How the known reference weakens the visual oblique effect: a Bayesian account of cognitive improvement by cue influence

This paper investigates the influence of a known cue on the oblique effect in orientation identification and explains how subjects integrate cue information to identify target orientations. We design the psychophysical task in which subjects estimate target orientations in the presence of a known oriented reference line. For comparison the control experiments without the reference are conducted. Under Bayesian inference framework, a cue integration model is proposed to explain the perceptual improvement in the presence of the reference. The maximum likelihood estimates of the parameters of our model are obtained. In the presence of the reference, the variability and biases of identification are significantly reduced and the oblique effect of orientation identification is obviously weakened. Moreover, the identification of orientation in the vicinity of the reference line is consistently biased away from the reference line (i.e., reference repulsion). Comparing the predictions of the model with the experimental results, the Bayesian Least Squares estimator under the Variable-Precision encoding (BLS_VP) provides a better description of the experimental outcomes and captures the trade-off relationship of bias and precision of identification. Our results provide a useful step toward a better understanding of human visual perception in context of the known cues.

general, gravitational, egocentric and visual reference frames may be considered in the visual task 3 and the three frames are often aligned 19 . In a fine direction discrimination task using moving random-dot stimuli, the subjects reported whether the direction of motion was clockwise or counterclockwise of a decision boundary indicated by a bar outside the edge of the dot-field. The observers' perception of the direction of motion is biased away from the boundary (the unknown explicit reference) 20 . In forced-choice orientation discrimination, when the reference and the stimulus are presented simultaneously and there is a sufficient distance between them to avoid orientation interaction, the oblique effect of discrimination is reduced considerably 12 . Rauber and Treue conducted an estimation task of direction, in which an unknown-direction line segments presented simultaneously with the moving random-dot pattern at the edge of motion area. There are the robust systematic biases which deviate away from the nearest horizontal cardinal direction. Their results also show that the subjects may have selected different references and their values to estimate goals according to experimental tasks and stimuli 18 . Wiese and Wenderoth 21 have reproduced Rauber and Treue's experiments in which the moving dots and the reference line presented simultaneously. Their experiment results also exhibit reference repulsion from the horizontal cardinal directions. But there is no unified statement about the influnce of the references. Zamboni et al. asked subjects estimated the direction of motion in the presence or absence of unknown reference lines. The unknown reference line was randomly chosen for each trial and presented at the edge of a circular aperture. The estimated mean directions were influenced by the position of the reference line 11 . Patten et al. used experiments to explore the behavioral and neural consequences of manipulating stimulus certainty in the context of orientation processing. The reference line on outside of the stimulus was chosen to prevent a biased representation of cardinal and primary oblique orientations and was not used as an independent variable 16 .
Meanwhile, the congnitive process of cues integration is a higher-level process and the vision perception can be described as a Bayesian inference process 11,15,[22][23][24] . Bayesian inference theories successfully explicate many visual neuropsychological phenomena [25][26][27][28][29] . The Bayesian inference framework provides the probabilistic way in which the target orientations are optimally estimated by integrating current sensory input and prior knowledge of these targets 30 . Once there exist the contextual cues in the visual task, humans can integrate available information of cues to achieve the optimal or suboptimal perception results in a complex environment, such as in sensorimotor learning 26 . Based on the framework of Bayesian inference, Ma and Jazayeri 27 studied how subjects integrate the contextual cues information to obtain measurements of task. Furthermore, Ma et al. 31 and Rich et al. 32 concluded that Bayesian inference can be implemented through the probabilistic population coding model.
Although there are many studies on orientation identification, the psychophysical experiments and the involving Bayesian orientation identification model under the condition that the known reference line intersects with the target has not been considered in previous studies. Here we investigate the influence of a known oriented reference line on orientation identification, and provide an explanation of the identification outcomes under the Bayesian inference framework. To address this issue, we design the orientation identification task. Eleven participants completed the orientation identification task with and without reference line. The experimental results suggest that the known tilted reference line significantly affects the identification accuracy and the oblique effect of orientation identification is obviously weakened.
We then provide an explanation of the identification outcomes in the presence of the reference line under the Bayesian inference framework. We model the orientation identification as a probabilistic encoding-decoding process. The Equal-Precision encoding (the same precision across stimuli and trials) and Variable-Precision encoding (precision vary across stimuli) patterns are considered in the encoding process. Comparing the predictions of model to the experimental data, we find that the data are best explained by Bayesian Least Squares estimator under the Variable-Precision (BLS_VP) encoding pattern.

Materials and methods
Ethics statement. The procedures were carried out in accordance with the National Commission for the Protection of Subjects of Behavioral Research guidelines for human studies. Before the study was initiated, informed consent was acquired for each subject and approval was obtained from the local Ethics Committee of Nanjing University of Aeronautics and Astronautics (IRB number 201807).
Participants. Eleven subjects participated in the orientation identification task (four females and seven males; age range 18-28 years old). All of them have normal or corrected-to-normal visions and they were naive to the purpose of the experiments.
Apparatus and stimuli. Line orientation stimuli and a known reference line were presented on a 21 in. liquid crystal display (LCD) with 120 Hz refresh rate and 1024 × 768 pixel resolutions. In order to suppress the influence of the edges of the screen on identification, all stimuli were presented in a central gray disk with 400 pixels in radius. Luminance was set to the middle range (16 cd/m 2 ). We kept the room's lights off for the entire duration of each trial session. Subjects' head positions were controlled using a chin and forehead rest located 54 cm away from the computer screen. The height was individually adjusted to hold the subjects' eyes level to the center of the screen in order to ensure the gravitational, egocentric or body-centered are aligned. The PsychoPy software package 33 was used to generate the stimuli and the reference line, and record timing and the estimates. The target stimulus was a green line (length 500 pixels; width 2 pixels; RGB [0, 255, 0]) and the reference line was a light gray dashed line (length 200 pixels; width 1 pixel; RGB [200,200,200]).
We chose the orientation of reference line at 45°. This is because the identification precisions of five oblique orientations (i.e., 15°, 30°, 45°, 60° and 75°) were lower than the precision of cardinal orientations 34  Experimental procedure. In this orientation identification task, the subjects identified a target orientation and entered estimated value via the keyboard. Each trial started with the appearance of a central fixed point in the gray disk for 700 ms. Then the target stimulus line and the known reference line were displayed simultaneously on the screen (Fig. 1a). The subjects were instructed to estimate and input the values of target orientations via the keyboard (if making a mistake, the subjects can press R to re-enter). The estimated values were recorded. To ensure that the subjects have enough time to adapt their responses to the range of target orientations and further obtain the prior knowledge of target orientations, each subject performed 400 trials before the experiment, which is called the Learning phase. In the Learning phase, the correct feedback of each stimulus was provided after the estimate was input, followed by 700 ms time interval (Fig. 1a). After completing all trials in the learning phase, each subject carried out the orientation identification experiments without feedback, which is called the Estimating phase. Each subject completed 800 trials in the estimating phase. These two phases of psychophysical tests were performed on different days. The subjects first performed control experiment without the known reference line. Other than the presence of the known reference line, the procedure of control experiment is the same as the one with reference. Prior to the experimental trials, each subject was informed about the degree of the reference line and underwent some practice to familiarize oneself with the tasks and procedure. www.nature.com/scientificreports/ The subjects were asked to rest for at least 10 s when the experiments were carried out in half. There was no time limit to identify a target stimulus for each subject.

Data analysis.
To evaluate the perceptual performance in each situation, we first calculated three metrics for each target orientation separately. These three metrics are defined as follows.
1) Identification bias (BIAS): the mean of the estimation errors of orientations. That is, the bias of the j th target stimulus s j is BIAS j = n i=1 x ji − s j n , where x ji is the i th estimate of s j and n is the number of the stimulus s j appears in the experiment. The value BIAS is signed and reflects the degree to which a subject's estimates deviates on average from the true stimulus value. If the sign is positive, it means that the true stimulus is overestimated, and vice versa.
x ji /n is the mean of estimates for s j . The SD value reflects the variability of the estimated orientations. The smaller the value SD, the higher the identification precision of the estimates, and vice versa.
is a measure of the overall quality of orientation identification. The smaller the value MSE, the better the quality, and vice versa.
In each situation with or without reference line, we computed these three metrics for each of 25 stimulus orientations (referred to by subscript j): BIAS j , SD j and MSE j (j = 1, . . . , 25). The three metrics satisfied: 25). The overall BIAS and SD of single subject's data are defined as: The corresponding overall mean squared error of identification is defined as: MSE S = BIAS 2 + SD 2 , which highlights the trade-off between bias and identification precision. We computed these three metrics for all subjects' data. A repeated-measures two-way ANOVA was used to determine any significant influences of the reference on the identification outcomes. www.nature.com/scientificreports/ Bayesian cue integration model of orientation identification. To quantitatively explain the psychophysical results, we put forward a cue integration model based on the Bayesian inference framework. The subjects are assumed to be Bayesian observers who can optimally estimate target orientations in a noisy environment by integrating prior knowledge of target orientations and current sensory input 26 . The identification process of orientation is modeled as Bayesian probabilistic inference, and consists of an encoding stage and a Bayesian decoding stage 36,37 (Fig. 2a). We assume that the observers consider the known reference line as a distinct sensory cue, which provides the subjects with lots of additional contextual information. Let θ be the target orientation, and R = 45 o be the known reference's orientation. Define α = θ − R as the cognitive angle (Fig. 1b). Thus, the estimation of a target orientation is transformed into the estimation of an cognitive angle α.
In the encoding stage, as a result of sensory noise, the subjects obtain the cognitive angle's sensory measurement or evidence α m , which may differ from the true angle α . The primary visual cortex implements the encoding process in brain 38 . Repeated appearance of the same target orientation resulted in slightly different measurements, which could be described by a conditional probability distribution p(α m |α, σ ) . We model the conditional probability distribution of sensory measurement as a Gaussian distribution with mean of α (Fig. 2b). Two types of encoding precision of the conditional probability distribution are considered 39-41 : • Equal-Precision (EP) encoding (Fig. 2b), in which sensory measurements have the same precision across trials and stimuli, σ = . In this case, the conditional probability distribution of sensory measurement α m can be denoted by p EP (α m |α, ) = N(α m ; α, 2 ). • Variable-Precision (VP) encoding (Fig. 2b), in which sensory measurement precision vary across stimuli.
In the decoding stage, the Bayesian observer can combine with the prior probability distribution of target angle p(α) and the conditional probability of sensory measurement p(α m |α, ) to compute the posterior distribution of target angle p(α|α m , ) (Fig. 2a). Then, given a cost function L(α, α e ) which quantifies the cost of estimating true orientation α as α e , the observer obtains the Bayesian estimator α e of true angle α by minimizing the expected posterior risk. According to the relationship between the target orientation θ and the cognitive angle α , the estimator of the stimulus orientation is derived, that is, θ e = R + α e .
To estimate the cognitive angle, the subjects need to compute the posterior distribution of target orientation p(α|α m , ) using Bayes' rule as follows: where α min and α max are the minimum and maximum value of supporting set of prior distribution of cognitive angle, respectively. The prior distribution of cognitive angle p(α) is a uniform distribution.
Given an objective cost function L(α, α e ) , Bayesian estimator of the angle α is chosen to minimize the expected cost averaged over the posterior distribution of target orientation. According to the relationship of the cognitive angle and target orientation, the observer could infer the estimator of target orientation, When the subjects chose the cost function L(α, α e ) = −δ(α − α e ) , where δ(·) denotes the Dirac delta function, the corresponding Bayesian estimator is obtained by maximizing a posteriori (MAP), then    www.nature.com/scientificreports/ Under the EP and VP encoding, we can also obtain the corresponding Bayesian estimators θ e,BLS_EP and θ e,BLS_VP , respectively. The two estimators have no analytical expressions because integrals of Eq. (6) are not analytically solvable. We can obtain the approximate values of two estimators using the Simpson rule to compute the two integrals in Eq. (6). Parameter estimation of model. Based on the experimental data in the presence of reference line, the maximum-likelihood estimates of parameter of model are obtained using Monte Carlo simulations. There is one parameter in model: the standard deviation of the sensory measurement distribution, σ = (equal precision pattern), or Weber fraction, σ = |α| (variable precision pattern). We wish to find the most probable parameter value which maximizes the likelihood function given the psychophysics experimental data set. Let α j be the j-th cognitive angle stimuli and α ji e be the i-th estimate of α j . The experimental data set is expressed as (α j , α where M denotes the one of four types of the estimators. By taking the logarithm of both sides, we can obtain Based on the conditional independence of Bayesian network, the variables (α, ) and the estimator of angle α e are conditionally independent given sensory measurement α m . The conditional probability p(α e |α, ) can be expressed as where δ(·) denotes the Dirac delta function.
Since the probability in Eq. (8) cannot be computed analytically, we estimate numerically the sum of the logarithm likelihood function by Monte Carlo simulations 41 . For a given parameter value , type M and the j th cognitive angle α j , we perform the following procedure to compute the probability p(α

Results
Experimental results. For each subject, except for trials data in the learning phase, we collected 800 trials data in the estimating phase. By analyzing experimental data, the following three features are revealed from the experimental data.
First, the identification precision is obviously improved when the known reference line is presented (Fig. 3a). In the absence of reference, the SD values of 25 target oblique orientations range from 4.43° to 6.15° and are obviously larger than the cardinal orientations' SD values (0° (0.501), 90° (1.114)) which is from the results of our preliminary experiment (Supplementary Table S1). In the presence of reference, as the target orientation moves closer to the reference line, the identification precision becomes higher. All subjects can accurately discriminate the stimuli of the 45°, 44° and 46° (Fig. 3a). We make 2 (reference situation) × 25 (orientation) repeated-measures two-way ANOVA for the SD. The results suggest that the known reference line significantly affects the precision  www.nature.com/scientificreports/ of identification ( F 1,500 = 373, p < 0.0001 ) and the interaction between "reference situation" and "orientations" has significant effects on the SD ( F 24,500 = 3.57, p < 0.0010). Second, the biases of identification across all subjects in two situations present different characteristics. The known reference line significantly affects the BIAS ( F 1,500 = 4, p = 0.0455 ). When the reference line is absent, the stimuli of less than 45° are overestimated and the ones more than 45° are underestimated (Figs. 3b, 4a left  panel). The phenomenon is known as the regression effect [45][46][47] . In the presence of reference, the reference line divides the stimulus interval into two parts and there exist two-regression effects in the two sub-intervals. Moreover, in the vicinity of the reference line, the stimuli of less than 45° are underestimated, while those over 45° are overestimated (Figs. 3b, 4a right panel). The sign of biases is opposite to the ones without the reference line. The biases of orientation perception show the reference repulsion phenomenon 18 . The biases of orientation identification in the vicinity of the orientation of 45° decrease dramatically. The results of repeated-measures two-way ANOVA illustrates the interaction between reference situation and orientations has significant effects on the BIAS ( F 24,500 = 106.93, p < 0.0001).
Third, in the presence of reference, experimental data show that there is a reasonable trade-off between the precision and the bias of identification (Fig. 4b). Suppose each subject has stable identification ability (MSE S ), according to the relationship among the three metrics, the BIAS decreases when the SD increases, and vice versa.  www.nature.com/scientificreports/ Figure 4b shows that the precision of identification tends to be improved by adjusting the bias of estimation. They reasonably consider the relationship of BIAS and SD in an identification task. In summary, the experimental results show that the known reference line has a remarkable impact on the orientation identification. The findings indicate that the subjects can integrate the information of the known reference into orientation identification. Moreover, the known reference line might be one of the important factors that affect the identification.
Model fitting and comparison. We put forward a Bayesian cue integration model to account for these identification results. Based on different assumptions of encoding precision of sensory stimulus and cost functions in decoding process, we obtain four Bayesian estimators: MAP_EP (Equal-Precision Maximum posteriori estimator), BLS_EP (Equal-Precision Bayesian least squares estimator), MAP_VP (Variable-Precision Maximum posteriori estimator), BLS_VP (Variable-Precision Bayesian least squares estimator). Each estimator has one parameter: the standard deviation of the sensory measurement distribution, = σ (equal precision pattern), or Weber fraction, = w (variable precision pattern). For each subject's data and total data across eleven subjects, the maximum likelihood estimate of parameter ( σ or w ) is approximately obtained. We compare each model's predictions to the experimental data using the trade-off between the BIAS, SD and MSE. At the same time, we also used the cross-entropy (CE) to compare the four models. It is computed where N is the number of samples, k is the number of classification labels, and p ik and p ik are the actual probability and estimated probability respectively. The values for three metrics and cross-entropy can be found in Supplementary informations (Table S4).
We can also theoretically compute the mean and variance of four Bayesian estimators through the Eqs. (4), (5) and (6). The mean and variance of MAP_EP estimator are as follows: The mean and variance of MAP_VP estimator are as follows: Table 1. The estimates of the parameter and the values for three metrics and cross-entropy baesd on the total data across eleven subjects. MLE is maximum likelihood estimates of parameter for the models. MSE S are the overall mean square errors. The BIAS and SD denote the overall bias and standard deviation of identification errors, respectively (in degrees). CE is the cross-entroy between the actual probability distribution and the predicted probability distribution under each model.    The Variable-Precision Bayesian models better fit the experimental data than the Equal-Precision models, and the BLS_VP estimator is better than the other three estimators. As we see from the Table 1, the MSE S and CE scores of Variable-Precision estimators obviously are less than that of Equal-Precision counterparts. Figure 5 shows that the Variable-Precision models provide a reasonable description of the precision of experimental data, especially in the vicinity of 45°. Figure 6 displays four contrast histograms for the biases of total data and the biases of model predictions. The MAP_EP estimator is theoretically optimal due to the unbiasedness (Eq. 10), but it clearly fails to capture the characteristic of experimental data (Figs. 5b, 6a). The prediction biases of the MAP_ VP and BLS_EP models also fail to account for the phenomenon of reference repulsion (Fig. 6b,c). The BLS_VP model best explains the reference repulsion and two-regression effects of the experimental data (Figs. 5e, 6d). Figure 7 illustrates that the BLS_VP model is obviously better than the others at capturing the trade-off relationship between the bias and precision of the data. The BIAS of the MAP estimator are obviously less than that of experimental data and their SDs are larger than that of the data (Fig. 7a,c,d). The BLS_EP model also underestimates the BIAS of orientations and overestimates the SDs of the data across all subjects (Fig. 7b left  panel, c,d). However, the BIAS and the SD of the BLS_VP estimator are consistent with the experimental results ( Fig. 7b right panel, c,d).

Discussion
Visual perception can be influenced by contextual cues. In this paper, we have investigated the influence of a known reference line on the orientation identification. We design the experiments in which the subjects estimate the orientations of target lines in the presence of a known reference line, and the control experiments without reference are also conducted. In the absence of reference, the oblique effect and the regression effect exist, which is consistent with the results in the previous studies 34,44,46 .
In the presence of a known reference, the orientation identification performance is obviously improved. The oblique effect is weakened with the improvement of the identification precision of the oblique orientations, especially in the vicinity of the reference line. But the presentation of known reference line may not abolish the type II oblique effect completely. Moreover, we observe the phenomena of two-regression in intervals of (22°, 45°) and (45°, 68°) respectively. A reasonable explanation may be that the stimulus interval is divided into two subintervals by the known reference line, and the orientation identification in the two subintervals respectively presents regression effect. If the orientation of reference is not 45°, the centers of regressions may be correspondingly shifted. Furthermore, there exist robust systematic biases which are deviated from the reference line (i.e., reference repulsion 11,18,20 ). The phenomenon may be caused by the overestimate of the angle between the stimulus and the reference.
At the same time, the observer's visual system can reasonably trade off the relationship of the bias and the precision to improve the overall identification performance. The observer wants to be able to accurately identify target orientations. But in most cases, there exist systematic deviation and uncertainty. We suppose the identification ability is limited (MSE is fixed). The observer may lower the requirements of bias to improve the identification precision. The trade-off relationships also appeared in judgment-time task 44 . From the framework of efficient coding the oblique effect may be interpreted as a result of the neural representations of different orientations in sensory systems 48,49 . The experimental phenomena in the presence of reference could be the results of efficient coding in preference-based decisions.
Bayesian inference provides successful description for the phenomena in many cognitive tasks 25,26,29,44,50 . In the paper a Bayesian cue integration models are proposed to describe the experimental results. The process of identification consists of two stages: encoding stage and Bayesian decoding stage. For a given stimulus, the encoding in brain is characterized by the conditional probability distribution of the sensory measurement. The equal precision and variable precision patterns are considered in encoding stage.
In decoding stage, a Bayesian estimator is obtained based on the posterior distribution of the angle α and the cost function. The results show that The Variable-Precision Bayesian models better fit the experimental data and the Variable-Precision Bayesian least squares estimator (BLS_VP) is superior to the others. Our results have revealed that human visual orientation identification can be reasonably described as the outcomes of a Bayesian inference process that optimally integrates the information of known reference into orientation identification. The findings provide an important step toward a better understanding of human visual perception given the known cues. (12) VAR θ e,MAP_VP |θ = 2 α 2 −1+ √ 1+4 2 2 2 , α min ≤ α ≤ α max 0, otherwise