Wiener drift diffusion model -- initialization errors

[Tor Tarantola]

Hi all,

I'm having a problem specifying a Wiener drift diffusion model. I'm attempting to specify the boundary separation using an intercept and a coefficient:

a ~ intercept + coefficient * value

The priors are specified as follows:

intercept_mean ~ normal(1,2)
intercept_sd ~ half-cauchy(0,1)
coefficient_mean ~ normal(0,1)
coefficient_sd ~ half-cauchy(0,0.5)

intercept_a <- pow((intercept_mean / intercept_sd),2)
intercept_b <- intercept_mean / pow(intercept_sd, 2)

intercept ~ gamma(intercept_a, intercept_b)
coefficient ~ normal(coefficient_mean, coefficient_sd)

"Value" is bounded (-1,1). The model is hierarchical, and allows all four drift parameters to vary freely. Initialization is failing because too many proposals are being rejected due to the boundary separation being negative. I've tried constraining the prior for the coefficient so it's less likely to exceed the intercept (see above), but I'm concerned about constraining it too much. Is there a good way to get around this problem (or override the 100 attempts limit)? Any advice much appreciated!

Best,
Tor


----

Tor Tarantola

PhD Candidate in Psychology
University of Cambridge

to...@cam.ac.uk, tor.ta...@gmail.com

+44 (0)7425 673663 (UK)  |  +1 (646) 450-4867 (US)
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Department of Psychology

Downing Street

Cambridge CB2 3EB
United Kingdom

[Daniel Lee]

Hi Tor,

Can you send the complete Stan program? It's hard to piece together what you're doing when it's not clear what are data and parameters and how the parameters are declared.

Daniel

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[Joachim Vandekerckhove]

You could try something like: log(a) ~ intercept + coefficient * value

[Tor Tarantola]

Hi both,

Thanks very much for your messages. I've tried a log transformation, but the errors are replaced with log(0) initialization errors. I'm attaching the full model. You'll see that it's a Q-learning model, so the threshold term (a) is a function of an intercept and the difference in learned values (q[2] - q[1]), which is weighted by a coefficient. This value difference is naturally bounded between -1 and 1.

Thanks again for any advice!

Best,

Tor

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[Guido Biele]

not sure if the following is useful, because it appears you defined

q[1] and q[2] as values for the wrong (low value) and correct 

(high value) options. 

still, stochastic payoffs will likely make that sometimes q[1] > q[2];

therefore, if boundary separation shall depend on the difficulty of 

the choice, the absolute of the value difference should work.

a <- threshold_int[s] + ( threshold_learning[s] * abs(q[2]-q[1]) );

guido

[Tor Tarantola]

Hi Guido,

Thanks very much for pointing this out! I've made the change, but unfortunately I'm still getting the same error due to the number of times the initial values for the intercept (threshold_int) and coefficient (threshold_learning) result in a negative value for 'a.' Is there any way to add a relative constraint to the parameters (e.g. threshold_learning has a lower bound of -threshold_int)?

Tor

[Guido Biele]

I guess the problem is that given how you have set up your model 

it easily happens that

threshold_int[s] < threshold_learning[s] * abs(q[2]-q[1])

I think you could either use priors to make sure that

threshold_int is relatively large.

Instead I would do what Joachim suggested;

threshold_int <- exp(threshold_int[s] + threshold_learning[s] * abs(q[2]-q[1]))

one other thing that can happen is that t_er gets larger

than your smallest RT, which will also lead log_prob values of -Inf.

One way to deal with this is to constrain t_er to be maximally the fastest RT.

Guido

[Bob Carpenter]

Absolute values are tricky on parameters because they wind up
providing multimodal answers at alpha and -alpha by definition.

If there's an actual constraint, it's better to put it in
the model rather than try to compensate with something like absolute
value. (I hope I understood the problem.)

- Bob

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[Bob Carpenter]

If you just want q[2] > q[1], then


ordered q[2];

will work, though it'll be q[1] > q[2]. We also have positive_ordered
if you also have q[2] > q[1] > 0.

- Bob

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[Guido Biele]

Bob:

q1 and q2 are not parameters. 

they are reward expectations learned in an experiment 

where  subjects learn from feedback how valuable two 

choice alternatives are.

in this context, abs(q[1]-q[2]) serves as an indicator of

decision difficulty, i.e. choices are more difficult when 

abs(q[1]-q[2]) is small.

It is a common assumption that people take more time 

to make choices when they are difficult (commonly 

referred to as speed accuracy trade-off). When modelling 

choices with the the wiener distribution (which should be 

renamed to wiener first passage tome distribution ...) this 

can be achieved by making the boundary separation 

parameter of the wiener fpt distribution dependent on 

ab(q[2]-q[1])

Guido

PS: 

Tor:

A student of mine has implemented combined RL-DDM models

(in jags) and he found that making boundary separation dependent 

on the absolute difference of Q values did not make the model fit

noticeably better.

[Tor Tarantola]

Hi all,

It looks like the main problem was the lack of constraint on the non-decision time (like you mentioned, Guido). I think too many initial proposals exceeded the total RTs for a few trials. I've made the non-decision time parameters non-hierarchical (one for each subject) and set the upper bound for each to the minimum RT for that subject. It seems to be working now, even without log transforming the threshold, and keeping the absolute value on the Q difference. Thanks very much for your help.

-Tor