How to Translate HDDM Priors in HSSM Setting?

[JoeSu112]

Hi all,

We are trying to replicate results from HDDM using the HSSM package. Specifically, we are modeling the parameters a, v, t, and z, with a regression structure on v: v ~ 1 + month_delayed + pay_diff
We are using the centered model. Here's our model specification:

HDDM_0 = hssm.HSSM(
    data=data_0,
    model="ddm",
    loglik_kind="approx_differentiable",
    prior_settings="safe",
    link_settings="log_logit",
    include=[
        {
            "name": "v",
            "formula": "v ~ 0 + (1 + month_delayed + pay_diff | participant_id)",
            "prior": {
                "1|participant_id": {
                    "name": "Normal",
                    "mu": {"name": "Normal", "mu": 1, "sigma": 1.0, "initval": 0},
                    "sigma": {"name": "HalfNormal", "sigma": 1, "initval": 0.25},
                    },
                "month_delayed|participant_id": {
                    "name": "Normal",
                    "mu": {"name": "Normal", "mu": 0, "sigma": 1.0, "initval": 0},
                    "sigma": {"name": "HalfNormal", "sigma": 1, "initval": 0.25},
                    },
                "pay_diff|participant_id": {
                    "name": "Normal",
                    "mu": {"name": "Normal", "mu": 0, "sigma": 1.0, "initval": 0},
                    "sigma": {"name": "HalfNormal", "sigma": 1, "initval": 0.25},
                    },
                },
            "link": "identity",
            },
        {
            "name": "a",
            "formula": "a ~ 0 + (1|participant_id)",
            },
        {
            "name": "z",
            "formula": "z ~ 0 + (1|participant_id)",
            },
        {
            "name": "t",
            "formula": "t ~ 0 + (1|participant_id)",
            },
    ],
    noncentered=False,
    p_outlier=0.05,
)

In HDDM, the priors are characterized like this:

We’ve read this helpful article: https://www.gingjehli.com/single-post/choosing-effective-samplers-and-setting-priors, but we are still unsure how to properly translate priors from HDDM above to the HSSM setting.

Moreover, in the article, the prior on v is suggested to be in the range [-5, 5]. However, since we have three regressors (Intercept, month_delayed, and pay_diff), we're not sure how to set priors so that the combined value of v stays roughly within [-5, 5].

Any guidance on how to translate these priors into the HSSM framework (and how to handle the regression case for v) would be greatly appreciated!

Thanks!

[AlexanderFengler]

Hi @JoeSu112,

there are a few subtleties on this one, specifically also on replicating the priors on regression betas.

Before attempting to replicate the HDDM priors in detail (one note of caution, I think those have their own problems and possibly shouldn't be taken as a 'gold-standard'), two things:

1. Which kind of discrepancy are you seeing between the HSSM and HDDM parameters, if you specify the priors as per your discussion post?
2. To 'make sure' that the parameters are within reasonable bounds, first, I would control specifically the initial points of the sampler so that this is guaranteed (if the sampler then tries to escape a reasonable region, one has to go back to the drawing board either way). Second, in HSSM we have boundary checks in the likelihoods, which should help on this front too. Are you in fact running into this problem?

Best,
Alex

[JoeSu112]

Hi @AlexanderFengler,

Thanks for your reply! The reason we're trying to replicate the HDDM priors is that we observed good convergence in HDDM, but our HSSM models are struggling to converge. Our thought was that using HDDM priors might help improve convergence in HSSM.

That said, since you mentioned the HDDM priors may not be a “gold standard,” I’m wondering if you have any recommendations on how to choose or tune priors in HSSM — specifically to improve convergence in a centered model with regression on v.

Any guidance would be much appreciated!

Best,
Joe

[AlexanderFengler]

Hey @JoeSu112,

let me try to give a more complete answer and sorry for the delay here:

1. Specifically concerning your point about the regression. I think when the HDDM paper came out, the toolbox didn't yet have all the regression functionality, which is why that information might be missing. For regressions in HDDM approximately two rules capture all cases.

• Rule 1, if you are dealing with an intercept, take the prior that you would have assigned to the parameter in isolation.
• Rule 2, if you are dealing with any other regressor, apply a normal centered around 0 with a pretty large standard deviation (I checked and it looks like it was set to 15). Note: This applies to the case for DDM with analytical likelihoods. I don't really recommend these setting per se, but you could try it.

2. Concerning the [-5,5] bounds, those are not strictly enforced if you are dealing with the analytical likelihood for the DDM. If you wanted to be roughly in that ballpark, you can e.g. use Truncated distributions on all your beta coefficients. You are using the approx_differentiable version in which case parameter boundaries are applied via boundary checks at the point of likelihood evaluation automatically. So you should be good here. Again, if you a priori strictly want to discourage leaving the region, Truncated distributions are probably your friend here.

Lastly, let me try to rewrite your model to setting that I consider a priori more robust:

HDDM_0 = hssm.HSSM(
    data=data_0,
    model="ddm",
    loglik_kind="approx_differentiable",
    prior_settings="safe",
    include=[
        {
            "name": "v",
            "formula": "v ~ 1 + (1 + month_delayed + pay_diff | participant_id)",
            "prior": {
                "1|participant_id": {
                    "name": "Normal",
                    "mu": {"name": "Normal", "mu": 0, "sigma": 1.0},
                    "sigma": {"name": "Weibull", "alpha": 1.5, "beta": 0.3},
                    },
                "month_delayed|participant_id": {
                    "name": "Normal",
                    "mu": {"name": "Normal", "mu": 0, "sigma": 0.5},
                    "sigma": {"name": "Weibull", "alpha": 1.5, "beta": 0.3},
                    },
                "pay_diff|participant_id": {
                    "name": "Normal",
                    "mu": {"name": "Normal", "mu": 0, "sigma": 0.5},
                    "sigma": {"name": "Weibull", "alpha": 1.5, "beta": 0.3},
                    },
                },
            "link": "identity",
            },
        {
            "name": "a",
            "formula": "a ~ 1 + (1|participant_id)",
            },
        {
            "name": "z",
            "formula": "z ~ 1 + (1|participant_id)",
            },
        {
            "name": "t",
            "formula": "t ~ 1 + (1|participant_id)",
            },
    ],
    noncentered=False,
    p_outlier=0.05,
)

Maybe you can try this. Some motivation for the changes:

• Smaller sigma on the Normal for mu on the regressors --> usually one doesn't expect these effects to be very large, so can benefit form a bit of a priori regularization without loosing a lot of fidelity
• Weibull instead of HalfNormal on the sigmas --> generally don't want to put large mass on 0 for sigmas in hierarchical setting. Invites degeneracies and is unrealistic.
• I turned off the log_logit link_settings for easier interpretation of what the priors mean in the actual target space (links make that generally more tricky)
• Generally initvals should be chosen reasonably by default, however if you check the documentation, you will find some utilities for getting and setting initial values after initializing the model. This is encourage by the PyMC library, instead of specifying it directly in the distribution. There is also this tutorial is the documentation giving some examples

Best,
Alex