, human randomness judgments were found to correspond to the predictions of a simple probabilistic model estimated from images of natural scenes. This model examined the frequency with which neighboring regions of an image had the same intensity value. It was found that neighboring regions tend to have similar intensity values, providing a potential explanation for why people consider binary arrays in which neighboring cells differ in their values more random. However, a full characterization of natural scene statistics is not feasible for large binary arrays, due to the exponentially large number of patterns of cell values that can be expressed in such arrays. In our present work we use small binary arrays, which allow us to directly estimate a probability distribution over all possible patterns of cell values from images of natural scenes. We then conduct an experiment with human participants to examine the relationship between this probability distribution and the subjective randomness of the image. Subjective randomness as Bayesian inference One explanation for human randomness judgments is to view them as the result of an inference on whether an observed stimulus, X , was generated by chance, or by some other more regular process (Griffiths & Tenenbaum, 2003, 2004; Feldman, 1996, 1997). If we let P(X |random) denote the probability of X being generated by chance, and P(X |regular) be the probability of X under the regular generating R626B – Randomness and Natural Scenes 5 process, then Bayes’ rule gives the posterior odds in favor of random generation as P(random|X) P(regular|X) = P(X |random) P(X |regular) P(random) P(regular) (1) where P(random) and P(regular) are the prior probabilities assigned to the random and regular processes respectively. Only the first term on the right hand side of this expression, the likelihood ratio, changes as a function of X , making it a natural measure of the amount of evidence X provides in favor of a random generating process. Hence, we can define the randomness of a stimulus X as random(X) = log P(X |random) P(X |regular) (2) where the logarithm simply places the result on a linear scale. The measure of subjective randomness defined in Equation 2 has been used to model human randomness judgments for single digit numbers and one-dimensional binary arrays (Griffiths & Tenenbaum, 2001, 2003, 2004). Following Schreiber and Griffiths (2007), we examine how subjective randomness might be applied to two-dimensional binary arrays of the kind shown in Figure 1. A reasonable choice of P(X |random) is to assume that each cell in the array takes on a value of 1 or 0 with equal probability, making P(X |random) = 1/2m, where m is the number of cells in the array. However, defining P(X |regular) is more challenging. Direct estimation of the probability of all 2m binary arrays becomes intractable as m becomes large. Thus, we use 4×4 binary arrays for which we can exhaustively tabulate the frequencies of all patterns that can appear, providing a non-parametric estimate of P(X |regular) that allows for a comprehensive test of our hypothesis. We estimated the values of P(X |regular) for a set of stimuli X corresponding to 4×4 binary arrays. Since there are only 216 possible patterns that can be expressed in such an array, we can count the frequency with which each pattern appears in natural images. These stimuli were extracted from a set of images of natural scenes that have been used in previous research (Doi, Inui, Lee, Wachtler, & Sejnowski, 2003). This set consisted of 62 still nature shots containing trees, flowers, and shrubs as shown in Figure 2. R626B – Randomness and Natural Scenes 6 There were no images of humans, animals, or cityscapes. Image patches of varying sizes were extracted from each natural image to measure statistics at a range of scales. A total of 700,000 patches were sampled at random from among the 62 images with dimensions n×n, for n = 4,8,16,32,64,128, and 256 pixels. All patches were then reduced through averaging down to 4×4 arrays, binarized by setting pixels with intensity greater than zero (the overall mean intensity) to 1 and, with all other pixels being set to 0. The resulting 4,900,000 binary arrays were then divided into the 216 possible patterns, and the frequency of each pattern was recorded. Normalizing these frequencies gives us an estimate for the probability distribution P(X |regular), which we can use to compare the Bayesian measure of randomness given in Equation 2 with human judgments.