Figure 1.
Demonstration of Inferences of Objects from Images
(A) and (B) show two images containing the same white patch, and (C) and (D) show the two possible inferred objects in the scene causing this white patch. The inferred causes for any particular input image patch is not unique, although some inferences are more likely than others. The difference in the most likely inferred object for the same image patch in (A) and (B) demonstrates that inference could be greatly influenced by the image context.
Figure 2.
Bayesian Inference for Target Perception
(A) Schematics of perceiving a weak vertical target bar in three different contexts. Colinear contexts give a higher prior belief P(yes) of the target present, as it could be grouped with the context. Higher contextual contrast Cc makes a low contrast input Ct at the would-be target location seem less likely to be caused by a target rather than noise, since observers expect a target to evoke a contrast similar to Cc, i.e., P(Ct | yes) peaks at Ct ≈ Cc, and P(Ct | yes) ≈ 0 if Ct ≪ Cc; see (B).
(B) the probability P(yes | Ct) of “yes” response depends on the ratio between the evidences P(Ct | yes) and P(Ct | no) for target present and absent, respectively, when the prior belief P(yes) = 0.5 is unbiased. This ratio should be multiplied by P(yes)/(1 − P(yes)) in general. Note that probability distributions P(Ct | yes) and P(Ct | no) peak at Ct = Cc and Ct = 0, respectively.
(C,D) Effects of the contextual contrast Cc (C) and of the prior P(yes) (D) by the Bayesian model. In (C) and (D), all curves have model parameters k = 2 and σn = 0.0015, the two red curves are identical, with P(yes) = 0.95 and Cc = 0.01. Comparing (C) and (D), a higher contextual contrast Cc has a similar effect as a lower prior P(yes).
Figure 3.
Results from Experiment 1, Where the Colinear Context Resembles the Two Left Ones in Figure 2A
The data points are the mean over six observers, and the error bars indicate the standard errors of the means (SEMs). On average and relative to the no-context condition, the weaker colinear contexts Cc = 0.01 and Cc = 0.05 raised the yes rates by CFI = 38% ± 8% and 15% ± 8%, respectively, whereas the stronger context Cc = 0.4 lowered it by −CFI = 17% ± 8%. The colored curves are Bayesian fits to data of the corresponding color, no fit is done for data without context. The root mean square normalized fitting error RMSNFE = 0.66 in the unit of SEM. The fitted parameters (and their 95% confidential intervals) are k = 1.9 (0.6, 3.2), σn = 0.0025 (0.0020, 0.0029), and P(yes) = 0.972 (0.967, 0.978).
Figure 4.
Results from Experiment 2 Averaged Over Five Observers
(A,B) Yes rates under colinear and orthogonal context (schematically like Figure 2A), respectively. The curves are the Bayesian fits. The four Bayesian parameters (and their 95% confidence intervals) are k = 3.8 (1.8, 5.8), σn = 0.0027 (0.0021, 0.0033), P(yes)colinear = 0.982 (0.974, 0.989), and P(yes)orthogonal = 0.88 (0.85, 0.92), giving a fitting quality of RMSNFE = 1.0.
(C,D) Yes rates under different contextual contrast Cc = 0.01 and Cc = 0.4, respectively, together with those under no context. For colinear context CFI = 0.23 ± 0.05 and −0.18 ± 0.15 for Cc = 0.01 and 0.4, respectively; for orthogonal context CFI = −0.018 ± 0.06 and −0.46 ± 0.055 for Cc = 0.01 and 0.4, respectively.
(E) Prior P(yes) for the two contextual configurations. The error bars denote SEMs in (A–D), and 95% confidence intervals in (E).
Figure 5.
Stimuli (Schematics) and Data for Experiment 3.
(A) The schematics of the stimuli.
(B–D) Yes rates (with SEM error bars) averaged over seven observers. The 2-sided context gives higher yes rates than other contexts for Cc = 0.01 (B) and Cc = 0.05 (C), but not significantly for Cc = 0.4 (D) when yes rates are all depressed relative to those under no context. The yes rates given a contextual configuration decrease with increasing Cc. Error bars indicate SEM. CFI under the 2-sided, 1-sided, and sparse contexts are respectively: CFI = 0.42 ± 0.06, 0.17 ± 0.04, and 0.19 ± 0.04, for Cc = 0.01, CFI = 0.204 ± 0.06, 0.016 ± 0.07, and 0.05 ± 0.06 for Cc = 0.05, and CFI = −0.11 ± 0.07, −0.18 ± 0.05, and −0.13 ± 0.06 for Cc = 0.4.
Figure 6.
Fit to Data in Experiment 3 by the Bayesian Model
(A–C) The red, magenta, and blue curves and data points indicate respective quantities associated with different contextual contrasts Cc = 0.01, 0.05, and 0.4, respectively. The fitted Bayesian parameters (and their 95% confidential intervals) are k = 3.91 (1.95, 5.87), σn = 1.60 × 10−3 (1.45 × 10−3, 1.73 × 10−3), and P(yes) = 0.97 (0.96, 0.98) for the 2-sided, P(yes) = 0.87 (0.83, 0.91) for the 1-sided, and P(yes) = 0.92 (0.89, 0.95) for the sparse context. RMSNFE = 1.07.
(D) The prior P(yes) for the three different contextual configurations. The error bars denote SEMs in (A–C), and 95% confidence intervals in (D).