Lopez Quintero et aL: Incorporating uncertainty into a length-based estimator of natural-mortality 
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Table 1 
Mean estimates, with standard deviations (SDs) and 
95% highest posterior density (HPD) intervals, of 
Bayesian log-normal model parameters from fitting 
the regression of natural mortality (M) to the von Ber- 
talanffy growth function parameters, the asymptotic 
length, the growth rate coefficient, and the theoretical 
age at length zero by using the Equation 4 and data in 
Charnov et al. (2013). 
Parameter 
Estimates 
SD 
95% HPD 
interval 
Po 
-0.050 
0.110 
-0.267, 0.163 
A 
-1.467 
0.125 
-1.714,-1.226 
A 
1.007 
0.075 
0.860, 1.153 
0.749 
0.041 
0.666, 0.825 
70 
0 5 10 15 20 25 
Age 
Figure 1 
Length-at-age composition for southern blue whiting 
(. Micromesistius australis) collected from Chilean con- 
tinental waters during 1997-2010 (open circles), fitted 
with the von Bertalanffy growth function: the solid 
line corresponds to the fit of the log-skew-^ model with 
a heteroscedastic power variance function (Table 2). 
Dashed lines correspond to the 95% highest posterior 
density for the fitted log-skew-£ model. Fish length was 
measured as total length in centimeters. 
burn-in period of the considered chains is 20,000 it- 
erations and 10,000, respectively. Last, a machine 
running on Linux (kernel 4.6), with an Intel Core 
i5 processor (Intel Corp., Santa Clara, CA) and 7.7 
GB of random-access memory, processed all these 
computations. 
Table 2 
Mean estimates, with standard deviations (SDs) and 
95% highest posterior density (HPD) intervals, of the 
von Bertalanffy growth function parameters from the 
power log-skew-f model. The parameters are asymptotic 
length (L„), growth rate coefficient ( K ), negative theo- 
retical age at length zero (-t 0 ), heteroscedasticity (p), 
dispersion (a 2 ), skewness (A), and degrees of freedom 
(v). 
95% HPD 
Parameter 
Estimates 
SD 
interval 
L„ 
59.573 
0.090 
59.386, 59.755 
K 
0.162 
0.001 
0.159, 0.165 
~t 0 
2.454 
0.042 
2.367, 2.541 
P 
-0.180 
0.009 
-0.197, -0.161 
o 2 
0.011 
0.001 
0.010, 0.013 
A 
-1.096 
0.053 
-1.200, -0.997 
V 
14.322 
1.047 
12.457, 16.586 
Results 
Table 1 shows the estimates from the M model with 
Bayesian inference. Differences between these results 
and those reported in Charnov et al. (2013) were 
caused by the method used to incorporate uncertainty. 
However, the actual values of the estimated parame- 
ters were very similar. The standard deviation, a v , is 
then a key parameter underpinning the method error, 
which is assumed to be a log-normal random variable 
rj in Equations 10 and 11. 
Figure 1 shows the application of the power log- 
skew-t model fitted to the observed length-at-age data 
(for southern blue whiting) by using the VBGF pa- 
rameters summarized in Table 2 and estimated with 
Bayesian inference. This curve was created by simu- 
lating 10,000 log-skew-^ random values from each age 
and then by taking the 95% highest posterior density 
interval. Further details regarding the Bayesian es- 
timation, such as residual diagnostic and sensitivity 
analysis, can be found in Lopez Quintero et al. (2017). 
Those authors also showed that the heteroscedastic 
parameter allows a better model with small variance 
across observed ages in southern blue whiting. In ad- 
dition, extreme values for younger and older fish (i.e., 
<9 and >16 years) were accounted for by the estimated 
degree of freedom parameter. Moreover, the credibility 
intervals showed that observations were affected by 
the negative heteroscedastic parameter, p. 
The marginal distributions estimated empirically 
with Equations 8 and 9 and the pseudosamples F and 
G from copula are shown in Figure 2. We first observed 
that points concentrate around the diagonal of unit 
square. The relationship between pseudosamples is 
linear and negative. Particularly, the Bayesian Markov 
chain simulation, shown in Figure 2A, which was ob- 
tained directly from the estimated VBGF parameters. 
