Tom's *provocative* post on Simplicity / Likelihood debate.
Naudojant „Google Reader" atsiųsta jums nuo Jonas:
Perhaps we can discuss a paradigm that I have been using to empirically measure the subjective priors involved in perceptual decisions. The results might help us to answer some of the questions addressed by Lee.
Some background:
The visual system is continually challenged to form accurate judgements based on noisy measurements of a perceptual environment. During the last decades, this challenge has often been formulated as a Bayesian inference problem. To maximize the accuracy of perceptual judgements, a Bayesian observer combines limited or unreliable sensory information with prior experience and knowledge of the perceptual environment. Formally, a Bayesian observer estimating a specific stimulus quantity, s, can be defined by two probability functions: the likelihood and prior function. The likelihood function, p(m|s), captures the probability of s underlying an observed set of noisy sensory measurements, m. The prior function, p(s), reflects the prior probability of the stimulus quantity s occurring in the natural environment or perceptual task at hand. The estimate of s corresponds to the mean or mode of the posterior distribution, p(s|m), which equals the (normalized) product of likelihood and prior functions. Note that we assume a continuous stimulus dimension here.
Measuring the prior:
Assuming human observers use Bayesian inference, the prior probability function can be estimated by measuring the perception of a specific stimulus quantity (spatial frequency, orientation, curvature, symmetry axis orientation, distance between the eyes, object aspect ratio, …) while varying the amount of visual information (by varying stimulus visibility through changes in contrast, presentation time, contour colinearity, phase scrambling, …). As the amount of visual information is reduced, a Bayesian observer will rely more on its prior probability function to estimate the stimulus quantity and, consequently, display a perceptual bias towards stimulus values favored by the prior. Formally, reducing amount of visual information by, e.g., lowering contrast broadens the likelihood function. Consequently, the likelihood function receives less weight in the posterior, which will mainly reflect the shape of the prior. Perceptual judgements based on posterior mean or mode will be biased towards the mean or mode of the prior.
Results:
Stocker and Simoncelli (2006) measured the perceived speed of a drifting grating as a function of grating contrast in a 2-AFC speed discrimination experiment. At low contrasts, observers not only respond more variably but also significantly underestimate the speed of the drifting grating. Such a contrast-induced bias is predicted by the Bayesian observer model when assuming a prior favouring low speeds. A prior favouring low speeds may be well matched to the statistics of motion in natural scenes as most objects in our environment are either stationary or moving at slow speeds. It should be noted that obtaining/implementing a subjective prior matching the objective prior may not be straightforward in the case of motion, as one may need to discount self-motion (including eye-movements).
A stimulus domain for which the statistics of natural scenes are very well defined and objectively measured is spatial frequency. Natural scenes have a highly-consistent 1/f spatial frequency amplitude spectrum: as spatial frequency increases, power decreases proportionally (Burton and Moorhead, 1987; Field, 1987; Ruderman and Bialek, 1994; Simoncelli and Olshausen, 2001; Tolhurst et al., 1992). The 1/f spatial frequency spectrum is a fundamental property, implying that natural scenes are scale-invariant fractal patterns: zooming or changing viewing distance does not affect their spectral composition.
Based on the 1/f spatial frequency spectrum of natural scenes, one could say that high spatial frequencies are relatively unlikely to occur. It seems reasonable to define the objective spatial frequency prior using a 1/f function: prior probability and spatial frequency are inversely related. This prior favours low spatial frequencies. Hence, a Bayesian observer model incorporating a 1/f prior predicts a perceptual bias towards low spatial frequencies (i.e., large spatial scales) as stimulus visibility is reduced. I measured perceived grating spatial frequency as a function of grating contrast (which is comparable to Stocker & Simoncelli measuring temporal frequency as a function of grating contrast). However, I did not find a bias towards low spatial frequencies as grating contrast is reduced. Instead, perceived spatial frequency is strongly biased towards high spatial frequencies at low contrasts. Assuming human observers are Bayesian, they are using a subjective prior favouring high spatial frequencies although these frequencies are objectively a-priori unlikely. It should be noted that spatial frequency tuning of early visual areas is contrast-independent because of contrast gain control and normalisation (the goal of these mechanisms is to avoid perceptual effects of contrast).
A lot of work has been done focussing on the adaptation of the visual system to the 1/f structure of natural scenes. A 1/f spatial frequency spectrum implies spatial correlations in the image and, consequently, redundancy in second-order statistics as the luminance value of a specific pixel can be predicted based on the luminance value of neighbouring pixels. According to the efficient coding hypothesis, early stages of the visual system have evolved to remove such higher-order redundancy in order to obtain a sparse code, representing a given natural image using a minimal number of independently-active neurons. In this context, it has been suggested that the early visual system amplifies high spatial frequencies in order to "whiten" the average response to 1/f images. Our measured subjective prior could reflect the effects of this high spatial frequency amplification.
Topics/issues/questions to discuss:
- Can simplicity be related to efficient coding (redundancy reduction, minimum energy consumption and spare coding) ?
- If so, likelihood vs. simplicity frameworks might generate different predictions in the context of this paradigm. Bayesian models predict a perceptual bias to stimuli that occur frequently in the world. The simplicity framework may predict the opposite under some circumstances.
- Attempts have been made to obtain objective or semi-objective priors in other stimulus domains such as curvature (cf. Geisler). An interesting question is whether perceptual biases can be found on these dimensions. For instance, does a curved line look straighter when presented at low contrast?
- Is the proposed 1/f spatial frequency prior an objective prior? Can we obtain similar well-defined and measurable priors for mid-level and high-level vision?
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