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25.5 30.6 35 38.7 41.8 44.5 46.7 48.7 50.3 51.7 52.9 53.9 54.7 55.4 56 56.6 57 57.4 57.7 58 58.2 58.4 58.6 58.7 58.8 58.9 
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 
Length and age 
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Length and age 
Figure S 
Boxplots of estimates of natural mortality in relation to total length (in centimeters) age in years 
(A) from direct simulation with Markov chains and (B) with the use of the copula method. The 
vertical black lines correspond to the observed median, the boxes represent the observed interval 
from the 25% residual quartile to the 75% residual quartile, the error bars are the observed inter- 
val from minimum to maximum residual value, and the dots are atypical residual values. 
to the trait error, we presented 2 ways to incorporate 
uncertainty in length-based M estimators, drawing on 
an empirical distribution of VBGF parameters. First, 
the applied Bayesian methods simulate the M esti- 
mator directly with the posterior distribution of the 
VBGF parameters, with the method of Lopez Quintero 
et al. (2017). This particular approach was preferred 
over that for traditional distributions (Siegfried and 
Sanso, 2006; Hamel, 2015), because this type of data 
usually contains a degree of asymmetry and extreme 
values. Additionally, an inadequate distribution may 
underestimate the real variance contained in the data. 
The model used gives great flexibility in modeling het- 
eroscedasticity by adding a function dependent on the 
scale cr 2 and a heteroscedastic parameter, p (Contreras- 
Reyes et al., 2014). In addition, a copula method was 
usedd to approximate the posterior distribution and 
calculate the M estimator. The method proposed in this 
study provides a way to incorporate uncertainty in the 
length-based M estimator proposed by Charnov et al. 
(2013), while acknowledging both method and traits er- 
rors. Furthermore, our scheme can easily be extended 
to generate values of uncertainty in indirect methods 
used to estimate mortality, and therefore has the poten- 
tial to improve actual ad-hoc methods for incorporating 
uncertainty, such as in Cubillos et al. (1999); Quiroz < 
et al. (2010) and Wiff et al. (2011). Furthermore, we 1 
recommend the copula method instead of the Bayesian 
Markov chain approach to incorporate uncertainty in \ 
the M-at-length estimates for 2 reasons: 1) the copula ; 
method conserves the underlying dependence in the 
posterior distribution (see Figs. 3 and 4) and 2) it uses [ 
less computing time than the Bayesian Markov chain 
approach. For example, in our case the copula method j 
required 1 s to compute each length class, whereas the 
same procedure takes at least 20 h with the Bayesian 
Markov chain approach. These differences in computa- 
tional time result in part from the Metropolis-Hastings ; 
algorithm that is derived from the kernel of likelihood 
function and discards, by construction, many proposed 
values of parameters. Other algorithms, such as the 
Gibbs sampler, can take the advantage of the known 
conditional distributions, making the sampling process • 
faster. 
Additionally, it is important to note that the pro- 
posed M-at-length estimators are not conceptually 
limited to use a Bayesian estimation. Researchers 
can simulate the entire M-at-length structure just by 
knowing the dependence between parameters L x and 
K. This can be achieved by reviewing specialized litera- i 
