Week 6 Homework

1. Play with the diffusion model, using the demo code. Vary each of the model parameters: z, mu, sigma, a. For each, describe how the model predictions change, and try to give an intuitive explanation based on the parameter's psychological interpretation.

2. Simulate the Bayesian random walk model, using equal-variance Gaussians for p_S and p_N. Define the model parameters Means and variance for the two distributions (p_S and p_N) mSignal = 1; %mean of signal distribution mNoise = 0; %mean of noise distribution sigma = 1; %standard deviation of both distributions
Decision threshold—corresponds to log-odds of accuracy, so try to define it that way alpha = log(.9/.1); %decision threshold; corresponds to 90% accuracy
Starting point—corresponds to prior log-odds, so try to define it that way E0 = log(2/1); %starting point; corresponds to prior belief of 2/3 signal, 1/3 noise
Time constant for converting number of samples to RT tau = .1; %time constant; 100 ms
Setup for simulation Choose a number of trials to simulate (can be large because model is fast) N = 10000; %number of trials to simulate
Create a sequence of correct categories (i.e., signal or noise on each trial) H = 1 + (rand(N,1) > 2/3); %1 for signal, 2 for noise; setting 2/3 baserate to match prior (though they don't have to match)
Create arrays for tracking response and RT across trials r = zeros(N,1); %response for each trial RT = zeros(N,1); %RT for each trial
Loop through trials and simulate each step of the model for i = 1:N %loop through trials Initialize things to be tracked this trial Evidence E = E0; %evidence begins at starting point Number of observations n = 0; %tracks number of observations taken on this trial Sample observations until reaching threshold (hint: use a while loop) while abs(E) < alpha Sample an observation (x) based on the correct category for this trial if H(i) == 1 %signal trial x = mSignal + randn*sigma; %observation else %noise trial x = mNoise + randn*sigma; %observation end Calculate the loglikelihood ratio for this observation; try to work through the algebra using the expressions for Gaussian distributions, and then compare to Eq 17 from this week's chapter L = (mSignal-mNoise)/sigma^2 * (x-(mSignal+mNoise)/2); %loglikelihood ratio Increment the evidence and the count of observations E = E + L; %increment evidence by loglikelihood of current observation n = n + 1; %increment count of observations end Record response and RT for this trial r(i) = 1 + (E<0); %response: 1 if E positive (hit upper threshold), 2 if E negative (hit lower threshold) RT(i) = n*tau; %RT for this trial end
Show results Response probabilities under both categories rProbs = zeros(2,2); %response probabilities; trial type (signal, noise) by response (signal, noise) for i=1:2,for j=1:2 %correct answer i, actual answer j rProbs(i,j) = mean(r(H==i)==j); %proportion of response on trials of type i that are response j end,end disp(' ') disp('Response probabilities on signal trials') disp(['Signal response: ' num2str(rProbs(1,1)) '; Noise response: ' num2str(rProbs(1,2))]) disp('Response probabilities on noise trials') disp(['Signal response: ' num2str(rProbs(2,1)) '; Noise response: ' num2str(rProbs(2,2))])
RT distributions for both responses under both categories figure(1),clf for i=1:2,for j=1:2 %correct answer i, actual answer j subplot(2,2,(i-1)*2+j) hist(RT(H==i & r==j),tau/2:tau:max(RT)+tau) %RTs for this trial type and this response, with bin size equal to tau set(gca,'XLim',[0 max(RT)],'Ylim',[0 max(histc(RT,0:tau:max(RT)))]) %put all graphs on same scale if i==1&&j==1,ylabel('Signal Trials'),end if i==2&&j==1,xlabel('Signal Responses'),ylabel('Noise Trials'),end if i==2&&j==2,xlabel('Noise Responses'),end end,end