The perceptual representation of our environment does not only involve what we actually can see, but also inferences about what is hidden from our sight. For example, in amodal completion, simple contours or surfaces are filled-in behind occluding objects allowing for a complete representation. This is important for many everyday tasks, such as visual search, foraging, and object handling. Although there is support for completion of simple patterns from behavioral and neurophysiological studies, it is unclear if these mechanisms extend to complex, irregular patterns. Here, we show that the number of hidden objects on partially occluded surfaces is underestimated. Observers did not consider accurately the number of visible objects and the proportion of occlusion to infer the number of hidden objects, although these quantities were perceived accurately and reliably. However, visible objects were not simply ignored: estimations of hidden objects increased when the visible objects formed a line across the occluder and decreased when the visible objects formed a line outside of the occluder. Confidence ratings for numerosity estimation were similar for fully visible and partially occluded surfaces. These results suggest that perceptual inferences about what is hidden in our environment can be very inaccurate und underestimate the complexity of the environment.

*s*,

*l*) was determined by the diameter of the holes, where

*s*and

*l*denote the diameter of the small and large holes in pixels, respectively. The holes in the occluder were arranged to be symmetric.

*I*) was changing gradually, from light blue (R = 49, G = 87, and B = 133) to dark blue (R = 10, G = 25, and B = 44) from the upper left to the lower right. The luminance of the original cloud image (

_{sky}*I*) was 255 (R = G = B = 255). There were two sizes and luminance of the original star image (

_{cloud}*I*). The size of the large stars was 3 × 3 pixels, and the luminance was 171 (R = G = 176, and B = 128). The size of the small stars was 2 × 2 pixels, and the luminance was 213 (R = G = 220, and B = 160). Large and small stars were arranged randomly on the sky.

_{star}*I*), the original cloud image (

_{sky}*I*), and the original star image (

_{cloud}*I*). The weight for the cloud image (

_{star}*w*) was obtained by adjusting a noise map. A Brownian (1/

_{cloud}*f*

^{2}) noise map of the same size as the cloud image was first generated. Then the noise map was shifted downward by subtracting 0.6, after which values smaller than 0 were clamped at 0. The weight for the star image (

*w*) was generated by applying Gaussian smoothing with a variance of 0.9 and 1.2 for small and large stars, respectively. In such a way, the night sky with clouds and stars was obtained by:

_{star}*t*-tests (two tailed). The alpha-level was set to 0.05. A nonlinear least-squares fitting was adopted to fit the models.

*a*is the scaling factor of the amplitude and modulates the minimum confidence judgment. The PSE of the confidence judgment was defined as the mean of the underlying Gaussian.

*f*(

*x*) =

*p*

_{1}

*x*+

*p*

_{2}, where

*p*

_{1}indicates the slope and

*p*

_{2}indicates the intercept.

*S*) was obtained by counting the grid cells where pieces would be completely hidden by the occluder. The visible area of the game board (

_{h}*S*) was obtained by subtracting the completely hidden cells from the total number of cells of the game board.

_{v}*n*) can be estimated based on the number of visible objects (

_{h}*n*) and the area of the visible (

_{v}*S*

_{v }) and occluded (

*S*) parts:

_{h}*N*) and hidden cells (

_{v}*N*) corresponds to the number of the two categories of balls in the urn. The total number of objects (

_{h}*n*) corresponds to the number of draws from the urn. The expected number of visible (

_{v+h}*n*) and hidden (

_{v}*n*) objects then corresponds to the resulting number of balls from the two categories. Upper- and lowercase characters denote quantities of the urn and the samples, respectively. A hypergeometric distribution allows to calculate the probability that a certain number of objects (

_{h}*n*) are visible given the total number objects (

_{v}*n*):

_{v+h}*P*(

*n*

_{v + h}|

*n*)) can be estimated by multiplying the likelihood distribution (

_{v}*P*(

*n*|

_{v}*n*

_{v + h})) with the prior probability (

*P*(

*n*

_{v + h})) of the total number of objects, divided by a normalization factor (

*P*(

*n*)):

_{v}*N*= 44), two game boards with variable numbers of game pieces appeared simultaneously. For each pair of numerosities, three conditions were investigated: (1) both game boards were fully visible (non-occluded), (2) both game boards were partially occluded (occluded), and (3) one of the game boards was fully visible, and the other was partially occluded (mixed; see Figure 1A). One of the game boards had a fixed number of visible pieces of 10 (standard) and the number of visible pieces on the other game board varied from two to 18 (comparison). In the mixed condition, the standard was occluded and the comparison was not occluded. We calculated the difference between standard and comparison numerosity so that zero indicates that both game boards had the same number of visible pieces. In each trial, observers had to perform two tasks. In the selection task, observers had to choose the game board with more game pieces. In the subsequent confidence task, observers had to report if their confidence about that numerosity selection was high or low.

*p*= 0.099) in the non-occluded (0.71 ± 0.19), occluded (0.73 ± 0.24), and mixed condition (0.76 ± 0.21). This indicates that observers were similarily confident in the mixed and the occluded condition as in the non-occluded condition, although they were objectively missing information due to the occlusion and should have been less confident about their responses than in the non-occluded condition.

*p*< 0.001), but also significantly larger than the expected number based on Equation 4 (t(43) = 7.80,

*p*< 0.001). This indicates that observers underestimated the number of hidden pieces. Similarly, the point of minimum confidence was −1.99 (±1.08), which was significantly smaller than zero (t(43) = −12.21,

*p*< 0.001), but also significantly larger than the expected number (t(43) = 4.33,

*p*< 0.001). The PSEs in the selection and the confidence task were highly correlated (r(42) = 0.56,

*p*< 0.001). The fact that both, the PSE in the selection task and the point of minimum confidence in the confidence task underestimated the number of hidden pieces is evidence that this underestimation was truly a perceptual effect and not merely a response bias (Gallagher et al., 2019; Luna et al., 2021; Maldonado Moscoso, Cicchini, Arrighi, & Burr, 2020).

*n*), which would also lead to an underestimation of the number of hidden pieces according to Equation 4. Because the selection task provides only relative numerosity judgments, this cannot be excluded based on the data of experiment 1. Third, observers might have been underestimating the proportion of the occluded relative to the visible area of the game board (

_{v}*S*/

_{h}*S*), which would also lead to an underestimation of the number of hidden pieces. Fourth, none of the reasons above holds and observers might not have been estimating the number of hidden pieces according to Equation 4 at all.

_{v}*n*in Equation 4), we asked observers to report the number of visible and hidden game pieces separately in experiment 2.

_{v}*N*= 30), only one game board with game pieces appeared in each trial. For each numerosity, three conditions were investigated: (1) the game board was fully visible (non-occluded), (2) the game board was partially occluded by a small occluder as used in experiment 1 (small-occluder; Figure 2A), and (3) the game board was partially occluded by a large occluder with smaller holes (large-occluder; Figure 2B). In each trial, observers had to perform two direct estimation tasks and a binary choice task. In the first direct estimation task, observers had to estimate the number of visible pieces. In the subsequent choice task, observers had to report if they perceived the game board as being occluded or not. Finally, in the second direct estimation task, observers had to estimate the number of hidden pieces.

*p*= 0.291; intercept: F(2,56) = 0.44,

*p*= 0.649) between the non-occluded (slope 0.81 ± 0.13, intercept 1.31 ± 0.89), small-occluder (slope 0.78 ± 0.15, intercept 1.57 ± 0.98), and large-occluder (slope 0.82 ± 0.17, intercept 1.36 ± 1.30) conditions. Slopes around 0.8 and intercepts around one in all three conditions indicate that observers slightly overestimated the number of visible pieces for small numerosities and slightly underestimated for large numerosities. Nevertheless, in general, observers were able to estimate the number of visible pieces quite accurately, making it clear that the underestimation of hidden objects in experiment 1 was not caused by the underestimation of visible objects.

*p*= 0.029) in the small-occluder (0.03 ± 0.20) and large-occluder (−0.02 ± 0.19) conditions, both were significantly smaller than expected (small-occluder: t(29) = −6.74,

*p*< 0.001; large occluder: t(29) = −30.39,

*p*< 0.001). The intercepts were significantly different from each other (t(29) = 2.58,

*p*= 0.015) and significantly larger than 0 in the small-occluder (1.80 ± 2.17, t(29) = 4.54,

*p*< 0.001) and large-occluder (2.44 ± 2.43, t(29) = 5.51,

*p*< 0.001) conditions. These intercepts were roughly consistent with the PSE shifts in experiment 1, of about −1.45 and −1.99 in the selection and the confidence task, respectively. In the non-occluded condition, slopes (0.02 ± 0.04, t(29) = 2.69,

*p*= 0.011) and intercepts (0.04 ± 0.54, t(29) = 0.36,

*p*= 0.723) were close to zero, indicating that observers did not hallucinate hidden pieces when there was no occlusion.

*n*) and the proportion of the occluded relative to the visible area of the game board (

_{v}*S*/

_{h}*S*). Experiment 2 showed that observers were able to accurately estimate the number of visible pieces

_{v}*n*but still were very inaccurate in the estimation of the number of hidden pieces. Possibly, they were unable to estimate the relative proportion of occluded and visible area of the game board (

_{v}*S*/

_{h}*S*) (Palmer, Brooks, & Lai, 2007; Scherzer & Ekroll, 2015). To investigate whether the underestimation was caused by underestimating the occlusion, we asked observers to report the proportion of the occluded area in experiment 3.

_{v}*N*= 28), the proportion of occluded area and the spatial scale of the game board was varied (Figures 4A, B).

*p*= 0.034), and the intercept (Figure 4E) was slightly smaller than 0 (−0.05 ± 0.13, t(27) = −2.22,

*p*= 0.035). This indicates that observers were able to accurately estimate the occluded area. Hence, the underestimation of hidden pieces in experiments 1 and 2 was not caused by underestimating the proportion of the occluded relative to the visible area of the game board (

*S*/

_{h}*S*). A remaining explanation for the inaccurate estimation of the number of hidden pieces would be that observers simply did not pay attention to the visible pieces for their estimation of hidden pieces. Therefore, we tested in experiment 4, if the arrangement of the visible pieces modulates the estimation of hidden pieces.

_{v}*N*= 29), one game board with game pieces appeared. For each numerosity, three conditions were investigated: (1) pieces arranged irregularly, like in experiments 1 and 2 (irregular), (2) pieces arranged regularly, forming a line across the occluded area (regular-across; Figure 5A), (3) pieces arranged regularly, forming a line outside of the occluded area (regular-outside; Figure 5B). To facilitate the generation of regular patterns across and outside the occluder, the shape of the occluder was changed to vertical or horizontal bars. In the irregular and regular-outside conditions, the expected number of hidden pieces was obtained according to Equation 1. In the regular-across condition, the expected hidden pieces are the ones along the straight path formed by the visible pieces. We arranged the visible pieces such that the expected number of hidden pieces was the same in both regular conditions across and outside the occluder. In each trial, observers had to perform two direct estimation tasks like in experiment 2 to estimate the number of visible and hidden pieces separately. Like in experiment 2, we analyzed the estimated number of visible and hidden pieces as a linear function of the actual number of visible pieces (Figures 5C, D).

*p*= 0.084). They were slightly smaller than 1 in the irregular condition (0.93 ± 0.11, t(29) = −3.43,

*p*= 0.002), but not in the regular-across (0.98 ± 0.12, t(29) = −0.80,

*p*= 0.428) and regular-outside (1.00 ± 0.08, t(29) = 0.24,

*p*= 0.816) conditions. The intercepts were significantly different in the three conditions (F(2,56) = 3.29,

*p*= 0.044). They were slightly larger than zero in the irregular condition (0.48 ± 0.71, t(29) = 3.67,

*p*< 0.001), but not in the regular-across (0.23 ± 0.78, t(29) = 1.64,

*p*= 0.111) and regular-outside (−0.02 ± 0.57, t(29) = −0.17,

*p*= 0.867) conditions. This indicates that the estimation of the number of visible pieces was accurate, for both irregular and regular patterns of pieces.

*p*= 0.038) and significantly smaller than 0.8 in the irregular (0.15 ± 0.20, t(29) = −17.88,

*p*< 0.001), regular-across (0.27 ± 0.24, t(29) = −12.36,

*p*< 0.001), and regular-outside (0.08 ± 0.29 , t(29) = −13.83,

*p*< 0.001) conditions. The intercepts in the irregular, regular-across and regular-outside conditions were not significantly different from each other (F(2,56) = 0.58,

*p*= 0.565). All intercepts were significantly larger than zero (irregular: 1.62 ± 3.00, t(29) = 2.95,

*p*= 0.006; regular-outside: 1.70 ± 3.32, t(29) = 2.81,

*p*= 0.009; and regular-across: 2.14 ± 2.60, t(29) = 4.51,

*p*< 0.001).

*N*= 28), an image of the night sky with stars and clouds appeared in each trial, and observers had to perform three direct estimation tasks: First, observers had to estimate the number of visible stars. Second, observers had to report the proportion of the sky that was covered by the clouds. Third, observers had to estimate the number of hidden stars.

*p*= 0.113) and the intercept was not significantly different from 0 (0.73 ± 1.73, t(27) = 2.224,

*p*= 0.305). This indicates that observers were able to estimate the number of visible stars accurately. For the estimation of the proportion of occlusion, the slope was 0.61 (±0.27) and significantly smaller than 1 (t(27) = −7.853,

*p*< 0.001). The intercept was around 0.13 (±0.13) and significantly larger than 0 (t(27) = 5.190,

*p*< 0.001). This indicates that observers underestimated the proportion of the sky covered by clouds.

*p*< 0.001). The intercept was 2.55 (±2.12) and significantly larger than 0 (t(27) = 6.347,

*p*< 0.001). This indicates that observers underestimated the number of hidden stars relative to the objectively expected number according to Equation 4.

*p*= 0.002) the explained variance from 0.15 (±0.13) with the objective prediction to 0.27 (±0.25) with the subjective prediction (see Figure 6F). However, the slope was with 0.21 (±0.23) still significantly smaller than 1 (t(27) = −18.572,

*p*< 0.001), and the intercept was with 2.37 (±2.06) still significantly larger than 0 (t(27) = 6.080,

*p*< 0.001). This indicates that observers still heavily underestimated the number of hidden stars even when taking into account their individual misestimations of the number of visible stars and the proportion of occlusion (Appendix).

**Authors’ contributions**: A.A., A.C.S., and H.M. designed the experiment. A.C.S. and H.M. analyzed the data. A.C.S. and H.M. visualized the data. H.M. wrote the first draft. A.A., A.C.S., and H.M. revised the manuscript.

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*p*< 0.001).

*p*< 0.001) and smaller than 0.84 in the large occluder condition (t(29) = −24.43,

*p*< 0.001).

*p*= 0.201) in the non-occluded (1.44 ± 0.62), occluded (1.60 ± 0.55) and mixed condition (1.71 ± 0.88). In the confidence task, the JND was significantly different (F(2,84) = 3.56,

*p*= 0.033) in the non-occluded (2.78 ± 1.66), occluded (3.62 ± 2.17), and mixed condition (3.43 ± 1.80). Hence, the precision was not affected by occlusion in the perceptual discrimination task, but in the confidence task. This indicates that observers were less certain about their confidence choices under conditions of occlusion.

*p*= 0.070), but in the typical range of Weber fractions of about 0.1 (Anobile et al., 2020; Pomè et al., 2019; non-occluded: 0.09 ± 0.04; smaller occluder: 0.10 ± 0.04; large-occluder: 0.10 ± 0.04). The CVs of the number estimation of hidden pieces (F(2,56) = 19.30,

*p*< 0.001) were significantly different from each other, but also close to 0.1 in the non-occluded (0.05 ± 0.07), small-occluder (0.12 ± 0.08), and large-occluder (0.13 ± 0.09) conditions.

*p*= 0.429) and again in the typical range in the irregular (0.16 ± 0.12), regular-across (0.15 ± 0.13), and regular-outside (0.17 ± 0.18) conditions (Figure A1C).

*p*< 0.001). The estimation of the number of visible stars was in the typical range (0.15 ± 0.05). The estimation of the hidden proportion was higher (0.27 ± 0.09) but very similar to the CV of the same task in experiment 3. This shows that the observers were consistent in their estimations of the number of visible stars and the proportion of occlusion in the naturalistic scene.