An experiment can be conducted to measure the effect of multiple visual cues on PEL—we will follow the experimental design from Matin and Li for comparison (Matin and Li, 2000). Subjects are seated upright with their heads stationary on a chin rest, one meter away from a blackened wall with pitched illuminated lines (visual cue). A laser point target is shown on the wall, along the subjects' midsagittal plane. The subject is asked to identify where their eye level is located by repositioning the laser target, without any visual cues present in the room. The PEL of the subject, due to internal stimuli, is recorded. After this initial measurement, a pair of vertical illuminated lines of different lengths are introduced on the wall. The subject is again tasked with lining up the laser target with their eye level. The PEL of the subject is recorded. This would be continuously repeated with the illuminated lines of different pitches and lengths until multiple measurements are collected for each subject. From this description, we apply a Bayesian framework to derive a model of PEL from first principles allowing us to predict the form of PEL expected prior to performing the experiment.

In order to use a Bayesian framework, we must define the prior, likelihood, and posterior distributions. Our prior in this experiment is the body-referenced mechanism for PEL estimation—the internal cues are what originally structure our belief about eye level before being presented with stimuli. We approximate this prior as a Gaussian distribution with a mean μ_b and standard deviation σ_b. There is also the estimate of eye level that comes directly from a geometric interpretation of visual information. In this case, lines can be pitched to give the appearance of a particular tilt from vertical, assuming the observed line is vertical but tilted due to geometric perspective. This purely visual contribution to the perception is μ_vi for individual visual cues and μ_v for the combination of visual cues—and similarly σ_vi and σ_v for the standard deviations. The posterior distribution, or the effect of n visual stimuli and internal cues on PEL, would be the combined effect of the body prior and the visual likelihood, described by μ_p and σ_p. This posterior distribution would then provide the estimate of the perceived of eye level. These distributions and their final combinations are demonstrated in Figure 1.

As a first step in deriving the equation for PEL in this experimental condition, we first combine the likelihoods of multiple pitched lines. Since we are using Gaussian distributions for our likelihoods, we can achieve this by multiplying them together. The resulting distribution is also Gaussian, with a mean μ_v and variance σv2: (1)μv  = [∑i=1nμviσvi2]σv2 (2)1σv2 = ∑i=1n1σvi2 As σv2 is in both equations, we can simplify through substitution to obtain the following likelihood estimate based only on visual information. This is the composite likelihood function which will be used in the derivation of μ_p. (3)μv  = [∑i=1nμviσvi2]*1∑i=1n1σvi2

To use Gaussian distributions for the priors and likelihoods, two variables are necessary for each stimulus—the mean and variance. The means for the body and individual visual stimuli can be assumed/easily measured and estimated based on geometric interpretation, respectively. However, the variances must be experimentally determined.

A common method to quantifying the effect of priors and likelihoods in Bayesian analysis is to observe the change in the percept as the mean of the likelihood is altered. When plotted, as done in Figure 2, the slope indicates the relative strength of the visual likelihood compared to the body-based prior. Analytically, this slope is directly related to the ratio of the variances, as shown in Equation (4). Using experimental data of true angle versus perceived angle from Matin and Li this slope, m, can be experimentally determined. (4)m= σb2σb2+ σv2 Figure 2

This graph depicts the experimental trends of average perceived eye level based on pitch. The prior (black), the body-referenced mechanism, is centered on the true eye level. The likelihood (blue), represents PEL if only the visual information was used. The experimentally measured PEL is shown for single short line (purple) and long line (green). The equation for the short-line regression is y = 0.1415x − 1.869, whereas the equation for the long-line regression is y = 0.3234x − 4.9429. These single line slopes are used to solve for the variances in our model, as shown in Equation (5).

Proceeding from Equation (3), if line length is constant, all visual stimuli will have equal variance (σ_vi are equal for all values of i). As such, σ_vi can be factored out of both summations and ∑i=1n1 becomes n. Further, the likelihood mean from each line, μ_vi, can be relabeled θ_i for clarity as the angle is known and does not need to be estimated. This allows us to create a new estimated likelihood mean based on visual cues of the same line length. (6)μv = [ 1σvi2∑i=1nθi] * 1nσvi2 We proceed to introduce the prior internal (“body”) mean, μ_b, and variance, σb2 (which are equivalent to the mean and variance of PEL in complete darkness). Using the same, multiplication-of-Gaussians approach to find the mean and variance of the posterior, as discussed in the Methods Section, we arrive at the following formulation for the mean PEL. (7)μp = [μb σb2+ 1σvi2∑i=1nθi]*11σb2+nσvi2 This equation can be algebraically rearranged and simplified to: (8)μp = μb1+ n σb2σvi2+ ∑i=1nθiσvi2σb2+ n However, we note that σ_vi depends on line length, and we would like a function independent of line length if possible. If we assume each additional segment of line length provides an independent sample to establish the slope of the line, we can approximate the relationship between line length and visual variance as an inversely proportional relationship. The validity of this independence assumption is clearly more questionable as line length increases, however, this approach provides a theoretically-justifiable approximation to parametrically estimate behavior when presented with variations in line lengths. Under this assumption, the form of that relationship is then similar to the standard error estimate from a sample mean, σvi=σvl/(l), where σ_vl is independent of line length. The result of this substitution yields: (9)μp = μb1+ nσb2lσvl2+ ∑i=1nθiσvl2σb2l+ n This is our final model, however, we can make a few substitutions to simplify use in an experimental setting. The first term in Equation (9) can be simplified as a function based on line length, abbreviated a(l), as given in Equation (11). Note, this indicates than an increase in the number of lines or line length should diminishes the value of a(l). Further, σvl2σb2 is composed of constants, so it can be substituted as a constant, k, as shown in Equation (12). This yields a simple form of the model to predict average PEL, given in the follow set of equations: (10)μc= a(l)+ ∑i=1nθikl+n (11)a(l)=μb1+ nlk (12)k= σvl2σb2

To test the validity of our derived Bayesian model in experimental circumstances we used data from the experiment outlined in Section Methods, performed by Matin and Li (2001). The experiment was performed with 2 line lengths (short 12° and long 64°) with pitches from −30 to 30° in 10° increments. According the Bayesian model as pitch angle (θ_i) increases, the mean PEL increases as well. However, PEL will never exceed the pitch angle. This is demonstrated in Figure 2. As such, PEL lies between estimates from internal cues (prior) and the pitch angle. Similarly, as the strength (line length) of a visual input increases, so does the influence of the visual stimuli on the perception of eye level, as PEL moves away from the prior. Both of these phenomena are demonstrated experimentally (Matin and Li, 2000, 2001).

The interpretation of multiple visual cues demonstrates an interesting perceptual trend. When two separate visual cues are observed independently, two distinct percepts of PEL are obtained. When the cues are given together, the resulting PEL appears to fall in two distinct regimes. Weak cues (e.g., short lines) PELs are additive, while strong cues (e.g., long lines) are averaging. This is a fact that have been observed and explained in previous psychophysical studies (Young et al., 1993; Gu et al., 2008) as well as neural cue integration (Fetsch et al., 2013). This was also demonstrated experimentally, long-line stimuli almost averaged in their influence on PEL and short-line stimuli appearing to be summed in the determination of PEL (Matin and Li, 1999, 2000). Figure 3 is an adaptation of the result from Matin and Li demonstrating this phenomenon. One can also observe from Equation (10) that as line length increases, the denominator tends toward “n” thus approaching an averaging effect, while shorter line lengths lead to the constant “k” dominating the denominator which leads to a summation effect. The compensatory effect of a(l) should be minimal given the tendency of the body prior to be near 0. In this way, this observed combination effect can be seen analytically in our model.

Note, this additive/averaging effect in cue combination is well understood in Bayesian literature. Our model also exhibits these effects straightforwardly by adjusting the likelihood variances based on line length. Weaker, short line stimuli lead to shallow, high-variance likelihoods which individually only marginally move the posterior from the body prior, but together the combined posterior moves farther than any one alone (“additive”). With the strong, low-variance likelihoods from long line stimuli, the posteriors are dominated by the likelihoods, leading to a combined posterior between the likelihoods when presented with two strong long line stimuli (“averaging”). The visual demonstration of this effect using the Bayesian framework is shown in Figure 4. The Bayesian approach successfully explains the effects observed with multiple cues.

From experimental results and fitting the behavioral data, a model was previously developed to explain the effects of multiple lines on the perception of eye level (Matin and Li, 2001).

Bayesian Model: (13)PEL=μp= a(l)+ ∑i=1nθikl+n Matin and Li Model: (14)PEL= a+ k1∑i=1nθik2l+n The Matin and Li model is very similar to our model. The notable difference between the previous model and the Bayesian model is the existence of k₁ in the Matin and Li model. Another small difference remains, as a is defined as a constant in the Matin and Li model, whereas it is derived as a function dependent on line length in our model. Our framework also allows the model to be derived for various different scenarios, such as for multiple lines of different lengths, as well as has the ability to be extended to other studies, perhaps even unrelated to PEL as studied here.

From experimental results shown in Figure 2 we calculated variance estimates for single line cues from a linear fit of the data. These results are presented in Table 1. With the prior and likelihood distributions' variance estimates, all necessary values are known for a complete Bayesian model. The fitted constants for Matin and Li's behavioral model (Equation 14) are displayed in Table 2. For each model we predicted the mean PEL given the experimental condition. Using the standard error in estimation we measured the accuracy of our predictions, displayed in Table 3. Both models are accurate (no standard errors over 3°) with the Matin and Li model slightly more accurate for long-line stimuli and the Bayesian model performing better with short-line stimuli. This is fitting with the independence assumption for how line segments contribute to variance estimates, which suggested the Bayesian model may not hold as well for longer line stimuli.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.