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Two-Parameter Mixture Model

Source: R/model_mixture2p.R
mixture2p.Rd

Two-Parameter Mixture Model

Usage

mixture2p(
  resp_error,
  set_size = NULL,
  version = c("simple", "slot", "slot_averaging"),
  variable_precision = FALSE,
  vp_nodes = 81L,
  links = NULL,
  ...
)

Arguments

resp_error

The name of the variable in the provided dataset containing the response error. The response Error should code the response relative to the to-be-recalled target in radians. You can transform the response error in degrees to radian using the deg2rad function.

set_size

Name of the variable in the dataset containing the set size, or a single number if the set size is constant. Required by the "slot" and "slot_averaging" versions and ignored by "simple". With a single set size the capacity rule cannot be tested: K is then estimated from the responses at that set size, and only below it.

version

A character string specifying which version of the model to use. Options are:

  • "simple" (default): the standard two-parameter mixture model, in which the probability of a memory response (thetat) and the precision of memory (kappa) are free parameters.

  • "slot": the fixed-resolution slot model of Zhang and Luck (2008). An item is either held at full precision or not held at all, so the probability of a memory response is min(1, K / set_size) and kappa does not vary with set size. Note that K is not identified above max(set_size), where every item is remembered whatever K is, and that the cap introduces a kink in the gradient at K = set_size.

  • "slot_averaging": the slots-plus-averaging model of Zhang and Luck (2008). K slots are distributed over the items, an item receives floor(K / set_size) or one more, and precision grows with the number of slots because averaging independent samples adds their Fisher information.

variable_precision

Logical; if TRUE, the precision of memory varies from trial to trial rather than being constant (van den Berg et al., 2012; Fougnie et al., 2012). This adds the parameter tau, and the likelihood integrates the Fisher information J of a memory representation over gamma(mean = J(kappa), scale = tau). kappa then describes the average trial: it is the concentration whose Fisher information J(kappa) is the mean of J, and tau -> 0 recovers the constant-precision model. Under version = "slot_averaging", the precision of an item held in j slots is the sum of j independent slot draws, gamma(shape = j * J(kappa) / tau, scale = tau), so its variance grows with j. The integral is evaluated by quadrature, which costs about ten times as much per gradient at the default vp_nodes; expect a larger factor in wall time, since the posterior geometry also needs more leapfrog steps. tau is the parameter that needs the most data, and is often only weakly identified in small designs.

vp_nodes

Number of quadrature nodes used when variable_precision = TRUE; must be an odd number of at least 41. With the default of 81, the log likelihood holds to about 1e-5 as long as the implied gamma shape J(kappa) / tau stays above 60 / vp_nodes (0.74 at the default), and to about 2e-6 above a shape of 1. Below that threshold the model rejects rather than returning a biased likelihood, and asks for more nodes; 121 nodes reach a shape of about 0.5, and 241 about 0.25.

A named list of link functions for the model parameters, e.g. list(kappa = "softplus"). Defaults are "log" for kappa, K and tau, and "logit" for thetat.

...

used internally for testing, ignore it

Value

An object of class bmmodel

Details

  • Domain: Working memory

  • Task: Continuous reproduction

  • Name: Two-Parameter Mixture Model

  • Citation:

    • Zhang, W., & Luck, S. J. (2008). Discrete fixed-resolution representations in visual working memory. Nature, 453(7192), 233-235. https://doi.org/10.1038/nature06860

    • van den Berg, R., Shin, H., Chou, W.-C., George, R., & Ma, W. J. (2012). Variability in encoding precision accounts for visual short-term memory limitations. Proceedings of the National Academy of Sciences, 109(22), 8780-8785. https://doi.org/10.1073/pnas.1117465109

  • Requirements:

    • The response variable should be in radians and represent the angular error relative to the target

    • The 'slot' and 'slot_averaging' versions additionally require a set_size variable, which should vary for the capacity rule to be testable

Version: simple

  • Parameters:

    • mu: Location of the von Mises distribution of memory responses (in radians). Fixed internally to 0 by default.

    • kappa: Concentration of the von Mises distribution of memory responses

    • thetat: Mixture weight for target responses, i.e. the probability that a response comes from memory rather than from guessing.

  • Fixed parameters:

    • mu = 0

  • Default parameter links:

    • mu = tan_half; kappa = log; thetat = logit

  • Default priors:

    • mu:

      • main: normal(0, 0.5)

      • effects: normal(0, 0.25)

      • sd: exponential(4)

    • kappa:

      • main: normal(2, 1)

      • effects: normal(0, 1)

      • sd: exponential(1)

    • thetat:

      • main: logistic(0, 1)

      • effects: normal(0, 2.5)

      • sd: exponential(1)

Version: slot

  • Parameters:

    • mu: Location of the von Mises distribution of memory responses (in radians). Fixed internally to 0 by default.

    • kappa: Concentration of the von Mises distribution of memory responses

    • K: Capacity: the number of items that can be held in memory. An item is either held at full precision or not held at all, so the probability of a memory response is min(1, K / set_size).

  • Fixed parameters:

    • mu = 0

  • Default parameter links:

    • mu = tan_half; kappa = log; K = log

  • Default priors:

    • mu:

      • main: normal(0, 0.5)

      • effects: normal(0, 0.25)

      • sd: exponential(4)

    • kappa:

      • main: normal(2, 1)

      • effects: normal(0, 1)

      • sd: exponential(1)

    • K:

      • main: normal(1, 0.5)

      • effects: normal(0, 0.3)

      • sd: exponential(2)

Version: slot_averaging

  • Parameters:

    • mu: Location of the von Mises distribution of memory responses (in radians). Fixed internally to 0 by default.

    • kappa: Concentration of the von Mises distribution for an item held in a single slot. An item holding several slots is more precise, because averaging independent samples adds their Fisher information.

    • K: Capacity: the number of memory slots distributed over the items in the array. An item receives floor(K / set_size) or one more slot, and an item that receives none is guessed.

  • Fixed parameters:

    • mu = 0

  • Default parameter links:

    • mu = tan_half; kappa = log; K = log

  • Default priors:

    • mu:

      • main: normal(0, 0.5)

      • effects: normal(0, 0.25)

      • sd: exponential(4)

    • kappa:

      • main: normal(2, 1)

      • effects: normal(0, 1)

      • sd: exponential(1)

    • K:

      • main: normal(1, 0.5)

      • effects: normal(0, 0.3)

      • sd: exponential(2)

Variable precision

With variable_precision = TRUE every version gains tau, and kappa becomes the precision of the average trial. Shown for "simple":

  • Parameters:

    • mu: Location of the von Mises distribution of memory responses (in radians). Fixed internally to 0 by default.

    • kappa: Concentration of the von Mises distribution of memory responses. With variable precision, kappa describes the average trial: it is the concentration whose Fisher information J(kappa) is the mean of J across trials.

    • tau: Scale of the trial-to-trial variability in memory precision. The Fisher information J of a memory representation is drawn from gamma(mean = J(kappa), scale = tau), so tau -> 0 recovers constant precision and larger values spread precision more widely across trials.

    • thetat: Mixture weight for target responses, i.e. the probability that a response comes from memory rather than from guessing.

  • Default parameter links:

    • mu = tan_half; kappa = log; tau = log; thetat = logit

  • Default priors:

    • mu:

      • main: normal(0, 0.5)

      • effects: normal(0, 0.25)

      • sd: exponential(4)

    • kappa:

      • main: normal(2, 1)

      • effects: normal(0, 1)

      • sd: exponential(1)

    • thetat:

      • main: logistic(0, 1)

      • effects: normal(0, 2.5)

      • sd: exponential(1)

    • tau:

      • main: normal(0, 1)

      • effects: normal(0, 0.5)

      • sd: exponential(2)

Priors from bmm 1.4

Since bmm 1.5.0, the parameters of this model are distributional parameters (dpar) of a custom family, and the location mu1 is called mu. Calls written for bmm 1.4 keep working until bmm 1.6.0 at the earliest: bmm translates their priors, formulas, links and emmeans(nlpar = ), and warns with the code to use instead.

# an intercept prior
set_prior("normal(2, 1)", class = "b", coef = "Intercept", nlpar = "kappa") # 1.4
set_prior("normal(2, 1)", class = "Intercept", dpar = "kappa")              # 1.5

# a prior on all coefficients
set_prior("normal(0, 0.5)", class = "b", nlpar = "kappa")                   # 1.4
set_prior("normal(0, 0.5)", class = "Intercept", dpar = "kappa") +          # 1.5
  set_prior("normal(0, 0.5)", class = "b", dpar = "kappa")

# a prior on the random-effect standard deviations
set_prior("exponential(2)", class = "sd", nlpar = "kappa")                  # 1.4
set_prior("exponential(2)", class = "sd", dpar = "kappa")                   # 1.5

# the location, and emmeans()
bmf(mu1 ~ 1 + cond, kappa ~ 1 + cond)                                      # 1.4
bmf(mu ~ 1 + cond, kappa ~ 1 + cond)                                       # 1.5
emmeans(fit, ~ cond, nlpar = "kappa")                                      # 1.4
emmeans(fit, ~ cond, dpar = "kappa")                                       # 1.5

In bmm 1.4 a class = "b" prior also covered the intercept, so it now becomes two priors with the same distribution, unless the intercept has a prior of its own. One prior changes its meaning. When a formula has an intercept and predictors, such as kappa ~ 1 + cond, brms centres the predictors of a distributional parameter, so an intercept prior applies at the average over the conditions, weighted by their number of trials, not at the reference condition. The slopes keep their meaning, and so do intercept-only formulas and formulas with one coefficient per condition (kappa ~ 0 + cond). If your intercept prior was meant for the reference condition, give it for the average instead.

Two things are not translated. The draws of the location are named after mu (b_Intercept instead of b_mu1_Intercept), and calls that go straight to brms need dpar =: posterior_linpred(fit, dpar = "kappa"), not nlpar = "kappa". Fits saved with bmm 1.4 work as before with summary(), prior_info(), emmeans(), native_parameters() and conditional_effects(). update() samples them again under the custom family, translates their priors and says so in a message.

References

Zhang, W., & Luck, S. J. (2008). Discrete fixed-resolution representations in visual working memory. Nature, 453(7192), 233-235.

van den Berg, R., Shin, H., Chou, W.-C., George, R., & Ma, W. J. (2012). Variability in encoding precision accounts for visual short-term memory limitations. PNAS, 109(22), 8780-8785.

Fougnie, D., Suchow, J. W., & Alvarez, G. A. (2012). Variability in the quality of visual working memory. Nature Communications, 3(1), 1229.

Examples

if (FALSE) { # identical(Sys.getenv("BMM_EXAMPLES"), "true")
# generate artificial data
dat <- data.frame(y = rmixture2p(n = 2000))

# define formula
ff <- bmmformula(kappa ~ 1, thetat ~ 1)

model <- mixture2p(resp_error = "y")

# fit the model
fit <- bmm(
  formula = ff,
  data = dat,
  model = model,
  cores = 4,
  iter = 500,
  backend = "cmdstanr"
)
}