tapas_ehgf_ar1_binary.m

function [traj, infStates] = tapas_ehgf_ar1_binary(r, p, varargin)
% Calculates the trajectories of the agent's representations under the HGF
%
% This function can be called in two ways:
% 
% (1) tapas_ehgf_ar1_binary(r, p)
%   
%     where r is the structure generated by tapas_fitModel and p is the parameter vector in native space;
%
% (2) tapas_ehgf_ar1_binary(r, ptrans, 'trans')
% 
%     where r is the structure generated by tapas_fitModel, ptrans is the parameter vector in
%     transformed space, and 'trans' is a flag indicating this.
%
% --------------------------------------------------------------------------------------------------
% Copyright (C) 2012-2020 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/>.


% Transform paramaters back to their native space if needed
if ~isempty(varargin) && strcmp(varargin{1},'trans')
    p = tapas_ehgf_ar1_binary_transp(r, p);
end

% Number of levels
try
    l = r.c_prc.n_levels;
catch
    l = (length(p)+1)/7;
    
    if l ~= floor(l)
        error('tapas:hgf:UndetNumLevels', 'Cannot determine number of levels');
    end
end

% Unpack parameters
mu_0 = p(1:l);
sa_0 = p(l+1:2*l);
phi  = p(2*l+1:3*l);
m    = p(3*l+1:4*l);
rho  = p(4*l+1:5*l);
ka   = p(5*l+1:6*l-1);
om   = p(6*l:7*l-2);
th   = exp(p(7*l-1));

% Add dummy "zeroth" trial
u = [0; r.u(:,1)];

% Number of trials (including prior)
n = length(u);

% Assume that if u has more than one column, the last contains t
try
    if r.c_prc.irregular_intervals
        if size(u,2) > 1
            t = [0; r.u(:,end)];
        else
            error('tapas:hgf:InputSingleColumn', 'Input matrix must contain more than one column if irregular_intervals is set to true.');
        end
    else
        t = ones(n,1);
    end
catch
    if size(u,2) > 1
        t = [0; r.u(:,end)];
    else
        t = ones(n,1);
    end
end

% Initialize updated quantities

% Representations
mu = NaN(n,l);
pi = NaN(n,l);

% Other quantities
muhat = NaN(n,l);
pihat = NaN(n,l);
v     = NaN(n,l);
w     = NaN(n,l-1);
da    = NaN(n,l);

% Representation priors
% Note: first entries of the other quantities remain
% NaN because they are undefined and are thrown away
% at the end; their presence simply leads to consistent
% trial indices.
mu(1,1) = tapas_sgm(mu_0(1), 1);
pi(1,1) = Inf;
mu(1,2:end) = mu_0(2:end);
pi(1,2:end) = 1./sa_0(2:end);

% Pass through representation update loop
for k = 2:1:n
    if not(ismember(k-1, r.ign))
        
        %%%%%%%%%%%%%%%%%%%%%%
        % Effect of input u(k)
        %%%%%%%%%%%%%%%%%%%%%%
        
        % 2nd level prediction
        muhat(k,2) = mu(k-1,2) +t(k) *rho(2) +t(k) *phi(2) *(m(2) -mu(k-1,2));
        
        % 1st level
        % ~~~~~~~~~
        % Prediction
        muhat(k,1) = tapas_sgm(ka(1) *muhat(k,2), 1);
        
        % Precision of prediction
        pihat(k,1) = 1/(muhat(k,1)*(1 -muhat(k,1)));

        % Updates
        pi(k,1) = Inf;
        mu(k,1) = u(k);

        % Prediction error
        da(k,1) = mu(k,1) -muhat(k,1);

        % 2nd level
        % ~~~~~~~~~
        % Prediction: see above
        
        % Precision of prediction
        pihat(k,2) = 1/(1/pi(k-1,2) +exp(ka(2) *mu(k-1,3) +om(2)));

        % Updates
        pi(k,2) = pihat(k,2) +ka(1)^2/pihat(k,1);
        mu(k,2) = muhat(k,2) +ka(1)/pi(k,2) *da(k,1);

        % Volatility prediction error
        da(k,2) = (1/pi(k,2) +(mu(k,2) -muhat(k,2))^2) *pihat(k,2) -1;

        if l > 3
            % Pass through higher levels
            % ~~~~~~~~~~~~~~~~~~~~~~~~~~
            for j = 3:l-1
                % Prediction
                muhat(k,j) = mu(k-1,j) +t(k) *rho(j) +t(k) *phi(j) *(m(j) -mu(k-1,j));
                
                % Precision of prediction
                pihat(k,j) = 1/(1/pi(k-1,j) +t(k) *exp(ka(j) *mu(k-1,j+1) +om(j)));

                % Weighting factor
                v(k,j-1) = t(k) *exp(ka(j-1) *mu(k-1,j) +om(j-1));
                w(k,j-1) = v(k,j-1) *pihat(k,j-1);

                % Mean update
                mu(k,j) = muhat(k,j) +1/2 *1/pihat(k,j) *ka(j-1) *w(k,j-1) *da(k,j-1);

                % Ingredients of precision update which depend on the mean
                % update
                vv = t(k) *exp(ka(j-1) *mu(k,j) +om(j-1));
                pimhat = 1/(1/pi(k-1,j-1) +vv); 
                ww = vv *pimhat;
                rr = (vv -1/pi(k-1,j-1)) *pimhat;
                dd = (1/pi(k,j-1) +(mu(k,j-1) -muhat(k,j-1))^2) *pimhat -1;
                
                % Precision update
                pi(k,j) = pihat(k,j) +max(0, 1/2 *ka(j-1)^2 *ww*(ww +rr*dd));

                % Volatility prediction error
                da(k,j) = (1/pi(k,j) +(mu(k,j) -muhat(k,j))^2) *pihat(k,j) -1;
            end
        end

        % Last level
        % ~~~~~~~~~~
        % Prediction
        muhat(k,l) = mu(k-1,l) +t(k) *rho(l) +t(k) *phi(l) *(m(l) -mu(k-1,l));
        
        % Precision of prediction
        pihat(k,l) = 1/(1/pi(k-1,l) +t(k) *th);

        % Weighting factor
        v(k,l)   = t(k) *th;
        v(k,l-1) = t(k) *exp(ka(l-1) *mu(k-1,l) +om(l-1));
        w(k,l-1) = v(k,l-1) *pihat(k,l-1);
        
        % Mean update
        mu(k,l) = muhat(k,l) +1/2 *1/pihat(k,l) *ka(l-1) *w(k,l-1) *da(k,l-1);
        
        % Ingredients of the precision update which depend on the mean
        % update
        vv = t(k) *exp(ka(l-1) *mu(k,l) +om(l-1));
        pimhat = 1/(1/pi(k-1,l-1) +vv); 
        ww = vv *pimhat;
        rr = (vv -1/pi(k-1,l-1)) *pimhat;
        dd = (1/pi(k,l-1) +(mu(k,l-1) -muhat(k,l-1))^2) *pimhat -1;
                
        pi(k,l) = pihat(k,l) +max(0, 1/2 *ka(l-1)^2 *ww*(ww +rr*dd));
 
        % Volatility prediction error
        da(k,l) = (1/pi(k,l) +(mu(k,l) -muhat(k,l))^2) *pihat(k,l) -1;
    else

        mu(k,:) = mu(k-1,:); 
        pi(k,:) = pi(k-1,:);

        muhat(k,:) = muhat(k-1,:);
        pihat(k,:) = pihat(k-1,:);
        
        v(k,:)  = v(k-1,:);
        w(k,:)  = w(k-1,:);
        da(k,:) = da(k-1,:);
        
    end
end

% Implied learning rate at the first level
sgmmu2 = tapas_sgm(ka(1) *mu(:,2), 1);
dasgmmu2 = u -sgmmu2;
lr1    = diff(sgmmu2)./dasgmmu2(2:n,1);
lr1(da(2:n,1)==0) = 0;

% Remove representation priors
mu(1,:)  = [];
pi(1,:)  = [];

% Remove other dummy initial values
muhat(1,:) = [];
pihat(1,:) = [];
v(1,:)     = [];
w(1,:)     = [];
da(1,:)    = [];

% Create result data structure
traj = struct;

traj.mu     = mu;
traj.sa     = 1./pi;

traj.muhat  = muhat;
traj.sahat  = 1./pihat;

traj.v      = v;
traj.w      = w;
traj.da     = da;

% Updates with respect to prediction
traj.ud = mu -muhat;

% Psi (precision weights on prediction errors)
psi        = NaN(n-1,l);
psi(:,2)   = 1./pi(:,2);
psi(:,3:l) = pihat(:,2:l-1)./pi(:,3:l);
traj.psi   = psi;

% Epsilons (precision-weighted prediction errors)
epsi        = NaN(n-1,l);
epsi(:,2:l) = psi(:,2:l) .*da(:,1:l-1);
traj.epsi   = epsi;

% Full learning rate (full weights on prediction errors)
wt        = NaN(n-1,l);
wt(:,1)   = lr1;
wt(:,2)   = psi(:,2);
wt(:,3:l) = 1/2 *(v(:,2:l-1) *diag(ka(2:l-1))) .*psi(:,3:l);
traj.wt   = wt;

% Create matrices for use by the observation model
infStates = NaN(n-1,l,4);
infStates(:,:,1) = traj.muhat;
infStates(:,:,2) = traj.sahat;
infStates(:,:,3) = traj.mu;
infStates(:,:,4) = traj.sa;

end