**Abstract**:

**Abstract**
An object's retinal image size is determined by a combination of its physical size and its distance, so judgments of an object's size and distance from its retinal image size are coupled. Since one does not have direct access to information about the object's physical size, people may make assumptions about how large it is likely to be. Here we investigated whether the sizes of similar, previously encountered objects influence the assumptions about the physical size of an object and therefore the interpretation of its retinal image size in terms of its distance. Subjects moved their unseen index finger to the positions of binocular simulations of red cubes. For identical target cubes at the same position, they indicated a nearer position of the cube when the preceding cube was small than when it was big. This is in agreement with a tendency to expect the cube to be the same size as that on the previous trial. However, if the expectation were simply adjusted slightly on each trial, the cube would be judged to be nearer when preceded by two consecutive smaller cubes than when preceded by only one smaller cube. It was not, so there must be a more direct influence of the size in the previous trial on distance judgments.

*perceived*size of the previous one. If the perceived size of the previous object was partly based on assumptions about the size, and therefore recursively on the perceived size of objects that were presented before, the assumed size for the target will be based on a range of presented objects. Thus, the previously experienced sizes can constitute a likelihood distribution of possible sizes: a size prior that is given a weight that corresponds with its reliability (Mamassian & Landy, 2001) and of which the position of the peak shifts with experience (Adams, Graf, & Ernst, 2004).

*recent*experience, one can make two predictions: that the variability of recently perceived sizes will influence the reliability of the size prior and consequently the weight given to image size when judging distance, and that recently experienced sizes will influence the position of the peak of the prior and consequently the interpretation of image size in terms of judged distance.

*single condition*) and one condition in which each trial with a 1.5-cm cube was preceded by a pair of trials with 1- or 2-cm cubes (

*double condition*). In the single condition there were 100 possible positions, 50 for the 1.5-cm cube and 50 for the 1- and 2-cm cubes. The 1.5-cm cube was presented twice at each position, once preceded by a 1-cm cube and once preceded by a 2-cm cube. The preceding 1-cm cubes were also presented at the same positions as the preceding 2-cm cubes (which was a different position than that of the 1.5-cm cube). Altogether, the

*single condition*had 200 trials that were presented in one session. The

*double condition*was very similar to the single condition except that two 1- or 2-cm cubes were presented before each 1.5-cm cube. The same positions were used for the first of the two as for the one directly preceding the 1.5-cm cube (so each cube appeared twice at each of these positions), but the first positions were presented in a different random order so the two consecutive cubes of the same size were not also at the same distance. The

*double condition*was presented in two sessions. In each session there were 26 positions for each 1.5-cm cube, so there were 156 trials per session and in total 52 positions for the 1.5-cm cube. In both conditions, the order of the positions of the 1.5-cm cubes and the order of the cube sizes between the trials with 1.5-cm cubes was random. Note that although we only used three simulated cube sizes, they were at many simulated distances, so our subjects could not identify the stimulus set (Keefe & Watt, 2009).

*single*condition we subtracted the judged distance for the 1.5-cm cube preceded by a 1-cm cube from the judged distance for the matched 1.5-cm cube preceded by a 2-cm cube. For each subject we then averaged these values and determined the associated standard error. This gave us an estimate of the influence of the cube size on the previous trial. We also subtracted the judged distances for the 2-cm cubes from those for the 1-cm cubes at the same positions, which gave us an estimate of the influence of the current size.

*single condition*to make a prediction for the same measure in the

*double condition*. In this prediction it was assumed that after every trial the size prior is updated in the direction of the size in that trial (Equation 3 of the Appendix). The rate at which size is updated was estimated from the average influences of the current and previous object size on the pointing distance in the

*single condition*(Equation 9 of the Appendix). If consecutive trials of the same size are presented, and the assumed size is updated in the manner described above, the difference in pointing for the 1.5-cm cubes preceded by a pair of either 1-cm cubes or 2-cm cubes should be bigger than the difference in pointing for the 1.5-cm cubes preceded by only one 1-cm cube or one 2-cm cube. A quantitative prediction for the effect of having two preceding small or large cubes is given in Equation 15 of the Appendix. We checked whether the influence of the preceding cubes' sizes on subjects' average judged distances in the

*single*condition was significantly different from that in the

*double condition*with a paired

*t*test. A similar test was used to examine whether judged distance in the

*double condition*was significantly different from the prediction.

*double condition*, we subtracted the judged distance for the 1-cm cube preceded by another 1-cm cube (i.e., the second cube of each pair of 1-cm cubes) from the judged distance for the 2-cm cube preceded by another 2-cm cube (second cube of each pair of 2-cm cubes) that was presented at the same position. We then averaged these values and determined the associated standard error. The difference between judged distances for the 1- and 2-cm cubes preceded by a 1.5-cm cube (current size) should be larger than the difference between the judged distances for the 1- and 2-cm cubes preceded by another 1- or 2-cm cube (repeating current size), because when the second cube of the same size is presented, the assumed size will already have shifted slightly towards that size. The predicted difference in pointing distance is given in Equation 12 of the Appendix. We checked whether the effect of repeating the current size on judged distance was significantly different from the prediction with a paired

*t*test.

*single condition*and the open dots are the data for the

*double condition*. The positive values for the effect of the previous size show that the size on the previous trial influences distance judgments, which is consistent with shifting a size prior on the basis of recent experience. Equation 15 of the Appendix was used to predict how much larger the effect would be if there were two preceding 1- or 2-cm cubes, rather than only one, when the 1.5-cm cube was presented (open squares). The measured values do not fit the prediction. The influence of the previous trial for the 1.5-cm cubes should have been bigger in the

*double condition*than in the

*single condition*That is not the case. The vertical positions of the open and filled dots do not differ significantly (

**.***p*= 0.37). The open dots' vertical positions are significantly different (

*p*= 0.02) from those of the open squares, showing that the reasoning behind Equation 15 does not hold. No difference was expected between the

*current size*effects in the two conditions (horizontal positions of the open and filled dots), because these are always the large or small cubes that are preceded by a 1.5-cm cube, and indeed no systematic difference is found (

*p*= 0.32). The similarity of the effect of current size for the two conditions demonstrates that the differences between subjects were consistent across sessions.

*p*= 0.13).

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*d*) is related linearly to a weighted average of various cues to distance,

_{p}*d*, one of which (

_{i}*d*) is based on retinal image size (others could include vergence and accommodation): The distance indicated by the retinal image size cue on trial

_{s}*t*depends on the object's distance on that trial (

*d*) and on the relationship between the object's size (

_{t}*s*) and its assumed size (

_{t}*ŝ*) on that trial: Our hypothesis in the present study is that the sizes encountered on previous trials influence the assumed size. A straightforward method to model this is by updating the assumed size on the basis of the encountered size by a fraction

_{t}*u*on each trial: On the basis of these equations, and a few approximations, we can make and test two predictions for our experiment. The first is about the effect of presenting two rather than one cube of a different size (

**X**; which is either 1 or 2 cm) before the standard cube (

**Y**; 1.5 cm). Since we will test the prediction on the basis of average values, our first approximation is to assume that when the first cube of a different size is presented (at

*t*= 1), the former assumed size (

*ŝ*

_{0}) is

**Y**, because this is both the size at that moment and the average across preceding trials each step back in time. Thus, the assumed size when the first cube of a different size is presented is: Similarly, when returning to the standard cube after that, the assumed size will be: So, if we look at the difference between the expected pointing distance for a 1 and a 2-cm cube (Δ

_{1C}; effect of one current target of a different size), assuming that all other cues are identical in both cases and that all the distances are the same (as they are in our experiment), we can combine Equations 1 and 2 to get: which can be combined with Equation 4, filling in the actual sizes, to give: Similarly, for pointing at 1.5-cm cubes after being shown a 1 or a 2-cm cube (Δ

_{1P}; effect of one previous target of a different size), we get: If we average the values of Δ

_{1C}and Δ

_{1P}across trials, the only parameter that differs across trials, the distance, can be removed from the equations, because

*d̄*

_{1}=

*d̄*

_{2}. Subsequently, combining Equations 7 and 8 yields: The value of

*u*can thus be estimated from the measured values of Δ

_{1P}and Δ

_{1C}and used to make predictions for the condition in which there were two successive large or small cubes. For the second such cube, the assumed size will be: Note that the value of

*ŝ*is different than in Equation 5 because of the different cube size. Filling in this value for the first stage of Equation 8, we now get (for Δ

_{2}_{2C}; the effect of two consecutive targets of a different size on pointing distance): For the average values (i.e.,

*d̄*

_{1}=

*d̄*

_{2}), we can combine Equations 11, 7, and 9 to give: Finally, when pointing at 1.5-cm cubes after being shown either two 1 cm or two 2-cm cubes (Δ

_{2P}; the effect of two previous targets of a different size), the assumed size will be: So, for the effect on pointing distance we get:

*d̄*

_{1}=

*d̄*

_{3}), we can combine Equations 7, 8, 9, and 14 to give: