Geoscience Reference
In-Depth Information
Outcomes from the BN model are expressions of probability
that each outcome state will occur (e.g., X% extinct, y% rare,
and Z% smaller). It is important here to understand that these
probability values are provided without error bars and should
not, in themselves, be interpreted as absolute measures of the
certainty of any particular outcome. Rather, probabilities of
outcome states of the model should be viewed in terms of
their general direction and overall magnitudes. When predic-
tions result in high probability of one outcome state and low
or zero probabilities of all other states, there is low overall
uncertainty of predicted results. When projected probabilities
of various states are more equally distributed or when two or
more states have large probability, there is greater uncertainty
in the outcome. In these cases, careful consideration should
be given to large probabilities representing particular states
even if those probabilities are not the largest.
vide qualitatively different outcome patterns than those we
obtained.
We ran overall sensitivity analyses to determine the de-
gree to which each input and summary variable influenced
the outcome variables. For discrete and categorical variables,
sensitivity was calculated in the modeling shell Netica® as
the degree of entropy reduction (reduction in the disorder or
variation) at one node relative to the information represented
in other nodes of the model. That is, the sensitivity tests indi-
cate how much of the variation in the node in question is ex-
plained by each of the other nodes considered. The degree of
entropy reduction,
I
, is the expected reduction in mutual infor-
mation of an output variable
Q
, with
q
states, due to a finding
of an input variable
F
, with
f
states. For discrete variables,
I
is
measured in terms of information bits and is calculated as
I
H
Q
H
Q
_
F
¦
q
¦
f
P
q
f
log
2
>
P
q
f
@
P
q
P
f
2.7. Sensitivity of the Bayesian Network Model
where
H
(
Q
) is the entropy of
Q
before new findings are ap-
plied to input node
F
and
H
(
Q
|
F
) is the entropy of
Q
after
new findings are applied to
F
. In Netica®, entropy reduction
is also termed mutual information.
For continuous variables, sensitivity is calculated as var-
iance reduction
VR
, which is the expected reduction in vari-
ation,
V
(
Q
), of the expected real value of the output variable
Q
due to the value of input variable
F
, and is calculated as
knowledge of polar bears, their dependence on sea ice,
the ways in which sea ice changes have been observed to
affect polar bears, and professional judgment regarding how
ecological and human factors may differ if sea ice changes
occur as projected were used to populate the conditional
probability tables in the BN model. Because our model in-
corporated the professional judgment of only one polar bear
expert, it is reasonable to ask how robust the results might be
to input probabilities which could vary among other experts.
It also is appropriate to ask whether it is likely that future
sea ice change, to which model outcomes are very sensi-
tive, could fall into ranges that would result in qualitatively
different outcomes than our BN model projects. Finally, it
is appropriate to ask the extent to which model outcomes
may be altered by active management of the states of nodes
which represent variables humans could control.
We addressed questions about the ability of changes in
human activities to alter the BN output states by fixing in-
puts humans could control and examining differences in the
overall outcomes. We evaluated the extent to which sea ice
projections would have to differ to make qualitative dif-
ferences in outcomes by holding all non-ice variables at
uniform priors and allowing ice variables only to vary at
future time steps. Comparing those results to the range of
ice conditions projected by our GCMs provided a sense of
just how much the realized future ice conditions would have
to vary from those projected to make a difference in popu-
lation outcomes. Finally, although we cannot second guess
how other polar bear experts may recommend parameteriz-
ing and structuring a BN model, comparison of model runs
with preset values provides some sense of how much dif-
ferently the model would have to be parameterized to pro-
VR
V
Q
V
Q
_
F
where
V
Q
¦
q
P
q
>
X
q
E
Q
@
2
V
Q
_
F
¦
q
P
q
_
f
>
X
q
E
Q
_
f
@
2
E
Q
¦
q
P
q
X
q
and where
X
q
is the numeric real value corresponding to state
q
,
E
(
Q
) is the expected real value of
Q
before new findings
are applied,
E
(
Q
|
F
) is the expected real value of
Q
after new
findings
f
are applied to
F
, and
V
(
Q
) is the variance in the real
value of
Q
before any new findings [
Marcot
et al.
, 2006].
3. RESUlTS
3.1. Bayesian Network Model Outcomes
The most probable BN model outcome, for both the SIE
and PBDE, was “extinct” (Table 2 and Plate 4). In all but
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