function c = tapas_hgf_categorical_config
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Contains the configuration for the Hierarchical Gaussian Filter (HGF) for categorical inputs
% restricted to 3 levels, no drift, and no inputs at irregular intervals, in the absence of
% perceptual uncertainty.
%
% This model deals with the situation where an agent has to determine the probability of categorical
% outcomes. The tendencies of these outcomes are modeled as independent Gaussian random processes at
% the second level of the HGF. They are transformed into predictive probabilities at the first level
% by a softmax function (i.e., a logistic sigmoid). This amounts to the assumption that the
% probabilities are performing a Gaussian random walk in logit space. The volatility of all of
% these walks is determined by the same higher-level state x_3 in standard HGF fashion.
%
% The HGF is the model introduced in
%
% Mathys C, Daunizeau J, Friston, KJ, & Stephan KE. (2011). A Bayesian foundation for individual
% learning under uncertainty. Frontiers in Human Neuroscience, 5:39.
%
% and elaborated in
%
% Mathys, C, Lomakina, EI, Daunizeau, J, Iglesias, S, Brodersen, KH, Friston, KJ, & Stephan, KE
% (2014). Uncertainty in perception and the Hierarchical Gaussian Filter. Frontiers in Human
% Neuroscience, 8:825.
%
% This file refers to CATEGORICAL inputs (Eqs 1-3 in Mathys et al., (2011)); for continuous inputs,
% refer to tapas_hgf_config.m, for binary inputs, refer to tapas_hgf_binary.m
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% The HGF configuration consists of the priors of parameters and initial values. All priors are
% Gaussian in the space where the quantity they refer to is estimated. They are specified by their
% sufficient statistics: mean and variance (NOT standard deviation).
%
% Quantities are estimated in their native space if they are unbounded (e.g., omega). They are
% estimated in log-space if they have a natural lower bound at zero (e.g., sigma2).
%
% Kappa and theta are estimated in 'logit-space' because bounding them above (in addition to
% their natural lower bound at zero) is an effective means of preventing the exploration of
% parameter regions where the assumptions underlying the variational inversion (cf. Mathys et
% al., 2011) no longer hold.
%
% 'Logit-space' is a logistic sigmoid transformation of native space with a variable upper bound
% a>0:
%
% logit(x) = ln(x/(a-x)); x = a/(1+exp(-logit(x)))
%
% Parameters can be fixed (i.e., set to a fixed value) by setting the variance of their prior to
% zero. Aside from being useful for model comparison, the need for this arises whenever the scale
% and origin of x3 are arbitrary. This is the case if the observation model does not contain the
% representations mu3 and sigma3 from the third level. A choice of scale and origin is then
% implied by fixing the initial value mu3_0 of mu3 and either kappa or omega.
%
% Kappa and theta can be fixed to an arbitrary value by setting the upper bound to twice that
% value and the mean as well as the variance of the prior to zero (this follows immediately from
% the logit transform above).
%
% Fitted trajectories can be plotted by using the command
%
% >> tapas_hgf_categorical_plotTraj(est)
%
% where est is the stucture returned by fitModel. This structure contains the estimated
% perceptual parameters in est.p_prc and the estimated trajectories of the agent's
% representations (cf. Mathys et al., 2011). Their meanings are:
%
% est.p_prc.mu2_0 initial values of the mu2s
% est.p_prc.sa2_0 initial values of the sigma2s
% est.p_prc.mu3_0 initial value of mu3
% est.p_prc.sa3_0 initial value of sigma3
% est.p_prc.ka kappa
% est.p_prc.om omega
% est.p_prc.th theta
%
% est.traj.mu mu
% est.traj.sa sigma
% est.traj.muhat prediction mean
% est.traj.sahat prediction variance
% est.traj.v inferred variances of random walks
% est.traj.w weighting factor of informational and environmental uncertainty at the 2nd level
% est.traj.da prediction errors
% est.traj.ud updates with respect to prediction
% est.traj.psi precision weights on prediction errors
% est.traj.epsi precision-weighted prediction errors
% est.traj.wt full weights on prediction errors (at the first level,
% this is the learning rate)
%
% Tips:
% - When analyzing a new dataset, take your inputs u and use
%
% >> est = tapas_fitModel([], u, 'tapas_hgf_categorical_config', 'tapas_bayes_optimal_categorical_config');
%
% to determine the Bayes optimal perceptual parameters (given your current priors as defined in
% this file here, so choose them wide and loose to let the inputs influence the result). You can
% then use the optimal parameters as your new prior means for the perceptual parameters.
%
% - If you get an error saying that the prior means are in a region where model assumptions are
% violated, lower the prior means of the omegas, starting with the highest level and proceeding
% downwards.
%
% - Alternatives are lowering the prior mean of kappa, if they are not fixed, or adjusting
% the values of the kappas or omegas, if any of them are fixed.
%
% - If the log-model evidence cannot be calculated because the Hessian poses problems, look at
% est.optim.H and fix the parameters that lead to NaNs.
%
% - Your guide to all these adjustments is the log-model evidence (LME). Whenever the LME increases
% by at least 3 across datasets, the adjustment was a good idea and can be justified by just this:
% the LME increased, so you had a better model.
%
% --------------------------------------------------------------------------------------------------
% Copyright (C) 2013-2014 Christoph Mathys, TNU, UZH & ETHZ
%
% This file is part of the HGF toolbox, which is released under the terms of the GNU General Public
% Licence (GPL), version 3. You can redistribute it and/or modify it under the terms of the GPL
% (either version 3 or, at your option, any later version). For further details, see the file
% COPYING or <http://www.gnu.org/licenses/>.
% Config structure
c = struct;
% Model name
c.model = 'hgf_categorical';
% Number of states
c.n_outcomes = 3;
% Upper bound for kappa and theta (lower bound is always zero)
c.kaub = 2;
c.thub = 0.1;
% Sufficient statistics of Gaussian parameter priors
% Initial mu2
c.mu2_0mu = repmat(tapas_logit(1/c.n_outcomes,1),1,c.n_outcomes);
c.mu2_0sa = zeros(1,c.n_outcomes);
% Initial sigma2
c.logsa2_0mu = repmat(log(1),1,c.n_outcomes);
c.logsa2_0sa = zeros(1,c.n_outcomes);
% Initial mu3
% Usually best kept fixed to 1 (determines origin on x3-scale).
c.mu3_0mu = 1;
c.mu3_0sa = 0;
% Initial sigma3
c.logsa3_0mu = log(0.1);
c.logsa3_0sa = 1;
% Kappa
% This should be fixed (preferably to 1) if the observation model
% does not use mu3 (kappa then determines the scaling of x3).
c.logitkamu = 0; % If this is 0, and
c.logitkasa = 0; % this is 0, and c.kaub = 2 above, then kappa is fixed to 1
% Omega
c.ommu = -4;
c.omsa = 5^2;
% Theta
c.logitthmu = 0;
c.logitthsa = 2;
% Gather prior settings in vectors
c.priormus = [
c.mu2_0mu,...
c.logsa2_0mu,...
c.mu3_0mu,...
c.logsa3_0mu,...
c.logitkamu,...
c.ommu,...
c.logitthmu,...
];
c.priorsas = [
c.mu2_0sa,...
c.logsa2_0sa,...
c.mu3_0sa,...
c.logsa3_0sa,...
c.logitkasa,...
c.omsa,...
c.logitthsa,...
];
% Model function handle
c.prc_fun = @tapas_hgf_categorical;
% Handle to function that transforms perceptual parameters to their native space
% from the space they are estimated in
c.transp_prc_fun = @tapas_hgf_categorical_transp;
return;