Estimation tasks (continuous reports)¶

bauer.models.EstimationTaskModel is a hierarchical Bayesian observer for tasks in which a participant sees a magnitude n and reports it on a bounded scale (a slider, a typed number). It is built from three components.

Component	Options
Encoding space	`LogSpace()` (Weber), `LinearSpace()` (constant noise), `PowerSpace()` (Box-Cox exponent `alpha` fitted per participant; `PowerSpace(a)` fixes it)
Prior	`UniformPrior(lo, hi)` (e.g. the task’s range) or `GaussianPrior()` (Gaussian in the encoding space: log-normal for log space; centre and width fitted)
Lapse	uniform over the response options (`lapse=True`)

The readout is the posterior mean of n. It is rounded to the response options and censored at the scale ends (response_range). The free noise nu is expressed in log units at n_ref (a Weber fraction) for every encoding space.

Quick start¶

from bauer.models import EstimationTaskModel, LogSpace, UniformPrior

# data: columns n (presented) and response (integer), index level 'subject'
model = EstimationTaskModel(data, encoding_space=LogSpace(),
                            prior=UniformPrior(4.5, 24.5), response_range=(5, 25))
model.build_estimation_model(data)
idata = model.sample(draws=1500, tune=1500, target_accept=0.95)

ppc = model.posterior_predictive(idata, n_posterior_samples=200)  # simulated datasets
sim = model.simulate(data, parameters_by_subject)                 # synthetic data
table, _ = model.parameter_recovery(data, parameters_by_subject)  # simulate -> refit

Because the likelihood is an observed pm.Categorical over the response options, pm.compute_log_likelihood(idata, model=model.estimation_model) gives pointwise log-likelihoods for az.loo / az.compare. Subject-level covariates on a parameter’s group mean: covariates={'nu': 'column'}.

How the likelihood is computed¶

Closed form (log space + Gaussian prior, i.e. log-normal): the posterior over log n is Gaussian, the posterior-mean estimate is log-normal in n, and the probability of each response option is a difference of normal CDFs.

Grid (all other combinations, e.g. the uniform prior or a free alpha): for every participant the estimate n_hat(m) is computed on a measurement grid (posterior mean over a hypothesis grid) and inverted at the edges of the response options. Because n_hat is monotone in m, the probability of each option is again a difference of normal CDFs of the measurement. This is exact up to linear interpolation, and smooth in the parameters, so NUTS gets usable gradients. The two paths agree to < 2e-3 per option probability (tested).

Efficient coding and power spaces¶

An efficient (infomax) encoder for a stimulus density p_enc uses the CDF of p_enc as its internal space, with constant noise there (Wei & Stocker, 2015). For a power-law density p_enc ~ n^-kappa that is PowerSpace with alpha = 1 - kappa. So the log space is efficient for a 1/n (“natural”) prior, and the linear space is efficient for a uniform prior. A fitted alpha can be read as the exponent of the stimulus statistics the encoding is adapted to. It is related to, but not the same as, Stevens’ exponent of the reports: the Bayesian readout undoes most of the transform and adds compression toward the prior.