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;
