Biomedical Engineering Reference
In-Depth Information
pdpd p
(|)
q
∝
( |)(),
q
q
(4.2)
since
p
(
d
) does not depend on
θ
. This last equation also makes clear that the Bayesian
framework uses both the likelihood and the prior distribution of
θ
to arrive at an informed
estimate
q
. To render these general remarks more concrete, we turn to the example of
sound localization in barn owls as recently described in Fischer and Peña [
2011
].
4.3
Sound Localization in Barn Owls
In the wild, owls use sound localization to detect and locate prey in the dead of
night. Psychophysically, the time lag between a sound picked up in each ear but
generated by a single source allows the owl to reconstruct the horizontal direction
(or azimuth) to the source (Fig.
4.2a
). A second and distinct cue, the interaural level
difference allows the owl to reconstruct the elevation of the source but will not be
considered further here (Konishi
2003
). The time lag between sound arrival at both
ears is called the interaural time difference (ITD), and is related to the azimuth
direction of the sound source as shown in Fig.
4.2b
. The horizontal axis shows the
source direction centred on the owl's sagittal plane, while the vertical axis shows the
corresponding ITD. This relationship is obtained by itting the function
ITD( )
q
=
A
sin()
wq
(4.3)
to head related transfer function data as a function of the source angle
θ
. Such a it
yields
A
= 260 μs and ω = 0. 0143 rad/
°
as the itted parameters (Fischer and
Peña
2011
). From this graph, one notices immediately that the inverse mapping
from ITD to source direction is not always one to one. Hence, the owl must some-
how pick one of the possible states of the world consistent with the observed ITD.
Ethologically, we know that it does so by biasing its choice to the one straight ahead
(Hausmann et al.
2009
, Knudsen et al.
1979
). This bias can be quantiied using
Bayesian statistics.
We begin by asking what knowledge of the world the owl already has and what
it wishes to know. In the sound localization problem, it knows approximately (see
below) the ITD of the source, but wishes to know its associated direction
θ
. In
Bayesian terms, we say that the owl wishes to infer the probability of
θ
given that it
knows the ITD, or equivalently, the probability distribution of
θ
given the ITD,
p
(
θ
| ITD). This is exactly the posterior distribution in Eq. (
4.1
) with ITD replacing
d
. In order to use Bayes' rule, we need a generative model and a prior distribution.
The sound reaching each ear may be corrupted by noise in the environment, like
that caused by wind; in addition, the neural computation of ITD is noisy as well.
Thus, we use
ITD( )
q
=
A
sin() ,
wq
+
W
(4.4)
