Simulations of the Autocorrelated Bayesian Sampler

A normal distribution for posterior of hypothesis

The fixed stopping rule means that a fixed number of samples are drawn to complete the tasks such as estimations and confidence intervals. This rule applies to tasks such as estimation tasks.

Estimation tasks involve participants providing estimates, such as the number of stimuli (e.g., dots) on a screen or offering confidence intervals of that counts at a specified level.

In this vignette, we will begin by generating several random numbers to represent the stimuli counts. We will assume the estimation task consists of 10 trials, wherein participants are tasked with estimating the number of dots displayed on the screen in each trial.

require(samplr)
#> Loading required package: samplr
set.seed(123)

trial_stim <- sample(20:25, 10, replace=TRUE)
print(trial_stim)
#>  [1] 22 25 22 21 21 25 22 24 23 25

ABS employs the [R6][R6::R6Class] object-oriented programming (OOP) system. Thus we need to construct a new object before running simulations. In the initialising step, we need to provide the values of the following arguments:

• the proposal width of the MC3 sampler, width;
• the number of chains of the MC3 sampler, n_chains;
• the lower bound of the non-decision time, nd_time;
• the variability of the non-decision time, s_nd_time;
• the rate parameter of the Erlang distribution for the response time, lambda.

In this section, we employ the normal posterior distribution, which requires specifying the values of distr_name and distr_params. The distr_name argument should be set to “norm” to indicate the normal distribution. The distr_params argument specifies the standard deviation of the normal distribution. This can be either a single numeric value, indicating a fixed standard deviation across all trials, or a numeric vector of the same length as the stimuli, specifying the standard deviation for each trial. It is important to note that ABS assumes the stimulus value is the mean of the normal distribution, thus there is no need to specify the mean separately.

abs_model <- Zhu23ABS$new(
  width = 1, n_chains = 3, nd_time = 0.3, 
  s_nd_time = 0.2, lambda = 10, distr_name = 'norm', distr_params = 1
)

The simulation process is conducted using the simulate function, which requires two arguments: stopping_rule and start_point. Based on the assumptions of ABS, individuals employ a fixed stopping rule for the estimation task, meaning a predetermined number of samples are drawn. Therefore, stopping_rule is set to 'fixed'.

The start_point argument is set to NA by default, indicating that the starting point of the first trial for each MC3 chain is randomly selected from the posterior of hypotheses, and the starting points of subsequent trials for each chain are set to the last sample of the same chain in the previous trial. Alternatively, users have the option to specify starting points for each trial, ensuring that the length of starting points matches the length of trial_stim. It is important to note that when specified, all chains share the same starting point in each trial, and it will break the dependency of samples between adjacent trials.

To demonstrate the usage of start_point, we will run the simulation twice: once with the default settings and again with specified starting points.

Additionally, when stopping_rule = 'fixed', two further arguments are required:

• the fixed number of samples, n_sample;
• the stimuli of the estimate task, trial_stim;

abs_model$simulate(stopping_rule = 'fixed', 
                   n_sample = 5, trial_stim = trial_stim)

The results of the simulate method save in the field sim_results. Users can get access to the results by running abs_model$sim_results. It is important to note that ABS assumes the response time for each trial follows an Erlang distribution, with the shape parameter equal to the length of the samples that is determined by n_sample and the rate parameter lambda has been specified above. The response time value provided in the table is a random number drawn from the Erlang distribution.

The table below presents the simulation results, displaying only the samples from the cold chain of the MC3 sampler in the samples column. These samples will be used for generating the responses of ABS. With the default setting of start_point = NA, the starting point of the first trial was randomly drawn from the normal distribution N(22, 1). From the second trial, the starting point was set to the last point of the previous trial. For instance, the starting point of the second trial was 23.19899 in this simulation. It is important to note that when start_point = NA, only the starting point of the first trial was included in the samples of ABS. The starting points of subsequent trials served merely as initializers for the sampler and were excluded from the ABS samples.

knitr::kable(abs_model$sim_results)

In the upcoming simulations, we will run the simulation with specified start_point. To proceed, let us generate some starting points for the simulation.

start_point <- runif(length(trial_stim), 20, 25)
print(start_point)
#>  [1] 23.15387 22.60421 23.29811 23.64680 22.43411 21.92228 20.03417 20.01842
#>  [9] 24.97468 20.53943

It is worth noticing that before rerunning the simulation, users should either create a new object or reset the sim_results using the reset_sim_results method.

abs_model$reset_sim_results()
abs_model$simulate(stopping_rule = 'fixed', 
                   start_point = start_point, 
                   n_sample = 5, 
                   trial_stim = trial_stim)

The following table shows the simulation results with specified starting points. We notice that the first sample of each trial follows the order of start_point and is included in the samples of ABS.

knitr::kable(abs_model$sim_results)

In addition to performing point estimation, ABS can also simulate confidence interval estimation by the confidence_interval method.

abs_model$confidence_interval(0.5)

The following table shows the interval estimation on the level of 0.5. conf_interval_l and conf_interval_u represent the lower and the upper bounds.

knitr::kable(abs_model$sim_results)

An advantage of R6 is that it allows method chaining, which means that we can simulate the point and confidence interval estimation in one line of code.

abs_model$reset_sim_results()
abs_model$simulate(
  stopping_rule = 'fixed', 
  n_sample = 5, 
  trial_stim = trial_stim, 
  start_point=start_point)$confidence_interval(0.5)
knitr::kable(abs_model$sim_results)

A custom posterior of hypotheses

In this section, we will illustrate how to employ custom distributions to ABS with a fixed stopping rule, using the same experimental scenario and the same stimuli. First, we specify our custom posterior function.

 custom_post_func <- function(x){
  if (x >= 19 & x < 22){
    return(0.3)
  } else if (x >= 22 & x < 24) {
    return(0.6)
  } else if (x >= 24 & x < 26) {
    return(0.1)
  } else {
    return(0)
  }
}

To employ a custom posterior of hypotheses, two special arguments are required in the initialisation step: custom_distr and custom_start. The custom_distr argument accepts a list of custom posterior functions, one for each trial, matching the length of the stimuli. The custom_start argument specifies the first starting point of the Zhu23ABS sampler, i.e., the initial sample of the entire simulation.

It is important to distinguish custom_start from start_point in the simulate() function. custom_start is only necessary when providing a custom posterior of hypotheses, whereas start_point can be used for both Gaussian and custom posteriors. custom_start initializes the Zhu23ABS sampler without breaking the dependency of samples between trials. In contrast, start_point sets the starting point for each trial, thus breaking the dependency between samples of adjacent trials. Additionally, if users provide a vector of start_point, a value for custom_start is still required as a placeholder and will be overwritten by start_point.

custom_func_list <- replicate(
  length(trial_stim), custom_post_func, simplify = FALSE
)
abs_model <- Zhu23ABS$new(
  width = 1, n_chains = 3, nd_time = 0.3, 
  s_nd_time = 0.2, lambda = 10, 
  custom_distr = custom_func_list, custom_start = 23
)
abs_model$simulate(
  stopping_rule = 'fixed', 
  n_sample = 5, 
  trial_stim = trial_stim
)

The following table shows the simulation results with a custom posterior. We notice that the first sample of the first trial equals to the value of custom_start.

knitr::kable(abs_model$sim_results)

A normal distribution for posterior of hypothesis

The relative stopping rule means that the model counts the difference in evidence between the two hypotheses, and terminates the sampling process whenever the accumulated difference exceeds a threshold. This rule applies to tasks such as 2AFC.

2AFC is a cognitive task that asks participants to make judgments between two alternatives. For instance, in the random dot motion (RDM) task, participants are presented with a screen where most dots move coherently in either the left or right direction, and they’re asked to perceive the correct direction.

ABS is able to describe and simulate this cognitive process. Similarly, we will begin by randomly generating 10 directions from the set c('left', 'right') to represent the stimuli in the RDM task.

trial_stim <- factor(sample(c('left', 'right'), 10, TRUE))

In 2AFC, ABS employs a sampling process and converts the samples into “evidence” supporting either the left or right responses. Specifically, if the sample falls below the decision boundary, it supports the first level in trial_stim, which in our example is “left”; otherwise, the sample will support the second level, which is “right”. According to the assumptions of ABS, it employs a ‘relative’ stopping rule: It counts the difference in evidence between the two responses, and terminates the sampling process whenever the accumulated difference exceeds a threshold.

To simulate the 2AFC of ABS, we need to initialize a new ABS model and then use the simulate method with stopping_rule = 'relative'. The posterior of hypotheses will be normal distribution. The following arguments are required:

• the threshold of the accumulated difference, delta;
• the decision boundary that split the posterior of hypotheses, dec_bdry;
• the discriminability, discrim, which determine the mean values of the distributions for the posterior of hypotheses. It is analogous to the concept of “sensitivity” in signal detection theory, representing the distance between the mean values of two distributions: one for “left” stimuli and the other for “right” stimuli in 2AFC. Under the assumption of ABS, these two distributions are symmetric around 0, so their mean values are -discrim/2 and discrim/2, respectively.
• the stimuli, trial_stim;
• the prior on responses, prior_on_resp, which determines the prior preference for the stimuli. The default setting is c(1, 1), representing an unbiased prior Beta(1, 1). Users can modify this to reflect different prior preferences.
• the prior dependency, prior_depend, which controls whether the prior on responses changes based on the previous stimulus. The default setting is TRUE, which adjusts the prior by adding 1 to the prior_on_resp according to the previous stimulus.
• the maximum length of the MC3 sampler, max_iterations, which determines the maximum length of the MC3 sampler. The sampler will stop when the length of the samples exceeds this value, even if the samples have not met the relative stop rule. The default setting is 1000, suitable for most cases, but users can adjust it based on the discriminability and the relative stopping rule.

To demonstrate the usage of these arguments, we will also run the simulation twice: once with the default settings and again with some of the settings modified.

abs_model2 <- Zhu23ABS$new(
  width=1, n_chains = 3, nd_time = 0.3, s_nd_time = 0.2, 
  lambda = 10, distr_name = 'norm', distr_params = 1
)
abs_model2$simulate(
  stopping_rule = 'relative', delta = 4, dec_bdry = 0, 
  discrim = 1, trial_stim = trial_stim
)

The table below presents the simulation results, including the simulated response, response time, confidence, and point estimates. It is important to note that in the simulation of 2AFC, the length of the sample sequences may vary due to ABS utilizing a relative stopping rule. To illustrate its mechanism, let us examine the sequences in the first two trials as an example.

knitr::kable(abs_model2$sim_results)

The prior on responses, set to c(1, 1), corresponds to an unbiased Beta distribution Beta(1, 1). Let us consider the first trial: the initial sample, 0.4275131, falls above the decision boundary of 0, supporting “right”. Consequently, the posterior on responses shifts to Beta(1, 2). With a relative difference of 1 between the amounts of evidence supporting both stimuli, which is lower than the relative stopping rule, the sampling process continues. Subsequent samples are analysed similarly: the posterior adjusts according to the samples until the relative difference meets the stopping rule. In this case, the last sample supporting “right” results in a posterior of Beta(1, 5), satisfying the stopping rule and prompting ABS to return a “right” response.

With the prior_depend=TRUE argument, the prior on responses for the second trial depends on the stimulus of the first trial. Given that “right” was the correct response in the first trial, the prior on responses for the second trial is Beta(1, 2). Since no starting points are provided, ABS began with 0.2739435, the last sample of the first trial, but this sample is excluded from the samples and from the calculation of the posterior on responses. The process then proceeds similarly to the first trial. In this instance, after eight samples in total, the posterior reaches Beta(6, 2), satisfying the stopping rule and resulting in a “left” response.

It is important to clarify two points. Firstly, the prior on responses is not accumulated when prior_depend=TRUE. In the example above, the prior on responses for the third trial is Beta(2, 1) rather than Beta(2, 2). Secondly, it’s crucial to distinguish the starting points of ABS from those in the drift diffusion model (DDM; Ratcliff (1978), Ratcliff & McKoon (2008)). The bias of the starting points in ABS is independent from the bias of the responses, which is captured by the prior on responses.

In the upcoming simulations, we will run the simulation again with two arguments changed: start_point and prior_depend. To proceed, let us generate some starting points for the simulation.

start_point <- runif(length(trial_stim), -3, 3)
print(start_point)
#>  [1]  2.49825567 -2.57474596  1.86057651 -0.08457317 -1.16859165  0.70724300
#>  [7] -1.08447977  0.91402088  2.49508984  0.41447419

Next, we will put these starting points into the ABS model and rerun the simulation. It is worth noting that in this simulation, the starting point of each trial precisely matches what we provided, and all starting points are included in the calculation of the posterior of responses. Additionally, it is important to observe that the prior on responses resets to Beta(1, 1) at the start of every trial.

abs_model2$reset_sim_results()
abs_model2$simulate(
  stopping_rule = 'relative', delta = 4, dec_bdry = 0, 
  discrim = 1, trial_stim = trial_stim, start_point = start_point, 
  prior_depend = FALSE
)
knitr::kable(abs_model2$sim_results)

A custom posterior of hypotheses

In this section, we will illustrate how to employ custom distributions to ABS with a relative stopping rule, using the RDM task with the same stimuli. First, we specify two custom posterior functions for the “left” and “right” stimuli, and create a list of the custom posterior function according to the stimuli.

custom_post_left <- function(x){
  if (x >= -3 & x < -1){
    return(0.25 * x + 0.75)
  } else if (x >= -1 & x < 0) {
    return(-0.25 * x + 0.25)
  } else {
    return (0)
  }
}

custom_post_right <- function(x){
  if (x >= -1 & x < 1){
    return(0.25 * x + 0.25)
  } else if (x >= 1 & x < 3) {
    return(-0.25 * x + 0.75)
  } else {
    return (0)
  }
}

custom_func_list <- lapply(trial_stim, function(stim) ifelse(stim=='left', custom_post_left, custom_post_right))

Then, we initialise a Zhu23ABS object with a list of custom_func_list and a value of custom_start.

abs_model2 <- Zhu23ABS$new(
  width=1, n_chains = 3, nd_time = 0.3, s_nd_time = 0.2, 
  lambda = 10, custom_distr = custom_func_list, custom_start = -0.1
)
abs_model2$simulate(
  stopping_rule = 'relative', delta = 4, dec_bdry = 0, 
  discrim = 1, trial_stim = trial_stim
)

The following table shows the simulation results with a custom posterior. We notice that the first sample of the first trial equals to the value of custom_start.

knitr::kable(abs_model2$sim_results)