Prior knowledge can help observers in various situations. Adults can simultaneously learn two location priors and integrate these with sensory information to locate hidden objects. Importantly, observers weight prior and sensory (likelihood) information differently depending on their respective reliabilities, in line with principles of Bayesian inference. Yet, there is limited evidence that observers actually perform Bayesian inference, rather than a heuristic, such as forming a look-up table. To distinguish these possibilities, we ask whether previously learned priors will be immediately integrated with a new, untrained likelihood. If observers use Bayesian principles, they should immediately put less weight on the new, less reliable, likelihood (“Bayesian transfer”). In an initial experiment, observers estimated the position of a hidden target, drawn from one of two distinct distributions, using sensory and prior information. The sensory cue consisted of dots drawn from a Gaussian distribution centered on the true location with either low, medium, or high variance; the latter introduced after block three of five to test for evidence of Bayesian transfer. Observers did not weight the cue (relative to the prior) significantly less in the high compared to medium variance condition, counter to Bayesian predictions. However, when explicitly informed of the different prior variabilities, observers placed less weight on the new high variance likelihood (“Bayesian transfer”), yet, substantially diverged from ideal. Much of this divergence can be captured by a model that weights sensory information, according only to internal noise in using the cue. These results emphasize the limits of Bayesian models in complex tasks.

_{pl}= 1% of screen width) or wide (SD of σ

_{ph}= 2.5% of screen width) priors. The side of the screen associated with each prior was counterbalanced across participants. One was always 35% of the way across the screen (from left to right), and the other 70%. When the narrow prior was centered on 35%, for example, the wide prior had a mean in the opposite side of the screen (i.e. to the right, centered on 70%). When the octopus appeared on the left-hand side (drawn from the prior centered on 35%) it was white, and when it appeared on the right (drawn from the prior centered on 70%) it was black.

_{ll}= 0.6% screen width), medium (σ

_{lm}= 3% screen width), or high (σ

_{lh}= 6% screen width) SD (in the following referring to as low, medium, and high variance likelihood conditions). The horizontal positions of the dots were scaled so that their SD was equal to the true SD (σ

_{ll}, σ

_{lm},

*or*σ

_{lh}) on each trial while preserving the mean of the dots. We performed this correction so that participants would “see” the same variability across trials for each likelihood condition. This ensures that an observer who computes the reliability for the likelihood information trial by trial would always calculate the same value within likelihood trial types. The vertical positions of the dots were spaced at equal intervals from the vertical center of the screen, with half of the dots appearing above, and the other half below the center. The vertical distance between each dot was fixed and equal to 1% screen width. Given that the vertical positions of the dots were fixed, only the horizontal position of the target was relevant. Participants estimated location only along the horizontal axis by moving a vertical green rectangle (measuring 1% of screen width in width and 3% of screen width in height) left or right, making this a one-dimensional estimation task. Participants received feedback in the form of a red dot (0.5% of screen width in diameter) that represented the true target position.

*n*is the number of dots that indicate the likelihood (in this case, there were 8 dots), and \(\sigma _p^2\) is the variance of the prior.

*R*

^{2}) was calculated by linearly regressing participants’ responses against each estimate participants could have taken from the cue (i.e. arithmetic mean, robust average, median, or mid-range). This was done for the combined data of all subjects in each experiment, across all blocks and trial types (prior and likelihood pairings). The estimate with the highest

*R*

^{2}value was taken to be the estimate participants had most likely used.

*R*

^{2}= 0.996), relative to the robust average (

*R*

^{2}= 0.995), median (

*R*

^{2}= 0.995), or the mid-range of the dots (

*R*

^{2}= 0.992). This suggests that the mean of the dots is the estimate that participants take from the sensory cue.

*p*< 0.001). There was also a main effect of likelihood variance (

*p*< 0.001), where participants relied less on the medium variance likelihood. However, there was also a significant interaction effect of likelihood and prior (

*p*= 0.001). When the prior was narrow, the decrease in reliance on the likelihood was smaller as the likelihood variance increased (

*t*(25) = 3.57,

*p*= 0.001).

*p*< 0.001) and an interaction between block and likelihood (

*p*= 0.014), with the medium variance likelihood weighted significantly differently across blocks (simple main effect of block, \(F ( {4,100} ) = 5.84, p < 0.001, \eta _{p}^2 = 0.189\), weights decrease with increasing exposure), but not the low variance likelihood (no simple main effect of block, \(F ( {4,100} ) = 1.64, p = 0.169,\eta _{p}^2 = 0.063\)). This suggests that participants adjusted, through practice, their weights on the medium variance likelihood, getting closer to optimal.

*t*(25) = − 0.77,

*p*= 0.450). They were significantly different in the narrow prior condition (

*t*(25) = − 2.78,

*p*= 0.010), although the bias was extremely small (95% confidence interval [CI]: 0.06, 0.41 percent of the screen width to the left). The median SD of responses for all subjects was 1.4% (narrow prior) and 2.5% (wide prior): almost identical to the true prior SDs of 1.3% and 2.5%, respectively.

*p*< 0.001 in all blocks for the medium likelihood when paired with either prior). This over-reliance on the likelihood is in line with previous studies (e.g. Bejjanki et al., 2016), although stronger in the present study. Participants, therefore, accounted for changes in the probabilities involved in the task (e.g. weighted the likelihood less when it was more variable), but did not perform exactly as predicted by the optimal strategy.

*p*< 0.001). However, post hoc analyses showed that the weight placed on the high likelihood was not significantly lower than the weight placed on the medium likelihood (

*p*= 0.103). Only the comparison of the weights placed on the likelihood in low and high variance trial types was significant (

*p*< 0.001). Moreover, there was no main effect of block (

*p*= 0.28), nor an interaction between block and likelihood (

*p*= 0.48), suggesting that the weight placed on the newly introduced likelihood variance did not decrease with increasing exposure.

*t*(25) = .784,

*p*= 0.440); low variance likelihood, wide prior:

*t*(25) = − 1.12,

*p*= 0.270). Subjects’ weights differed significantly from optimal in all other conditions (

*p*< 0.001 in all cases).

*R*

^{2}= 0.991 for both), followed by the median (

*R*

^{2}= 0.990) and the mid-range (

*R*

^{2}= 0.989). We, thus, proceed with the mean as the estimate from the likelihood.

*p*< 0.001) and as the prior uncertainty decreased (main effect of prior,

*p*= 0.002). However, unlike in Experiment 1, there was no significant interaction of these factors (

*p*= 0.123) (see Table 3).

*p*=0.02) and an interaction between block and likelihood (

*p*=0.01), with participants weighting the likelihood significantly less with increasing task exposure (regardless of prior) when its variance was medium (\(F ( {2.21,25.34} ) = 3.81, p = 0.03, \eta _p^2 = 0.257\), with a Greenhouse-Geisser correction), but not when it was low (\(F ( {4,44} ) = 0.70, p = 0.60, \eta _p^2 = 0.060\)).

*t*(11) = − 0.002,

*p*= 0.999; wide prior:

*t*(11) = 0.35,

*p*= 0.734). The median SD of responses was also remarkably similar to the true prior SDs (narrow prior: 1.6% vs. 1.3% in screen units; wide prior: 2.5% for both).

*p*< 0.001 in both cases, for both priors). However, it is worth noting that the weights placed on the medium and high likelihoods are closer to optimal than they were in Experiment 1 (compare bar heights in Figures 2 and 3).

*p*< 0.001). Unlike in Experiment 1, post hoc analysis showed that the weight placed on the high likelihood was significantly lower than the weight placed on the medium likelihood (

*p*= 0.034). The weights placed on the likelihood in the low variance trial type were significantly lower than those in the medium and high variance trial types (

*p*< 0.001 for both). Moreover, there was no main effect of block (

*p*= 0.64), or an interaction effect of block and likelihood (

*p*= 0.15), meaning that the weight placed on the newly added likelihood information did not vary with increasing exposure.

*p*< 0.001 in both cases), but not when it was low, irrespective of prior variance (low likelihood, narrow prior:

*t*(11) = .120,

*p*= 0.907); low likelihood, wide prior:

*t*(11) = − 1.29,

*p*= 0.163).

*R*

^{2}= 0.990). The amount of variance explained decreased for the robust average (

*R*

^{2}= 0.989), median (

*R*

^{2}= 0.988), and the mid-range of the dots (

*R*

^{2}= 0.985). We, thus, proceed with the mean as the estimate from the likelihood.

*p*< 0.001). Specifically, subjects placed significantly more weight on the low likelihood than on the medium (

*p*= 0.001) or high likelihood (

*p*< 0.001), and more weight on the medium likelihood than the high likelihood (

*p*= 0.005). No other main effects or interactions were significant (see Table 5 for a summary of results).

*t*(11) = − 1.14,

*p*= 0.278), but were significantly different from the mean of the narrow prior (

*t*(11) = − 3.91,

*p*= 0.002) (although we note that the bias was small (95% CI: 0.24, 0.87 percent of the screen width to the left). The median SD of responses was 2.2% for the narrow prior condition and 2.6% for the wide prior condition; the SD of responses was, therefore, only close to the true variance of the wide prior (which was 2.5%). Together, these findings suggest that subjects had not learned either the mean, or the variance of the narrow prior condition. This may explain the lack of difference in performance between the narrow and wide prior conditions in this task.

*p*< 0.001), with the exception of the wide prior/ low likelihood condition (

*t*(11) = − .362,

*p*= 0.724).

*p*< 0.01). Indeed, Figure 5 shows that the internal noise model still predicts less weight on the sensory cue than we see in our data (compare bars and dotted lines). This could reflect participants downweighing the prior because it is, in fact, subject to additional internal noise, stemming from a need to remember and recall the correct prior from memory. Even so, empirical weights were closer to the internal noise predictions, compared to those predicted by the optimal strategy (with experimentally controlled cue variance, dashed lines).

*p*< 0.001). No significant difference was observed when the likelihood variance was low, and the prior variance was wide (

*p*= 0.79). This means that accounting for the added internal variability fails to explain our results as observers are placing more weight on the sensory cue than is optimal, not less.

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