Prior distributions

Optional priors/penalties on model parameters. This results in penalized likelihood within TMB or can be used as priors if the model is passed to tmbstan (see the Bayesian vignette).

Note that Jacobian adjustments are only made if bayesian = TRUE when the sdmTMB() model is fit. I.e., the final model will be fit with tmbstan and priors are specified then bayesian should be set to TRUE. Otherwise, leave bayesian = FALSE.

pc_matern() is the Penalized Complexity prior for the Matern covariance function.

Usage

sdmTMBpriors(
  matern_s = pc_matern(range_gt = NA, sigma_lt = NA),
  matern_st = pc_matern(range_gt = NA, sigma_lt = NA),
  phi = halfnormal(NA, NA),
  ar1_rho = normal(NA, NA),
  tweedie_p = normal(NA, NA),
  b = normal(NA, NA),
  sigma_G = halfnormal(NA, NA)
)

normal(location = 0, scale = 1)

halfnormal(location = 0, scale = 1)

mvnormal(location = 0, scale = diag(length(location)))

pc_matern(range_gt, sigma_lt, range_prob = 0.05, sigma_prob = 0.05)

Arguments

matern_s: A PC (Penalized Complexity) prior (pc_matern()) on the spatial random field Matérn parameters.
matern_st: Same as matern_s but for the spatiotemporal random field. Note that you will likely want to set share_fields = FALSE if you choose to set both a spatial and spatiotemporal Matérn PC prior since they both include a prior on the spatial range parameter.
phi: A halfnormal() prior for the dispersion parameter in the observation distribution.
ar1_rho: A normal() prior for the AR1 random field parameter. Note the parameter has support -1 < ar1_rho < 1.
tweedie_p: A normal() prior for the Tweedie power parameter. Note the parameter has support 1 < tweedie_p < 2 so choose a mean appropriately.
b: normal() priors for the main population-level 'beta' effects.
sigma_G: halfnormal() priors for the random intercept SDs.
location: Location parameter(s).
scale: Scale parameter. For normal()/halfnormal(): standard deviation(s). For mvnormal(): variance-covariance matrix.
range_gt: A value one expects the spatial or spatiotemporal range is greater than with 1 - range_prob probability.
sigma_lt: A value one expects the spatial or spatiotemporal marginal standard deviation (sigma_O or sigma_E internally) is less than with 1 - sigma_prob probability.
range_prob: Probability. See description for range_gt.
sigma_prob: Probability. See description for sigma_lt.

Value

A named list with values for the specified priors.

Details

Meant to be passed to the priors argument in sdmTMB().

normal() and halfnormal() define normal and half-normal priors that, at this point, must have a location (mean) parameter of 0. halfnormal() is the same as normal() but can be used to make the syntax clearer. It is intended to be used for parameters that have support > 0.

See https://arxiv.org/abs/1503.00256 for a description of the PC prior for Gaussian random fields. Quoting the discussion (and substituting the argument names in pc_matern()): "In the simulation study we observe good coverage of the equal-tailed 95% credible intervals when the prior satisfies P(sigma > sigma_lt) = 0.05 and P(range < range_gt) = 0.05, where sigma_lt is between 2.5 to 40 times the true marginal standard deviation and range_gt is between 1/10 and 1/2.5 of the true range." Also see INLA::inla.spde2.pcmatern().

Keep in mind that the range is dependent on the units and scale of the coordinate system. In practice, you may choose to try fitting the model without a PC prior and then constraining the model from there. A better option would be to simulate from a model with a given range and sigma to choose reasonable values for the system or base the prior on knowledge from a model fit to a similar system but with more spatial information in the data.

References

Fuglstad, G.-A., Simpson, D., Lindgren, F., and Rue, H. (2016) Constructing Priors that Penalize the Complexity of Gaussian Random Fields. arXiv:1503.00256

Simpson, D., Rue, H., Martins, T., Riebler, A., and Sørbye, S. (2015) Penalising model component complexity: A principled, practical approach to constructing priors. arXiv:1403.4630

Examples

normal(0, 1)
#>      [,1] [,2]
#> [1,]    0    1
#> attr(,"dist")
#> [1] "normal"
halfnormal(0, 1)
#>      [,1] [,2]
#> [1,]    0    1
#> attr(,"dist")
#> [1] "normal"
mvnormal(c(0, 0))
#>      [,1] [,2] [,3]
#> [1,]    0    1    0
#> [2,]    0    0    1
#> attr(,"dist")
#> [1] "mvnormal"
pc_matern(range_gt = 5, sigma_lt = 1)
#> [1] 5.00 1.00 0.05 0.05
#> attr(,"dist")
#> [1] "pc_matern"
plot_pc_matern(range_gt = 5, sigma_lt = 1)


if (inla_installed()) {

d <- subset(pcod, year > 2011)
pcod_spde <- make_mesh(d, c("X", "Y"), cutoff = 30)

# - no priors on population-level effects (`b`)
# - halfnormal(0, 10) prior on dispersion parameter `phi`
# - Matern PC priors on spatial `matern_s` and spatiotemporal
#   `matern_st` random field parameters
m <- sdmTMB(density ~ s(depth, k = 3),
  data = d, mesh = pcod_spde, family = tweedie(),
  share_range = FALSE, time = "year",
  priors = sdmTMBpriors(
    phi = halfnormal(0, 10),
    matern_s = pc_matern(range_gt = 5, sigma_lt = 1),
    matern_st = pc_matern(range_gt = 5, sigma_lt = 1)
  )
)

# - no prior on intercept
# - normal(0, 1) prior on depth coefficient
# - no prior on the dispersion parameter `phi`
# - Matern PC prior
m <- sdmTMB(density ~ depth_scaled,
  data = d, mesh = pcod_spde, family = tweedie(),
  spatiotemporal = "off",
  priors = sdmTMBpriors(
    b = normal(c(NA, 0), c(NA, 1)),
    matern_s = pc_matern(range_gt = 5, sigma_lt = 1)
  )
)

# You get a prior, you get a prior, you get a prior!
# (except on the annual means; see the `NA`s)
m <- sdmTMB(density ~ 0 + depth_scaled + depth_scaled2 + as.factor(year),
  data = d, time = "year", mesh = pcod_spde, family = tweedie(link = "log"),
  share_range = FALSE, spatiotemporal = "AR1",
  priors = sdmTMBpriors(
    b = normal(c(0, 0, NA, NA, NA), c(2, 2, NA, NA, NA)),
    phi = halfnormal(0, 10),
    tweedie_p = normal(1.5, 2),
    ar1_rho = normal(0, 1),
    matern_s = pc_matern(range_gt = 5, sigma_lt = 1),
    matern_st = pc_matern(range_gt = 5, sigma_lt = 1))
)

}