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Defining Priors

k_defining_priors.Rmd

In some cases, prior information might be available for use to inform model estimation. Priors can either serve as informative priors that have a strong influence on model estimation (e.g., natural mortality, catchability), but can also serve as regularizing priors that help stabilize model estimation. These regularizing priors can be helpful particularly for stabilizing tag reporting, movement, and selectivity parameters because some parameters are not well informed by data (e.g., slope of selectivity when it approaches infinity) such that the gradient is zero with respect to the likelihood surface.

In SPoRC, priors can be placed on several parameters. These include:

  1. Steepness,
  2. Natural Mortality,
  3. Catchability,
  4. Selectivity,
  5. Movement,
  6. Tag Reporting Rates,
  7. Recruitment Regional, Seasonal Proportions, and R0,
  8. Stray rates

In the following, we will demonstrate how priors can be specified for each of these parameters. For further mathematical details on the formulation of these priors, refer to the vignette describing model equations.

Let us first load in any required packages, data we may use, and define model dimensions.

library(SPoRC) 
#> Loading required package: RTMB
data("sgl_rg_sable_data")

input_list <- Setup_Mod_Dim(years = 1:length(sgl_rg_sable_data$years),
                            ages = 1:length(sgl_rg_sable_data$ages),
                            lens = seq(41, 99, 2),
                            n_regions = 1,
                            n_sexes = sgl_rg_sable_data$n_sexes,
                            n_fish_fleets = sgl_rg_sable_data$n_fish_fleets,
                            n_srv_fleets = sgl_rg_sable_data$n_srv_fleets,
                            n_seas = 1,
                            n_pop = sgl_rg_sable_data$n_pop,
                            verbose = TRUE)
#> Number of Years: 65
#> Number of Seasons: 1
#> Duration of season 1: 1
#> Number of Projection Years for Dev Pars: 0
#> Number of Regions: 1
#> Number of Populations: 1
#> Number of Age Bins: 30
#> Number of Length Bins: 30
#> Number of Sexes: 2
#> Number of Fishery Fleets: 2
#> Number of Survey Fleets: 3

Steepness

To specify priors on the steepness of a Beverton-Holt stock-recruitment relationship, we use the Setup_Mod_Rec function. First, we define a dataframe that includes the population (pop), region, mean (mu), and standard deviation (sd) of the steepness prior. These values are specified in normal space but are internally transformed to a beta distribution bounded between 0.2 and 1. To activate steepness priors, set Use_h_prior = 1 and provide the dataframe to the h_prior argument:

steepness_prior <- data.frame(
  pop    = 1,   # population 1
  region = 1,   # region 1
  mu     = 0.6, # prior mean
  sd     = 0.2  # prior sd
)

input_list <- Setup_Mod_Rec(
  input_list = input_list,
  rec_model = "bh_rec",
  Use_h_prior = 1,
  h_prior = steepness_prior,
  h_spec = "est_shared_pop_r"
)
#> Recruitment is specified as: bh_rec
#> Recruitment Density Dependence is specified as: global
#> Recruitment and SSB lag is specified as: 1
#> Recruitment regional proportion priors are: Not Used
#> Recruitment seasonal proportion priors are: Not Used
#> Recruitment seasonal proportions is: fix
#> Stray rates for population 1 specified with 1 block(s).
#> Stray rate prior is: Not Used
#> Steepness priors are: Used
#> Sex Ratios specified with 1 block for population 1 and region 1
#> Recruitment Bias Ramp is: Off
#> Initial Age Structure is: Movement and Matrix Geometric Series
#> Recruitment deviations for every year are estimated
#> Spawning season occurs in season 1
#> Recruitment Variability is specified as: est_all
#> Initial Age Deviations is stochastic for all ages, but the plus group follows equilibrium calculations.
#> Recruitment Deviations is estimated for all dimensions
#> Steepness is specified as: est_shared_pop_r
#> Sex ratio is specified as: fix
#> Stray rates fixed (n_pop == 1, straying not applicable).

In a spatial model where region-specific steepness priors are required, the prior dataframe can include multiple entries. Each row corresponds to a population–region combination along with its associated mean and standard deviation:

steepness_prior <- data.frame(
  pop    = 1,
  region = c(1, 2, 3),
  mu     = c(0.6, 0.8, 0.3),
  sd     = c(0.2, 0.2, 0.2)
)

steepness_prior
#>   pop region  mu  sd
#> 1   1      1 0.6 0.2
#> 2   1      2 0.8 0.2
#> 3   1      3 0.3 0.2

Natural Mortality

To define natural mortality priors, we use the Setup_Mod_Biologicals function. Setting Use_M_prior = 1 activates the prior, and a dataframe of values is provided via the M_prior argument. Each row corresponds to a specific combination of population, region, year, age, and sex blocks, along with their associated mean and standard deviation.

M_prior <- data.frame( 
  popblk    = 1,
  regionblk = 1,
  yearblk   = 1,
  ageblk    = 1,
  sexblk    = c(1, 2),
  mu        = 0.085,
  sd        = 0.1
)

input_list <- Setup_Mod_Biologicals(
  input_list = input_list,
  WAA          = sgl_rg_sable_data$WAA,
  MatAA        = sgl_rg_sable_data$MatAA,
  AgeingError  = as.matrix(sgl_rg_sable_data$age_error),
  SizeAgeTrans = sgl_rg_sable_data$SizeAgeTrans,
  Use_M_prior  = 1,
  M_prior      = M_prior,
  M_spec       = "est_ln_M",
  M_sexblk_spec = list(1, 2)
)
#> Length Composition data are: Not Used
#> Natural Mortality priors are: Used
#> WAA_fish was specified at NULL. Using the spawning WAA for WAA_fish
#> WAA_srv was specified at NULL. Using the spawning WAA for WAA_srv
#> Ageing Error is specified to be time-invariant
#> Natural Mortality specified as: est_ln_M
#> Natural Mortality Population Blocks is specified as: 1
#> Natural Mortality Region Blocks is specified as: 1
#> Natural Mortality Year Blocks is specified as: 1
#> Natural Mortality Age Blocks is specified as: 1
#> Natural Mortality Sex Blocks is specified as: 2

Catchability

Defining catchability priors for the fishery and survey follows the same process. For demonstration purposes, the following section will illustrate how catchability priors can be defined for the survey using the Setup_Mod_Srvsel_and_Q function. A dataframe of catchability priors is first defined that determines the region, fleet, and block to which the prior is applied. The dataframe also includes columns defining the mean (mu) and standard deviation (sd) of the catchability prior. These priors are provided in normal space, but are internally transformed to a lognormal density. Setting Use_srv_q_prior = 1 activates the prior.

srv_q_prior <- data.frame(
  region = 1,
  fleet  = c(1, 2, 3),
  block  = 1,
  mu     = c(6, 0.5, 4),
  sd     = c(0.1, 0.35, 0.1)
)

srv_q_prior
#>   region fleet block  mu   sd
#> 1      1     1     1 6.0 0.10
#> 2      1     2     1 0.5 0.35
#> 3      1     3     1 4.0 0.10

input_list <- Setup_Mod_Srvsel_and_Q(
  input_list = input_list,
  cont_tv_srv_sel        = c("none_Fleet_1", "none_Fleet_2", "none_Fleet_3"),
  srv_sel_blocks         = c("none_Fleet_1", "none_Fleet_2", "none_Fleet_3"),
  srv_sel_model          = c("logist1_Fleet_1", "exponential_Fleet_2", "logist1_Fleet_3"),
  srv_q_blocks           = c("none_Fleet_1", "none_Fleet_2", "none_Fleet_3"),
  srv_fixed_sel_pars_spec = c("est_all", "est_all", "est_all"),
  srv_q_spec             = c("est_all", "est_all", "est_all"),
  Use_srv_q_prior        = 1,
  srv_q_prior            = srv_q_prior
)
#> Survey Catchability priors are: Used
#> Survey Selectivity priors are: Not Used
#> Survey Selectivity is aged-based.
#> Continuous survey time-varying selectivity specified as: none for survey fleet 1
#> Continuous survey time-varying selectivity specified as: none for survey fleet 2
#> Continuous survey time-varying selectivity specified as: none for survey fleet 3
#> Survey Selectivity Time Blocks for survey 1 is specified at: 1
#> Survey Selectivity Time Blocks for survey 2 is specified at: 1
#> Survey Selectivity Time Blocks for survey 3 is specified at: 1
#> Survey selectivity functional form specified as:logist1 for survey fleet 1
#> Survey selectivity functional form specified as:exponential for survey fleet 2
#> Survey selectivity functional form specified as:logist1 for survey fleet 3
#> Survey Catchability Time Blocks for survey 1 is specified at: 1
#> Survey Catchability Time Blocks for survey 2 is specified at: 1
#> Survey Catchability Time Blocks for survey 3 is specified at: 1
#> srv_fixed_sel_pars_spec is specified as: est_all for survey fleet 1
#> srv_fixed_sel_pars_spec is specified as: est_all for survey fleet 2
#> srv_fixed_sel_pars_spec is specified as: est_all for survey fleet 3
#> srv_q_spec is specified as: est_all for survey fleet 1
#> srv_q_spec is specified as: est_all for survey fleet 2
#> srv_q_spec is specified as: est_all for survey fleet 3

Selectivity

Defining selectivity priors for the fishery and survey follows the same process. Here, we will focus on defining selectivity priors for the survey for brevity, using the Setup_Mod_Srvsel_and_Q function. We first define a dataframe with the following columns: region, par, block, sex, fleet, mu, and sd, which defines the region, parameter, time block, sex, and fleet combination to apply the prior to. Given the complexity of the number of selectivity parameters that can be estimated across model partitions (regions, sexes, blocks, and fleets), users will need to be cautious about how the selectivity prior dataframe is defined to ensure they are correctly applied.

In the following, we will apply priors to the logistic selectivity parameters for both sexes in fleet 1, and priors to the power parameter describing exponential selectivity in fleet 2.

srv_selex_prior <- data.frame(
  region = 1,
  par    = c(1, 2, 1, 2, 1, 1),
  sex    = c(1, 1, 2, 2, 1, 2),
  fleet  = c(1, 1, 1, 1, 2, 2),
  block  = 1,
  mu     = rep(0, 6),
  sd     = rep(5, 6)
)

srv_selex_prior
#>   region par sex fleet block mu sd
#> 1      1   1   1     1     1  0  5
#> 2      1   2   1     1     1  0  5
#> 3      1   1   2     1     1  0  5
#> 4      1   2   2     1     1  0  5
#> 5      1   1   1     2     1  0  5
#> 6      1   1   2     2     1  0  5

In the example provided above, the first row of the dataframe indicates that a prior is applied to region 1, parameter 1, sex 1, fleet 1, and block 1, which corresponds to the b50b_{50} parameter for logistic selectivity specified for the first survey fleet. The last row indicates that a prior is applied to region 1, parameter 1, sex 2, fleet 2, and block 1, which corresponds to the male ϕ\phi power parameter for exponential selectivity specified for the second survey fleet.

Next, we can activate selectivity priors by setting Use_srv_selex_prior = 1 and providing the associated selectivity prior dataframe to srv_selex_prior.

input_list <- Setup_Mod_Srvsel_and_Q(
  input_list = input_list,
  cont_tv_srv_sel         = c("none_Fleet_1", "none_Fleet_2", "none_Fleet_3"),
  srv_sel_blocks          = c("none_Fleet_1", "none_Fleet_2", "none_Fleet_3"),
  srv_sel_model           = c("logist1_Fleet_1", "exponential_Fleet_2", "logist1_Fleet_3"),
  srv_q_blocks            = c("none_Fleet_1", "none_Fleet_2", "none_Fleet_3"),
  srv_fixed_sel_pars_spec = c("est_all", "est_all", "est_all"),
  srv_q_spec              = c("est_all", "est_all", "est_all"),
  Use_srv_selex_prior     = 1,
  srv_selex_prior         = srv_selex_prior
)
#> Survey Catchability priors are: Not Used
#> Survey Selectivity priors are: Used
#> Survey Selectivity is aged-based.
#> Continuous survey time-varying selectivity specified as: none for survey fleet 1
#> Continuous survey time-varying selectivity specified as: none for survey fleet 2
#> Continuous survey time-varying selectivity specified as: none for survey fleet 3
#> Survey Selectivity Time Blocks for survey 1 is specified at: 1
#> Survey Selectivity Time Blocks for survey 2 is specified at: 1
#> Survey Selectivity Time Blocks for survey 3 is specified at: 1
#> Survey selectivity functional form specified as:logist1 for survey fleet 1
#> Survey selectivity functional form specified as:exponential for survey fleet 2
#> Survey selectivity functional form specified as:logist1 for survey fleet 3
#> Survey Catchability Time Blocks for survey 1 is specified at: 1
#> Survey Catchability Time Blocks for survey 2 is specified at: 1
#> Survey Catchability Time Blocks for survey 3 is specified at: 1
#> srv_fixed_sel_pars_spec is specified as: est_all for survey fleet 1
#> srv_fixed_sel_pars_spec is specified as: est_all for survey fleet 2
#> srv_fixed_sel_pars_spec is specified as: est_all for survey fleet 3
#> srv_q_spec is specified as: est_all for survey fleet 1
#> srv_q_spec is specified as: est_all for survey fleet 2
#> srv_q_spec is specified as: est_all for survey fleet 3

Movement

Defining priors for movement, tag reporting rates, and recruitment proportions (see sections below) is best illustrated with a spatial model. Here, we will load in data from the multi-region sablefish case study.

data(mlt_rg_sable_data)

input_list <- Setup_Mod_Dim(years = 1:length(mlt_rg_sable_data$years),
                            ages = 1:length(mlt_rg_sable_data$ages),
                            lens = mlt_rg_sable_data$lens,
                            n_regions = mlt_rg_sable_data$n_regions,
                            n_sexes = mlt_rg_sable_data$n_sexes,
                            n_fish_fleets = mlt_rg_sable_data$n_fish_fleets,
                            n_srv_fleets = mlt_rg_sable_data$n_srv_fleets,
                            n_seas = 1,
                            n_pop = mlt_rg_sable_data$n_pop,
                            verbose = TRUE)
#> Number of Years: 62
#> Number of Seasons: 1
#> Duration of season 1: 1
#> Number of Projection Years for Dev Pars: 0
#> Number of Regions: 5
#> Number of Populations: 1
#> Number of Age Bins: 30
#> Number of Length Bins: 30
#> Number of Sexes: 2
#> Number of Fishery Fleets: 2
#> Number of Survey Fleets: 2

Because movement parameters are correlated and must sum to 1, SPoRC uses a Dirichlet prior to model movement probabilities. Movement priors are enabled by setting Use_Movement_Prior = 1. The Movement_prior argument must be provided as a dataframe where each row corresponds to a combination of population, origin region, year, age, season, and sex indices, and includes a list-column alpha specifying the Dirichlet concentration parameters that define the prior over movement probabilities to all destination regions.

prior <- expand.grid(
  pop         = 1,
  region_from = 1:input_list$data$n_regions,
  year        = 1,
  age         = 1,
  seas        = 1,
  sex         = 1,
  alpha       = I(list(rep(2, input_list$data$n_regions)))
)

input_list <- Setup_Mod_Movement(
  input_list           = input_list,
  Movement_ageblk_spec = "constant",
  Movement_yearblk_spec = "constant",
  Movement_sexblk_spec = "constant",
  Use_Movement_Prior   = 1,
  Movement_prior       = prior
)
#> Movement is: Estimated
#> Movement priors are: Used
#> Recruits are: Not Moving
#> Continuous movement specification is: none
#> Continuous movement process error specification is: none
#> Movement type is: Unstructured Markov
#> Movement fixed effect blocks are population-invariant
#> Movement fixed effect blocks are season-invariant
#> Movement fixed effect blocks are sex-invariant
#> Movement fixed effect blocks are time-invariant
#> Movement fixed effect blocks are age-invariant

Tag Reporting Rates

Defining priors for tag reporting rates is done by constructing a dataframe and passing it to the Setup_Mod_Tagging function. The dataframe must contain the following columns: region, block, fleet, mu, sd, and type. These columns define the region and time block the prior applies to, the mean and standard deviation of the prior distribution, and the type of beta prior to use. Two prior types are supported: a symmetric beta prior (type = 0), which penalizes extreme values near 0 and 1, and a standard beta prior (type = 1), which allows users to specify both the mean and standard deviation directly.

In the example below, symmetric beta priors are applied to two time blocks in region 1. Because the symmetric form does not require a mean, the mu column is set to NA:

tag_prior <- data.frame(
  region = 1,
  block  = c(1, 2),
  fleet  = 1,
  mu     = NA,
  sd     = 5,
  type   = 0
)
tag_prior
#>   region block fleet mu sd type
#> 1      1     1     1 NA  5    0
#> 2      1     2     1 NA  5    0

The first row of the dataframe indicates that a symmetric beta prior is applied to tag reporting rates in region 1 and block 1, with a standard deviation of 5. The second row applies the same prior structure to block 2. Because type = 0, the prior is symmetric around 0.5 and discourages extreme reporting rate estimates. If type = 1 were used instead, the mu column would need to be specified with the prior mean.

Tag reporting rate priors are activated by setting use_conv_tag_fishrep_prior = 1 and providing the dataframe to the conv_tag_fishrep_prior argument.

input_list <- Setup_Mod_Tagging(
  input_list = input_list,
  use_conv_fish_tagging      = c(1, 0),
  conv_tag_max_liberty       = 15,

  # Data inputs
  conv_tag_release_indicator = mlt_rg_sable_data$conv_tag_release_indicator,
  conv_tagged_fish           = mlt_rg_sable_data$conv_tagged_fish,
  obs_conv_tag_fish_recap    = mlt_rg_sable_data$obs_conv_tag_fish_recap,

  # Model options
  conv_fish_tag_like           = "Poisson",
  use_conv_tag_fishrep_prior   = 1,
  conv_tag_fishrep_prior       = tag_prior,
  init_conv_tag_mort_spec      = "fix",
  conv_tag_shed_spec           = "fix",
  conv_tagrep_spec             = "est_shared_r_f",
  conv_tag_fish_reporting_blocks = c(
    apply(expand.grid(1:input_list$data$n_regions, 1:input_list$data$n_fish_fleets), 1, function(x)
      paste0("Block_1_Year_1-35_Region_", x[1], "_Fleet_", x[2])),
    apply(expand.grid(1:input_list$data$n_regions, 1:input_list$data$n_fish_fleets), 1, function(x)
      paste0("Block_2_Year_36-terminal_Region_", x[1], "_Fleet_", x[2]))
  ),

  # Starting / fixed values
  ln_init_conv_tag_mort = log(0.1),
  ln_conv_tag_shed      = log(0.02)
)
#> Tagging priors are used
#> Conventional Tag Likelihood specified as: Poisson
#> Conventional Tagging data are fit to 1 population groups
#> Conventional Tagging data are fit to 30 age groups
#> Conventional Tagging data are fit to 2 sex groups
#> Conventional Tag Reporting estimated with 2 blocks for region 1 and fleet 1
#> Conventional Tag Reporting estimated with 2 blocks for region 2 and fleet 1
#> Conventional Tag Reporting estimated with 2 blocks for region 3 and fleet 1
#> Conventional Tag Reporting estimated with 2 blocks for region 4 and fleet 1
#> Conventional Tag Reporting estimated with 2 blocks for region 5 and fleet 1
#> Conventional Tag Reporting estimated with 2 blocks for region 1 and fleet 2
#> Conventional Tag Reporting estimated with 2 blocks for region 2 and fleet 2
#> Conventional Tag Reporting estimated with 2 blocks for region 3 and fleet 2
#> Conventional Tag Reporting estimated with 2 blocks for region 4 and fleet 2
#> Conventional Tag Reporting estimated with 2 blocks for region 5 and fleet 2
#> Conventional Initial Tag Mortality is specified as: fix
#> Conventional Tag Reporting is specified as: est_shared_r_f

Recruitment Proportions and R0

Recruitment Regional Apportionment

In the context of a spatial model, Dirichlet priors can also be placed on recruitment proportions. This is done through the Setup_Mod_Rec function by setting use_rec_region_prop_prior = 1 and providing a dataframe to rec_region_prop_prior. Each row of the dataframe corresponds to a population and contains a list-column alpha specifying the Dirichlet concentration parameters for movement among destination regions. If a uniform Dirichlet prior is desired, all concentration parameters can be set to the same value. Providing different values for each region will apply a weighted Dirichlet prior.

input_list <- Setup_Mod_Rec(
  input_list = input_list,
  do_rec_bias_ramp = 0,
  rec_model = "mean_rec",
  use_rec_region_prop_prior = 1,
  rec_region_prop_prior = data.frame(
    pop   = 1,
    alpha = I(list(rep(5, mlt_rg_sable_data$n_regions)))
  )
)
#> Recruitment is specified as: mean_rec
#> Recruitment Density Dependence is specified as: global
#> Recruitment regional proportion priors are: Used
#> Recruitment seasonal proportion priors are: Not Used
#> Recruitment seasonal proportions is: fix
#> Stray rates for population 1 specified with 1 block(s).
#> Stray rate prior is: Not Used
#> Sex Ratios specified with 1 block for population 1 and region 1
#> Sex Ratios specified with 1 block for population 1 and region 2
#> Sex Ratios specified with 1 block for population 1 and region 3
#> Sex Ratios specified with 1 block for population 1 and region 4
#> Sex Ratios specified with 1 block for population 1 and region 5
#> Recruitment Bias Ramp is: Off
#> Initial Age Structure is: Movement and Matrix Geometric Series
#> Recruitment deviations for every year are estimated
#> Spawning season occurs in season 1
#> Recruitment Variability is specified as: est_all
#> Initial Age Deviations is stochastic for all ages, but the plus group follows equilibrium calculations.
#> Recruitment Deviations is estimated for all dimensions
#> Sex ratio is specified as: fix
#> Stray rates fixed (n_pop == 1, straying not applicable).

Recruitment Seasonal Apportionment

Likewise, priors on recruitment seasonal apportionment can be applied in an analogous manner. This is done by setting use_rec_seas_prop_prior = 1 and providing a dataframe to rec_seas_prop_prior. Note that seasonal recruitment proportion priors are only relevant when n_seas > 1 and use_fixed_rec_seas_prop = 0 (i.e., seasonal proportions are being estimated rather than fixed). Each row of the dataframe corresponds to a population and contains a list-column alpha specifying the Dirichlet concentration parameters for the seasonal apportionment of recruitment.

R0

Lognormal priors on R0R_0 can be applied per population via use_r0_prior and r0_prior. Each row of r0_prior corresponds to a population, with mu specifying the prior mean on the natural scale and sd the standard deviation on the log scale.

input_list <- Setup_Mod_Rec(
  input_list = input_list,
  use_r0_prior = 1,
  do_rec_bias_ramp = 0,
  rec_model = "mean_rec",
  r0_prior = data.frame(
    pop = 1,
    mu  = 1e6,
    sd  = 1
  )
)
#> Recruitment is specified as: mean_rec
#> Recruitment Density Dependence is specified as: global
#> Recruitment regional proportion priors are: Not Used
#> Recruitment seasonal proportion priors are: Not Used
#> Recruitment seasonal proportions is: fix
#> Stray rates for population 1 specified with 1 block(s).
#> Stray rate prior is: Not Used
#> Sex Ratios specified with 1 block for population 1 and region 1
#> Sex Ratios specified with 1 block for population 1 and region 2
#> Sex Ratios specified with 1 block for population 1 and region 3
#> Sex Ratios specified with 1 block for population 1 and region 4
#> Sex Ratios specified with 1 block for population 1 and region 5
#> Recruitment Bias Ramp is: Off
#> Initial Age Structure is: Movement and Matrix Geometric Series
#> Recruitment deviations for every year are estimated
#> Spawning season occurs in season 1
#> Recruitment Variability is specified as: est_all
#> Initial Age Deviations is stochastic for all ages, but the plus group follows equilibrium calculations.
#> Recruitment Deviations is estimated for all dimensions
#> Sex ratio is specified as: fix
#> Stray rates fixed (n_pop == 1, straying not applicable).

Stray Rates

When stray rates are estimated (stray_rate_spec is not "fix"), a beta prior should almost always be used to keep estimation stable. Estimation of stray rates are only applicable when n_pop > 1. The prior is specified via a dataframe passed to stray_rate_prior, with columns pop, block, mu, and sd. The mean (mu) and standard deviation (sd) are supplied in probability space and are internally converted to beta shape parameters via method-of-moments. Note that sd must satisfy σ<μ(1−μ)\sigma < \sqrt{\mu(1-\mu)} to ensure a valid (unimodal) beta density.

In the example below, stray rates are estimated independently per population using a single time block, with population 1 given a prior centred at 0.1 and population 2 centred at 0.5:


input_list$data$n_pop <- 2 # modify to two populations for demo purposes

stray_prior <- data.frame(
  pop   = c(1, 2),
  block = c(1, 1),
  mu    = c(0.1, 0.5),
  sd    = c(0.1, 0.1)
)

input_list <- Setup_Mod_Rec(
  input_list       = input_list,
  do_rec_bias_ramp = 0,
  rec_model        = "bh_rec",
  rec_dd           = "local",
  stray_rate_blocks = c(
    "Block_1_Year_1-terminal_Pop_1",
    "Block_1_Year_1-terminal_Pop_2"
  ),
  stray_rate_spec      = "est_all",
  use_fixed_stray_rate = 0,
  use_stray_rate_prior = 1,
  stray_rate_prior     = stray_prior
)

Tighter values of sd produce a stronger regularizing effect and are recommended when data contain little information about cross-population spawning contributions. If sd is set too large relative to mu * (1 - mu), the beta density becomes U-shaped (mass near 0 and 1), which can cause numerical instability during optimization; SPoRC guards against this internally by squishing the logistic transform away from the boundaries, but specifying a sensible sd is still good practice.

Time-varying stray rates

Stray rates can also be allowed to vary over time by defining multiple blocks. The stray_rate_blocks argument follows the same blocking syntax used elsewhere in SPoRC. For example, to allow stray rates to differ between an early and late period for each population:

stray_prior_tv <- data.frame(
  pop   = c(1, 1, 2, 2),
  block = c(1, 2, 1, 2),
  mu    = c(0.1, 0.2, 0.4, 0.5),
  sd    = c(0.05, 0.05, 0.1, 0.1)
)

input_list <- Setup_Mod_Rec(
  input_list       = input_list,
  do_rec_bias_ramp = 0,
  rec_model        = "bh_rec",
  rec_dd           = "local",
  stray_rate_blocks = c(
    "Block_1_Year_1-20_Pop_1",
    "Block_2_Year_21-terminal_Pop_1",
    "Block_1_Year_1-20_Pop_2",
    "Block_2_Year_21-terminal_Pop_2"
  ),
  stray_rate_spec      = "est_all",
  use_fixed_stray_rate = 0,
  use_stray_rate_prior = 1,
  stray_rate_prior     = stray_prior_tv
)

Each unique block index in stray_rate_blocks requires a corresponding row in stray_rate_prior, one per population × block combination.