**Abstract**:

**Abstract**
One of the key features of active perception is the ability to predict critical sensory events. Humans and animals can implicitly learn statistical regularities in the timing of events and use them to improve behavioral performance. Here, we used a signal detection approach to investigate whether such improvements in performance result from changes of perceptual sensitivity or rather from adjustments of a response criterion. In a regular sequence of briefly presented stimuli, human observers performed a noise-limited motion detection task by monitoring the stimulus stream for the appearance of a designated target direction. We manipulated target predictability through the hazard rate, which specifies the likelihood that a target is about to occur, given it has not occurred so far. Analyses of response accuracy revealed that improvements in performance could be accounted for by adjustments of the response criterion; a growing hazard rate was paralleled by an increasing tendency to report the presence of a target. In contrast, the hazard rate did not affect perceptual sensitivity. Consistent with previous research, we also found that reaction time decreases as the hazard rate grows. A simple rise-to-threshold model could well describe this decrease and attribute predictability effects to threshold adjustments rather than changes in information supply. We conclude that, even under conditions of full attention and constant perceptual sensitivity, behavioral performance can be optimized by dynamically adjusting the response criterion to meet ongoing changes in the likelihood of a target.

*i*is the index for interval and

*p*is the probability of a target signal. As we took the mean to be 4,

*p*was 0.25. In case the draw from the geometric distribution produced a number larger than 7, no target signal was presented and the trial was considered a no-go, in which subjects had to withhold the response.

Interval | ||||||

1 | 2 | 3 | 4 | 5 | 6 | |

Increasing hazard rate | 450 | 357 | 265 | 179 | 111 | 55 |

Constant hazard rate | 450 | 282 | 174 | 102 | 58 | 34 |

*d*′) and response criterion (

*C*). We consider the point at which an observer's responses are neither biased toward “yes” nor “no” as the zero bias point.

*C*gives the number of standard deviation units that the criterion is above or below the zero bias point. Defined in this way, the criterion

*C*is expressed in the same units as the sensitivity

*d*′ (Gescheider, 1997). These two measures were determined for every combination of subject, signal interval, and hazard rate condition. Statistical significance was evaluated with a two-way Analysis of Variance (ANOVA), separately for

*d*′ and

*C*, involving the within-subject factors hazard rate condition (increasing vs. constant) and stimulus interval (1–6). Interval 7 was excluded from the analyses because, with an increasing hazard rate, false alarms cannot be made here. The resulting

*p*values were adjusted, whenever appropriate, for violations of the sphericity assumption using the Greenhouse-Geyser correction. To further check the validity of our results, we analyzed our data using nonparametric methods based on marginal effects analysis (Brunner & Pur, 2001) or Monte Carlo simulations (Anderson, 2001). Both nonparametric variants of the ANOVA yielded very similar results leading to identical conclusions.

*SD*= 190) under an increasing hazard rate and from 450 to 1,800 ms (mean = 799,

*SD*= 195) under a constant hazard rate. To assess statistical significance, we performed an ANOVA on RT analogous to the one on accuracy, except for the inclusion of the additional within-subject factor target presence (present vs. absent).

*r*, the standard deviation of the slope

*r*, and six independent parameters for distance to threshold (

*S*–

_{T}*S*), one for each presentation interval. In the shift variant, we only allowed the slope

_{0}*r*to vary between intervals, resulting in the same number of parameters: the distance to threshold (

*S*–

_{T}*S*), the standard deviation of slope

_{0}*r*, and six independent parameters for mean slope

*r,*one for each interval. As these two variants of the LATER model have identical degrees of freedom, model superiority was determined simply on the basis of the difference in log likelihoods (Reddi et al., 2003).

*d*′) and response criterion (

*C*). Whether stimulus predictability affects sensitivity, the criterion, or both can most easily be seen in ROC curves. An ROC curve commonly assumes a constant sensitivity and describes how hit and false alarm rates change with changes in criterion. Thus, if stimulus predictability only influences the criterion, plotting hit and false alarm rates across intervals will resemble an ROC curve.

*cti*(Figure 2A through F). As the target signal became more and more predictable, the hit rate strongly increased (Figure 2A, filled symbols). This increase in hit rate, however, was accompanied by an increase in the number of false alarms (Figure 2A, open symbols). Plotting hit rate versus false alarm rate across intervals (Figure 2C, numbers 1–6) yields a series of data points that nearly fall along a theoretical ROC curve (Figure 2C, gray curve), suggesting a constant sensitivity and a variable criterion. Conversely, in the condition in which the likelihood of a target signal coming next did not increase, hits and false alarm rates did not change as much with passing intervals (Figure 2B). As a consequence, rather than describing an ROC curve with a constant value of

*d*′, these data points tend to form a cluster (Figure 2D). To assess the reliability of these findings, we compared effects of interval on sensitivity and on criterion between the hazard rate conditions. Effects of hazard rate on the criterion were systematic (Figure 2E): At a criterion of zero (grey horizontal line), the subject is equally likely to respond with

*yes*or

*no*regarding the presence of a target. As the target signal became more and more likely (solid line), the subject's tendency for

*no*responses decreased, culminating in a bias toward

*yes*responses. Under the condition in which the target likelihood did not change, however, the tendency for

*no*responses remained on a fairly constant level (dotted line). Sensitivity, in contrast, seemed to fluctuate unspecifically both within and between hazard rate conditions (Figure 2F).

*kao*(Figure 2G through L). With an increasing target signal likelihood, hit and false alarm rates change concurrently, and these changes are well described by a theoretical ROC curve (Figure 2G and I). This well-ordered relation breaks down when target signal likelihood does not increase with passing intervals (Figure 2H and J). Indeed, effects of interval and hazard rate on sensitivity (Figure 2L) are rather variable, when compared to those on the criterion (Figure 2K).

*p*= 0.04) and of the interaction between target signal interval and hazard rate condition (

*p*= 0.006). Post-hoc comparisons indicated that the criterion dropped when the hazard rate increased (

*p*= 0.01), but did not change when the hazard rate remained constant (

*p*= 0.32). Thus, the main effect of target signal interval arose because of the relatively strong decrease of the criterion with an increasing hazard rate. From this we can conclude that subjects adjust their decision criterion over time, and they only do so when the target signal becomes more and more predictable.

*p*= 0.37) or target signal interval (

*p*= 0.41), nor their interaction (

*p*= 0.77) approached statistical significance. From this we can conclude that, across intervals, sensitivity remains unchanged, irrespective of the predictability of the target signal.

*p*< 0.0001), of hazard rate condition (

*p*= 0.01), and a strong trend toward a significant interaction between interval and hazard rate condition (

*p*= 0.06). Observers were faster, overall, for correct responses than for false alarms and for an increasing compared to constant hazard rate. The interaction effect arose because, from interval to interval, RTs strongly decreased under an increasing hazard rate (post-hoc comparison,

*p*= 0.02), but did not change under a constant hazard rate (

*p*= 0.5). None of the other interactions approached statistical significance, indicating that the relation between hazard rate condition and interval did not depend on whether a target was present or absent. Thus, irrespective of the presence of a target signal, increasing stimulus predictability decreases RTs, which is consistent with adjustments of the response criterion, rather than changes in sensory processing.

*S*(middle), which rises linearly from start level

*S*to a threshold

_{0}*S*. The crossing of the threshold initiates a behavioral response (top). The slope

_{T}*r*of the decision signal

*S*varies randomly from trial to trial following a Gaussian distribution, which produces the commonly observed skew in the distribution of RTs (shaded areas). The LATER model predicts that the reciprocal of RT is normally distributed, which can be verified in a

*reciprobit*graph (Figure 5B).

*S*–

_{T}*S*), rate of information supply (slope

_{0}*r*), or both. Pure changes in either parameter would be particularly simple to describe. If observers adjusted distance to threshold, it would effectively rotate the reciprobit line around the infinite time intercept (swivel variant, Figure 5C). In contrast, changes in slope

*r*would shift the line along the horizontal axis (shift variant, Figure 5D).

*p*> 0.12 and

*p*> 0.3 for increasing and constant hazard rates). We also fitted a shift variant of the model, in which only slope

*r*was allowed to vary. Under an increasing hazard rate, the swivel model outperformed the shift model (log likelihood difference of 6.71), indicating that the observed distributions of RT can largely be accounted for by adjustments of the distance to threshold.

*S*is taken to represent the prior log likelihood of a target signal being present. The decision signal

_{0}*S*represents the posterior log likelihood which, when high enough, triggers a response. According to Bayes' rule, the posterior log likelihood, and hence RT, should be linearly related to variations in prior log likelihood. Experiments manipulating prior target likelihood have documented this relationship (Carpenter & Williams, 1995), showing that, under such conditions, observers seem to behave in the Bayesian optimal way: Prior knowledge of target likelihood is incorporated by setting the appropriate starting level

*S*. In our paradigm, as the hazard rate increases, the prior likelihood of a target being shown changes from interval to interval, and Bayesian optimal performance would therefore require adjusting the distance to threshold accordingly.Indeed, we observed a linear relation between median RT and the logarithm of the hazard rate (Figure 5G). From this we can conclude that our observers optimize behavioral performance by taking into consideration step-wise changes in the prior likelihood of a target signal.

_{0}*N*, where

*N*is the end of the sequence. With continuous stimuli, in contrast, an observer's prediction will become less precise with passing time (Bausenhart, Rolke, Seibold, & Ulrich, 2010) leading some researchers to approximate an observer's presentation of the actual hazard rate by a subjective hazard rate (Bueti et al., 2010; Janssen & Shadlen, 2005). Though we instructed our subjects to aim for accuracy rather than speed, the pattern of RTs is very consistent with the results of pioneering RT studies. A reduction in RT with an increasing hazard rate has been taken as evidence that observers can anticipate the appearance of the target signal (Niemi & Näätänen, 1981). The fact that we can replicate this basic pattern of RT shows that our manipulation of expectancies was successful, indicating that effects of stimulus predictability can also be studied with classic signal detection paradigms.

*d*′. Indeed, predictability-related changes in

*d*′ have been reported in two recent behavioral studies (Correa et al., 2005; Rolke & Hofmann, 2007). How can this apparent discrepancy be explained?

*d*′. In contrast, in our task, observers were required to detect a weak coherent motion signal, which was embedded in noise. The neuronal mechanisms underlying stimulus detection might differ from those mediating discrimination (Hol & Treue, 2001; Jazayeri & Movshon, 2007) and might not leave a trace in common measures of perceptual sensitivity.

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