**Abstract**:

**Abstract**
**Looking for a target in a visual scene becomes more difficult as the number of stimuli increases. In a signal detection theory view, this is due to the cumulative effect of noise in the encoding of the distractors, and potentially on top of that, to an increase of the noise (i.e., a decrease of precision) per stimulus with set size, reflecting divided attention. It has long been argued that human visual search behavior can be accounted for by the first factor alone. While such an account seems to be adequate for search tasks in which all distractors have the same, known feature value (i.e., are maximally predictable), we recently found a clear effect of set size on encoding precision when distractors are drawn from a uniform distribution (i.e., when they are maximally unpredictable). Here we interpolate between these two extreme cases to examine which of both conclusions holds more generally as distractor statistics are varied. In one experiment, we vary the level of distractor heterogeneity; in another we dissociate distractor homogeneity from predictability. In all conditions in both experiments, we found a strong decrease of precision with increasing set size, suggesting that precision being independent of set size is the exception rather than the rule.**

^{2}. Stimuli consisted of Gabor patches with a spatial frequency of approximately 1.6 cycles/deg, a Gaussian envelope of approximately 0.29 deg, and a peak luminance of 122 cd/m

^{2}.

*N*stimuli (83 ms; five frames) (Figure 2a). Set size

*N*was pseudorandomly chosen to be 1, 2, 4, or 8. Stimuli were placed 45° apart on an imaginary circle at the center of the screen with a radius of 5° of visual angle. On target-present trials, the stimulus set consisted of one target stimulus and

*N*− 1 distractor stimuli. On target-absent trials, it consisted of

*N*distractor stimuli. The target stimulus was always vertical (denoted

*s*

_{T}and defined as 0°) and each distractor orientation

*s*was independently drawn from a Von Mises distribution centered at

_{i}*s*

_{T}: where

*T*= 0 refers to a distractor being present at the

_{i}*i*

^{th}location, and

*I*

_{0}is the modified Bessel function of the first kind of order 0. The concentration parameter of this distribution,

*κ*

_{D}, determined the level of heterogeneity in a display and was different across conditions:

*κ*

_{D}= 0 (high heterogeneity; uniform distribution),

*κ*

_{D}= 1 (medium heterogeneity), or

*κ*

_{D}= 8 (low heterogeneity; Figure 2b and c). The concentration parameter is inversely and monotonically related to the circular variance of the distractors. The observer reported by pressing a key whether or not the target was present and received immediate correctness feedback.

*κ*

_{D}= 0) to relate to our earlier results. Vincent et al. fixed set size at four, while our focus is on the effect of set size. We will also test a different set of models, specifically Bayes-optimal models and variable-precision models, which we will compare to Vincent et al.'s model.

*κ*

_{D}= 32.8 (corresponding to about 5°). The homogeneous condition was identical to the heterogeneous condition, except that on each trial a

*single*distractor orientation was drawn from the

*κ*

_{D}= 32.8 Von Mises distribution and assigned to all distractors. Set sizes were 1, 2, 4, and 8 in the homogeneous condition and 1, 2, 3, and 4 in the heterogeneous condition; the latter was different from Experiment 1 because the higher concentration parameter made this condition more difficult. Set size 1 trials had identical statistics in the two conditions. Each subject performed one session in the homogenous condition, followed by two sessions in the heterogeneous condition, followed by one session in the homogeneous condition. Each session consisted of four blocks of 175 trials.

*s*

_{1}, …,

*s*) the stimulus orientations on a given trial and by (

_{N}*x*

_{1}, …,

*x*) the corresponding noisy measurements. We assume that the measurements are drawn from independent Von Mises distributions on [0,

_{N}*π*),

*J*, which is a general measure of the amount of information that a random variable (

*x*in our case) carries about an unknown parameter upon which this variable depends (

_{i}*s*in our case). Fisher information is related to the Von Mises concentration parameter through where

_{i}*I*

_{1}is the modified Bessel function of the first kind of order 1 (Keshvari, van den Berg, & Ma, 2012; Mazyar et al., 2012; van den Berg, Shin, Chou, George, & Ma, 2012).

*J*, whose value we fit separately at each set size. In the VP model, we assume that

*J*is a gamma-distributed random variable with a mean that we fit separately for each set size and a scale parameter

*τ*that we assume to be constant across set size.

*C*= 0 denote the state of the target being absent,

*C*= 1 that of the target being present. The observer responds “target present” when, given the evidence in (

*x*

_{1}, …,

*x*), the posterior probability of target presence is greater than that of target absence,

_{N}*p*(

*C*= 1 |

*x*

_{1}, …,

*x*) >

_{N}*p*(

*C*= 0 |

*x*

_{1}, …,

*x*). A lengthy but straightforward derivation (see Appendix 1) reveals that this is equivalent to where

_{N}*p*

_{present}is the observer's prior belief of the target being present and is the likelihood ratio of target presence at location

*i*. To gain some intuition for Equation 3, consider the case that

*p*

_{present}= 0.5 and

*κ*= 0 for all

_{i}*i*; this means that no information is available. In this case, the posterior ratio (the left-hand side of Equation 3) equals 1. This is as expected because it means that the posterior probability that the target is present is 0.5. In another special case,

*κ*

_{D}= 0 (high-heterogeneity condition; uniform distractor distribution), the posterior ratio simplifies to which we studied before (Ma et al., 2011; Mazyar et al., 2012).

*p*

_{present}is not equal to 0.5, reflecting that the observer may not use the true frequencies of target-absent and target-present trials during inference. A value of

*p*

_{present}greater than 0.5 will make the observer respond “target present” in some cases when the evidence points towards target absence. This aspect of the model makes it strictly speaking suboptimal, and large deviations from 0.5 will make the observer stray far from performance maximization. However, since the observer otherwise accurately takes into account the structure of the generative model, we will for convenience still refer to this Bayesian model as optimal.

_{d}model

*i*

^{th}location, this ratio would be the summand in Equation 3), and responds “target present” if the largest of these ratios exceeds a fixed decision criterion

*k*: with

*d*as defined above. Contrary to the claims of Vincent et al. (2009), this is not an optimal model, because the decision rule is not equivalent to Equation 3 with

_{i}*p*

_{present}= 0.5. It is not clearly Bayesian either, because it is not obvious that there exists a generative model for which the decision rule maximizes performance (Ma, 2012). However, it is a very plausible model for human behavior.

*λ*that the observer produces a random guess instead of responding according to the decision rule of the respective model. This parameter can also capture unintended key presses.

*(M). This gives rise to a total of four models, which we will name EPO, EPM, VPO, and VPM. These models have six, six, seven, and seven free parameters, respectively: four (mean) precision parameters, a lapse rate parameter, a scale parameter in the VP models, a prior parameter in the optimal models, and a decision criterion in the max*

_{d}*models.*

_{d}*x*

_{0}=

*s*

_{T}and

*κ*

_{0}=

*κ*

_{D}. When

*p*

_{present}= 0.5 and

*κ*= 0 for all

_{i}*i*, the posterior ratio again equals 1.

*J̄*=

*J̄*

_{1}

*N*. Second, we fixed the lapse rate to 0. We fitted the homogeneous and heterogeneous conditions together, assuming that

^{α}*p*

_{present}and

*J̄*

_{1}are shared between conditions. For

*J̄*

_{1}, we make this assumption because at set size 1, the stimulus statistics are identical in the two conditions. The power

*α*and the scale parameter

*τ*can vary between the conditions. Thus, in this experiment, the VPO model has six free parameters to fit both conditions.

*κ*

_{D}) on performance,

*F*(2, 10) = 19.5,

*p*< 0.001, but no effect of session number,

*F*(1, 5) = 2.73,

*p*= 0.16, and no interaction,

*F*(2, 10) = 2.06,

*p*= 0.18 (Figure 3a). Moreover, at every level of heterogeneity, hit rate decreased (one-way repeated-measures ANOVA,

*F*(3, 15) > 14.7,

*p*< 0.001) and false-alarm rate increased (

*F*(3, 15) > 21.2,

*p*< 0.001), as a function of set size (Figure 3b).

*J̄*) as a function of set size in both VP models. For each level of heterogeneity, the relationship between set size and mean precision is fitted well by a power-law function with a power close to −1 in both models (−0.95 ± 0.26, −1.08 ± 0.20, and −0.91 ± 0.08 for

*κ*

_{D}= 0, 1, and 8, respectively in the VPO model, and −1.06 ± 0.17, −1.17 ± 0.13, and −1.34 ± 0.06 in the VPM model; mean and

*SEM*across subjects), similar to our previous results (Mazyar et al., 2012). We also tested a variant of the VPO model with the prior probability

*p*

_{present}fixed to 0.5. Paired

*t*tests showed no significant differences between the powers obtained with this model and those obtained with the “fixed prior” model (

*p*= 0.41,

*p*= 0.48, and

*p*= 0.71 for

*κ*

_{D}= 0, 1, and 8, respectively).

*F*(3, 36) = 15.7,

*p*< 0.001, and heterogeneous,

*F*(3, 36) = 30.2,

*p*< 0.001, conditions. There was also a significant effect of set size on the false alarm rates in both conditions (homogeneous:

*F*(3, 36) = 3.73,

*p*< 0.05; heterogeneous:

*F*(3, 36) = 6.90,

*p*< 0.01).

*I*

_{0}(

*κ*

_{C}_{=1,i})/

*I*

_{0}(

*κ*

_{C}_{=0}) reflects the “oddball effect.” The former is obvious: The smaller the difference between the measurement

*x*and the target orientation

_{i}*s*

_{T}, the larger $ e \kappa i \u2009 cos 2 ( x i \u2212 s T ) $. The latter requires a bit more thought. If we represent the

*i*

^{th}measurement as a vector in the plane with angle 2

*x*and length

_{i}*κ*, then

_{i}*κ*

_{C}_{=0}, given by the second line of Equation 6, is the length of the vector sum of all measurements, and

*κ*

_{C}_{=1,i}, given by the first line, is the length of the vector sum of all measurements except for the

*i*

^{th}one. These lengths are greater when the vectors in the sum are more aligned with each other, that is, when the measurements are closer to each other. Therefore, the factor

*I*

_{0}(

*κ*

_{C}_{=1,i})/

*I*

_{0}(

*κ*

_{C}_{=0}) is the answer to the question: If I were to leave out the

*i*

^{th}measurement, how strongly aligned would the remaining measurements be relative to the original alignment of all measurements? If the

*i*

^{th}measurement is different from all others, as will often be the case if it was produced by the target, then alignment would increase by leaving it out, and the factor

*I*

_{0}(

*κ*

_{C}_{=1,i})/

*I*

_{0}(

*κ*

_{C}_{=0}) would be greater than 1. Therefore, this ratio measures the strength of the evidence that the

*i*

^{th}stimulus is an odd element in the display. In Experiment 1, since the distractors always differed amongst each other, there is no such factor.

*t*(12) = 0.10,

*p*= 0.92).

*predictable*. Stated otherwise, unless visual displays are largely predictable across trials, the spreading of visual attention has detrimental effects on the quality of encoding of each stimulus.

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*C*(target presence, 0 or 1),

*T*(target presence at the

_{i}*i*

^{th}location, 0 or 1),

*s*(orientation at the

_{i}*i*

^{th}location), and

*x*(measurement of orientation at the

_{i}*i*

^{th}location). The homogeneous condition in Experiment 2 has an extra variable,

*s*

_{D}(common distractor orientation). We denote the vector (

*T*

_{1},…,

*T*) by

_{N}**T**, and similarly for

**s**and

**x**. A probability distribution is associated with each variable in the generative model. The equations are given in Figure A1. Some distributions are common to both the heterogeneous and the homogeneous condition. In the order of Figure A1, these distributions reflect that there is a probability that the target is present (

*p*

_{present}); that if the target is absent, it is absent everywhere; that if the target is present, it is only at one location, selected with equal probability; that the target always has value

*s*

_{T}; that the noise corrupting a measurement is independent across locations; and that it follows a Von Mises distribution centered at the true orientation (Equation 2). A few distributions are specific to the experiment. For heterogeneous distractors, these reflect that given target presence at a location, the orientation at that location is not influenced by other locations, and that each distractor orientation is drawn from a Von Mises distribution centered at

*s*

_{T}(Equation 1). For homogeneous distractors, the distributions reflect that a common distractor orientation

*s*

_{D}is drawn from a Von Mises distribution centered at

*s*

_{T}, that locations are “coupled” through this common orientation, and that every distractor orientation is equal to

*s*

_{D}. In both conditions, the optimal decision rule is to respond “target present” if

*p*(

*C*= 1|

**x**)/

*p*(

*C*= 0|

**x**) > 1. We will now work out this expression using the respective generative models.

*p*(

*s*|

_{i}*T*= 0), a single target with value

_{i}*s*

_{T}, and a general noise distribution

*p*(

*x*|

_{i}*s*), the posterior ratio is equal to (Ma et al., 2011)

_{i}*s*

_{D}is outside the product, indicating that the

*s*

_{D}is common to all distractor locations. In Experiment 1, there was no such outer integral. In both the numerator and denominator we find an integral of a product of Von Mises distributions over

*s*

_{D}. For convenience, we define

*x*

_{0}=

*s*

_{T}and

*κ*

_{0}=

*κ*

_{D}, so that

*p*(

*s*

_{D}) formally becomes a factor in the product. Then the posterior ratio simplifies to with

*) decision rule to a large number of simulated samples*

_{d}**x**drawn using the presented stimuli on a given trial. The result is a prediction for the probability of reporting “target present” on that trial for a given set of model parameter values. Maximum-likelihood parameter estimates were obtained by computing the joint response probabilities under a large number of parameter combinations (31 values per parameter). We verified that our results were insensitive to the range and discretization of the parameter space.

^{−2}), the power in the power law dependence of mean precision on set size, the scale parameter of the gamma distribution over precision (in rad

^{−2}), and the observer's prior probability that the target is present.

Condition | J_{1} | J_{2} | J_{4} | J_{8} | p_{present} | λ |

κ_{D} = 0 | 267 ± 99 | 64 ± 15 | 21.4 ± 4 | 27.2 ± 4.8 | 0.423 ± 0.024 | (6.2 ± 1.2)·10^{−2} |

κ_{D} = 1 | 205 ± 73 | 71 ± 12 | 31.3 ± 3 | 45 ± 13 | 0.482 ± 0.021 | 0.110 ± 0.010 |

κ_{D} = 8 | 331 ± 75 | 133 ± 27 | 75 ± 17 | 28 ± 15 | 0.508 ± 0.013 | 0.123 ± 0.018 |

Condition | J_{1} | J_{2} | J_{4} | J_{8} | p_{present} | λ |

κ_{D} = 0 | (31 ± 12)·10 | 53 ± 12 | 17.2 ± 3.6 | 5.44 ± 0.73 | 5.06 ± 0.32 | (1.22 ± 0.36)·10^{−2} |

κ_{D} = 1 | 175 ± 44 | 54.8 ± 9.7 | 18.3 ± 2.7 | 10.4 ± 1.5 | 3.53 ± 0.22 | (1.22 ± 0.58)·10^{−2} |

κ_{D} = 8 | 273 ± 49 | 92 ± 14 | 45.5 ± 9.4 | 37.2 ± 6 | 2.22 ± 0.12 | (1.11 ± 0.54)·10^{−2} |

Condition | J̄_{1} | J̄_{2} | J̄_{4} | J̄_{8} | τ | p_{present} | λ |

κ_{D} = 0 | 198 ± 45 | 131 ± 30 | 49 ± 11 | 31.2 ± 7.6 | 119 ± 28 | 0.508 ± 0.015 | (4.4 ± 2.8)·10^{−3} |

κ_{D} = 1 | 218 ± 23 | 118 ± 30 | 51.7 ± 6.1 | 25.8 ± 4.9 | 120 ± 27 | 0.522 ± 0.011 | (1.7 ± 1.1)·10^{−2} |

κ_{D} = 8 | 313 ± 33 | 205 ± 49 | 97 ± 16 | 31 ± 7 | 195 ± 66 | 0.525 ± 0.011 | 0 ± 0 |

Condition | J̄_{1} | J̄_{2} | J̄_{3} | J̄_{4} | τ | p_{present} | λ |

κ_{D} = 0 | 178 ± 24 | 99 ± 13 | 39.1 ± 7.4 | 18.7 ± 3.4 | 64 ± 10 | 3.47 ± 0.34 | (7.8 ± 2.0)·10^{−3} |

κ_{D} = 1 | 186 ± 30 | 88 ± 13 | 32.7 ± 3.6 | 17.1 ± 1.8 | 52.4 ± 6.1 | 2.56 ± 0.10 | (8.9 ± 4.1)·10^{−3} |

κ_{D} = 8 | 306 ± 49 | 105 ± 13 | 49.7 ± 9.1 | 37.2 ± 8.2 | 51 ± 16 | 1.89 ± 0.14 | (1.1 ± 1.1)·10^{−3} |

Condition | J̄_{1} | β (power) | τ | p_{present} |

Homogeneous κ_{D} = 32.8 | 339 ± 26 | −0.83 ± 0.11 | 147 ± 38 | 0.506 ± 0.005 |

Heterogeneous κ_{D} = 32.8 | −0.85 ± 0.13 | 208 ± 42 |