Priors for the Bayesian hierarchical SDT model

Specifies the prior distributions of the Bayesian model. Each prior is written as a string in Stan notation, "family(arguments)", with the families and parameterisations of Stan, e.g. "normal(0, 1)", "student_t(4, 0, 0.5)" or "gamma(2, 4)". Each argument takes the families that cover the values its parameter can take:

means of \(d'\) and of the criterion can take any real value, so their prior must be defined on the whole real line: normal, std_normal, student_t, cauchy, double_exponential, logistic, gumbel, skew_normal, exp_mod_normal or skew_double_exponential.
standard deviations take any of those, truncated at zero (with location 0, a student_t is a half-Student-\(t\)), or a family on the positive values: lognormal, exponential, gamma, inv_gamma, weibull, frechet, rayleigh, chi_square, inv_chi_square, scaled_inv_chi_square, loglogistic, or pareto_type_2 with mu <= 0.
correlations take "lkj(eta)", the LKJ prior on a correlation matrix and the only correlation prior brms accepts. On a 2 x 2 matrix the correlation follows a beta(eta, eta) distribution stretched from \((0, 1)\) to \((-1, 1)\): lkj(1) is uniform, and larger values of eta concentrate it around zero.

Usage

usdt_priors(
  dprime = "normal(0, 1)",
  sd_dprime = "student_t(4, 0, 0.5)",
  cor_dprime = "lkj(1)",
  criterion = "normal(0, 0.5)",
  sd_criterion = "student_t(4, 0, 0.5)",
  cor_criterion = "lkj(1)",
  sd_signal = "lognormal(0, 0.5)",
  intercept = "normal(0, 1)"
)

# S3 method for class 'usdt_priors'
print(x, ...)

Arguments

dprime: Prior of the mean \(d'\) of each task.
sd_dprime: Prior of the between-subject standard deviation of \(d'\) in each task.
cor_dprime: Prior of the correlation between awareness and influence \(d'\) across subjects.
criterion: Prior of the mean criterion \(c\) of each task, on the classical scale where positive values are conservative.
sd_criterion: Prior of the between-subject standard deviation of the criterion in each task.
cor_criterion: Prior of the correlation between the two criteria. It is used only when both criteria are estimated.
sd_signal: Prior of the standard deviation of the signal distribution, relative to the noise distribution. It is used only under unequal variances.
intercept: Prior of the intercept of the latent regression of influence on awareness \(d'\). It is used only by the latent regression parameterization, where it replaces the prior of the influence mean \(d'\).
x: A usdt_priors object.
...: Ignored.

Value

An object of class usdt_priors: a data frame with one row per prior, giving the parameter, the task, the family, its arguments (a list column) and the prior as a string.

Details

dprime, sd_dprime, criterion, sd_criterion and sd_signal take one prior for both tasks or one per task as list(awareness = ..., influence = ...). The correlations take a single prior.

The defaults are centred at zero and weakly informative on the probit scale: normal(0, 1) for the mean \(d'\), normal(0, 0.5) for the mean criterion, student_t(4, 0, 0.5) for every between-subject standard deviation and a uniform lkj(1) for every correlation. With them, the hit and false-alarm rates of a new subject spread over \((0, 1)\) without piling up at either end. See plot.usdt_priors(). With the same prior for both means, the prior of the difference between them (H1) is symmetric around zero, and every lkj prior is centred at zero (H2).

The model describes the two sensitivities by their means, standard deviations and correlation, so the slope and intercept of the latent regression (H3) have no priors of their own: their priors follow from those of the means, standard deviations and correlation. With the defaults, the intercept prior is symmetric around zero but has heavy tails. It does not change when both standard deviation priors are rescaled by the same factor, because the slope depends on their ratio.

The latent regression parameterization moves the prior of the influence mean to the intercept \(\beta_0\) of the latent regression, so that both the intercept (H3) and the correlation (H2) have the priors you give them. It keeps the priors of the awareness mean, of both standard deviations and of the correlation, and the influence mean follows from them as \(\mu_I = \beta_0 + \rho \, (\sigma_I / \sigma_A) \, \mu_A\). dprime then applies to the awareness task only. The intercept takes normal(0, 1) by default. Inspect its implications with plot.usdt_priors(). Two things follow:

the difference between the means (H1) has an induced prior, \(\beta_0 + (\rho \sigma_I / \sigma_A - 1) \mu_A\), with the heavy tails the intercept has under the correlation parameterization.
the intercept and the correlation are independent a priori but not a posteriori. The data fix the influence mean, the sum of \(\beta_0\) and \(\rho (\sigma_I / \sigma_A) \mu_A\), so a narrow intercept prior also narrows the posterior of \(\rho\), mostly when \(\sigma_A\) is small. The Bayes factor of a correlation hypothesis then depends on the intercept prior too.

Examples

# Default priors
usdt_priors()
#> ── Priors ────────────────────────────────────────────────────────────────────── 
#> 
#>   Parameter    Task       Prior
#>   mean d'      awareness  normal(0, 1)
#>   mean d'      influence  normal(0, 1)
#>   sd(d')       awareness  student_t(4, 0, 0.5)
#>   sd(d')       influence  student_t(4, 0, 0.5)
#>   cor(d')      both       lkj(1)
#>   mean c       awareness  normal(0, 0.5)
#>   mean c       influence  normal(0, 0.5)
#>   sd(c)        awareness  student_t(4, 0, 0.5)
#>   sd(c)        influence  student_t(4, 0, 0.5)
#>   cor(c)       both       lkj(1)
#>   sd(signal)   awareness  lognormal(0, 0.5)
#>   sd(signal)   influence  lognormal(0, 0.5)
#>   intercept    both       normal(0, 1)
#> 
#>   Standard deviation priors are truncated at zero. cor(c) applies when
#>   both criteria are estimated, and sd(signal) under unequal variances.
#>   Under the latent regression, intercept replaces the influence mean d'.

# A narrower prior for the mean influence sensitivity
usdt_priors(dprime = list(awareness = "normal(0, 1)",
                          influence = "normal(0, 0.5)"))
#> ── Priors ────────────────────────────────────────────────────────────────────── 
#> 
#>   Parameter    Task       Prior
#>   mean d'      awareness  normal(0, 1)
#>   mean d'      influence  normal(0, 0.5)
#>   sd(d')       awareness  student_t(4, 0, 0.5)
#>   sd(d')       influence  student_t(4, 0, 0.5)
#>   cor(d')      both       lkj(1)
#>   mean c       awareness  normal(0, 0.5)
#>   mean c       influence  normal(0, 0.5)
#>   sd(c)        awareness  student_t(4, 0, 0.5)
#>   sd(c)        influence  student_t(4, 0, 0.5)
#>   cor(c)       both       lkj(1)
#>   sd(signal)   awareness  lognormal(0, 0.5)
#>   sd(signal)   influence  lognormal(0, 0.5)
#>   intercept    both       normal(0, 1)
#> 
#>   Standard deviation priors are truncated at zero. cor(c) applies when
#>   both criteria are estimated, and sd(signal) under unequal variances.
#>   Under the latent regression, intercept replaces the influence mean d'.

# A correlation prior that concentrates near zero
usdt_priors(cor_dprime = "lkj(2)")
#> ── Priors ────────────────────────────────────────────────────────────────────── 
#> 
#>   Parameter    Task       Prior
#>   mean d'      awareness  normal(0, 1)
#>   mean d'      influence  normal(0, 1)
#>   sd(d')       awareness  student_t(4, 0, 0.5)
#>   sd(d')       influence  student_t(4, 0, 0.5)
#>   cor(d')      both       lkj(2)
#>   mean c       awareness  normal(0, 0.5)
#>   mean c       influence  normal(0, 0.5)
#>   sd(c)        awareness  student_t(4, 0, 0.5)
#>   sd(c)        influence  student_t(4, 0, 0.5)
#>   cor(c)       both       lkj(1)
#>   sd(signal)   awareness  lognormal(0, 0.5)
#>   sd(signal)   influence  lognormal(0, 0.5)
#>   intercept    both       normal(0, 1)
#> 
#>   Standard deviation priors are truncated at zero. cor(c) applies when
#>   both criteria are estimated, and sd(signal) under unequal variances.
#>   Under the latent regression, intercept replaces the influence mean d'.

# Standard deviations of d' expected around 0.3
usdt_priors(sd_dprime = "student_t(4, 0.3, 0.2)")
#> ── Priors ────────────────────────────────────────────────────────────────────── 
#> 
#>   Parameter    Task       Prior
#>   mean d'      awareness  normal(0, 1)
#>   mean d'      influence  normal(0, 1)
#>   sd(d')       awareness  student_t(4, 0.3, 0.2)
#>   sd(d')       influence  student_t(4, 0.3, 0.2)
#>   cor(d')      both       lkj(1)
#>   mean c       awareness  normal(0, 0.5)
#>   mean c       influence  normal(0, 0.5)
#>   sd(c)        awareness  student_t(4, 0, 0.5)
#>   sd(c)        influence  student_t(4, 0, 0.5)
#>   cor(c)       both       lkj(1)
#>   sd(signal)   awareness  lognormal(0, 0.5)
#>   sd(signal)   influence  lognormal(0, 0.5)
#>   intercept    both       normal(0, 1)
#> 
#>   Standard deviation priors are truncated at zero. cor(c) applies when
#>   both criteria are estimated, and sd(signal) under unequal variances.
#>   Under the latent regression, intercept replaces the influence mean d'.

# Any family that covers the positive values, e.g. a gamma distribution
usdt_priors(sd_dprime = "gamma(2, 4)")
#> ── Priors ────────────────────────────────────────────────────────────────────── 
#> 
#>   Parameter    Task       Prior
#>   mean d'      awareness  normal(0, 1)
#>   mean d'      influence  normal(0, 1)
#>   sd(d')       awareness  gamma(2, 4)
#>   sd(d')       influence  gamma(2, 4)
#>   cor(d')      both       lkj(1)
#>   mean c       awareness  normal(0, 0.5)
#>   mean c       influence  normal(0, 0.5)
#>   sd(c)        awareness  student_t(4, 0, 0.5)
#>   sd(c)        influence  student_t(4, 0, 0.5)
#>   cor(c)       both       lkj(1)
#>   sd(signal)   awareness  lognormal(0, 0.5)
#>   sd(signal)   influence  lognormal(0, 0.5)
#>   intercept    both       normal(0, 1)
#> 
#>   Standard deviation priors are truncated at zero. cor(c) applies when
#>   both criteria are estimated, and sd(signal) under unequal variances.
#>   Under the latent regression, intercept replaces the influence mean d'.