Behavioral Measures of Multisensory Integration: Bounds on Bimodal Detection Probability

Abstract

One way to test and quantify multisensory integration in a behavioral paradigm is to compare bimodal detection probability with bounds defined by some combination of the unimodal detection probabilities. Here we (1) improve on an upper bound recently suggested by Stevenson et al. (Brain Topogr 27(6):707–730, 2014), (2) present a lower bound, (3) interpret the bounds in terms of stochastic dependency between the detection probabilities, (4) discuss some additional assumptions required for the validity of any such bound, (5) suggest some potential applications to neurophysiologic measures, and point out some parallels to the ‘race model inequality’ for reaction times.

Appendix

Derivation of the Bounds in Eq. (4)

The Fréchet bounds (Fréchet 1951; Joe 1997) for the bivariate Bernoulli distribution with fixed marginal probabilities $p_{1+}, p_{+1}$ presented in Table 1 are

$$\max \{0, p_{1+}+p_{+1} -1\} \le p_{11} \le \min \{p_{1+}, p_{+1}\}.$$

(7)

We want an upper and a lower bound for

$$\begin{aligned}1-p_{00}&\equiv p_{11}+p_{01}+p_{10} \\&= p_{+1}+p_{1+}-p_{11}\end{aligned}$$

(8)

Inserting the upper and the lower Fréchet bound for $p_{11}$ yields

$$\begin{aligned} p_{+1}+p_{1+}-\min \{p_{1+}, p_{+1}\}&\le 1-p_{00} \le p_{+1}+p_{1+}- \max \{0, p_{1+}+p_{+1} -1\} \\ \text{ or, }& \\ \max \{ p_{1+}, p_{+1}\}&\le 1-p_{00} \le \min \{p_{1+}+p_{+1},1\}, \end{aligned}$$

(9)

with the last line following after simple algebraic transformations. Then Eq. (9) is identical to the bounds in Eq. (4).

Dependence Between$A$ and $V$

The correlation between $A$ and $V$ equals

$$\rho = \frac{p_{11}-\pi _1\pi _2}{\sqrt{\pi _1(1-\pi _1)\pi _2(1-\pi _2)}}.$$

Inserting the upper and lower Fréchet bound for $p_{11}$ into this expression results in

$$\begin{aligned} \max \left\{ -\sqrt{\frac{\pi _1 \pi _2}{\bar{\pi }_1 \bar{\pi }_2}},-\sqrt{\frac{\bar{\pi }_1 \bar{\pi }_2}{\pi _1 \pi _2}}\right\} \le \rho \le \sqrt{ \frac{\pi _{min}(1-\pi _{max})}{\pi _{max}(1-\pi _{min})}}, \end{aligned}$$

where $\bar{\pi }_i=1-\pi _i$, $i=1,2$, $\pi _{min}=\min \{\pi _1,\pi _2\}$, and $\pi _{max}=\max \{\pi _1,\pi _2\}$. The claims about perfect positive and negative dependence also follow by simple insertions into these bounds.