Are the parameters estimated by HSSM collectively following a joint distribution, or can they be considered to be independently distributed?

[JoeSu112]

Hi everyone,

I've been working on experimental data and using DDM (Drift Diffusion Model) for parameter estimation. Recently, I obtained the estimation results for each parameter and now I want to simulate data to recover the original dataset. Below is the code I used for the estimation process:

HDDM_0 = hssm.HSSM(
    data=data_0,
    model="ddm",
    loglik_kind="approx_differentiable",
    prior_settings="safe",
    include=[
        {
            "name": "v",
            "prior": {
                "Intercept": {"name": "Normal", "mu": 0, "sigma": 3},
                "month_delayed": {"name": "Normal", "mu": 0, "sigma": 3},
                "pay_diff": {"name": "Normal", "mu": 0, "sigma": 3},
                "time_advantage": {"name": "Normal", "mu": 0, "sigma": 3},
            },
            "formula": "v ~ 1 + month_delayed + pay_diff + time_advantage",
            "link": "identity",
        },
    ],
)

draws=25000
chains=4

infer_HDDM_0 = HDDM_0.sample(
    sampler="nuts_numpyro",  # type of sampler to choose,
    cores=2,  # how many cores to use
    chains=chains,  # how many chains to run
    draws=draws,  # number of draws from the markov chain
    tune=int(draws*0.2),  # number of burn-in samples
    idata_kwargs=dict(log_likelihood=False),  # return log likelihood
)  

And here are the estimation results:

parameter	mean	std
a	1.48326	0.01491
z	0.52278	0.00755
t	0.6727	0.01736
v_month_delayed	-0.21165	0.01298
v_pay_diff	0.41775	0.01352
v_Intercept	0.11768	0.01757

Now, I want to use these parameters to perform a simulation and recover the dataset.

My question is:

1. Do these parameters follow a joint distribution that I can extract from the InferenceData object (e.g., infer_HDDM_0.posterior)? If yes, how can I access this joint distribution so that I can draw a full set of parameters (e.g., v, a, z, t) for each simulated subject?

2. Or should I draw each parameter independently using their respective posterior means and standard deviations?

My intuition is that the Bayesian framework should account for correlations between parameters, and drawing from the joint distribution would better preserve this relationship. I would appreciate any guidance or code snippets that could help me proceed!

Thanks in advance!

[frankmj]

Hi, just a quick response here -- when you simulate data from the posterior samples (using a posterior predictive check - -see the HSSM functions that do this) - it simulates from the joint distribution of parameters. Doing the simulation from posteriors means and stds would not be as good - both because it doesn't take into account the covariance with other parameters and because it would assume the posteriors are gaussian which they need not be. M

…

On Wed, Dec 25, 2024 at 7:09 AM JoeSu112 ***@***.***> wrote: Hi everyone, I've been working on experimental data and using DDM (Drift Diffusion Model) for parameter estimation. Recently, I obtained the estimation results for each parameter and now I want to simulate data to recover the original dataset. Below is the code I used for the estimation process: HDDM_0 = hssm.HSSM( data=data_0, model="ddm", loglik_kind="approx_differentiable", prior_settings="safe", include=[ { "name": "v", "prior": { "Intercept": {"name": "Normal", "mu": 0, "sigma": 3}, "month_delayed": {"name": "Normal", "mu": 0, "sigma": 3}, "pay_diff": {"name": "Normal", "mu": 0, "sigma": 3}, "time_advantage": {"name": "Normal", "mu": 0, "sigma": 3}, }, "formula": "v ~ 1 + month_delayed + pay_diff + time_advantage", "link": "identity", }, ], ) draws=25000chains=4 infer_HDDM_0 = HDDM_0.sample( sampler="nuts_numpyro", # type of sampler to choose, cores=2, # how many cores to use chains=chains, # how many chains to run draws=draws, # number of draws from the markov chain tune=int(draws*0.2), # number of burn-in samples idata_kwargs=dict(log_likelihood=False), # return log likelihood ) And here are the estimation results: parameter mean std a 1.48326 0.01491 z 0.52278 0.00755 t 0.6727 0.01736 v_month_delayed -0.21165 0.01298 v_pay_diff 0.41775 0.01352 v_Intercept 0.11768 0.01757 Now, I want to use these parameters to perform a simulation and recover the dataset. My question is: 1. Do these parameters follow a joint distribution that I can extract from the InferenceData object (e.g., infer_HDDM_0.posterior)? If yes, how can I access this joint distribution so that I can draw a full set of parameters (e.g., v, a, z, t) for each simulated subject? 2. Or should I draw each parameter independently using their respective posterior means and standard deviations? My intuition is that the Bayesian framework should account for correlations between parameters, and drawing from the joint distribution would better preserve this relationship. I would appreciate any guidance or code snippets that could help me proceed! Thanks in advance! — Reply to this email directly, view it on GitHub <#615>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AAG7TFG3PTZUKHPQBFRVXTL2HKU6ZAVCNFSM6AAAAABUF7C3FGVHI2DSMVQWIX3LMV43ERDJONRXK43TNFXW4OZXG42DGMBWGM> . You are receiving this because you are subscribed to this thread.Message ID: ***@***.***>

[JoeSu112]

Thank you so much for your response!

I've found the sample_posterior_predictive function, but I'm a bit confused about the "chain" dimension in the sampling result. I had initially expected there to be one rt and response result for each trial in the data. Could you help me understand how to interpret this "chain" dimension and its role in the output?

Thank you again for your help!

[frankmj]

The chain gives you samples from the posterior distribution. If you just took one value of parameters you could simulate a response and RT for each trial but you want to take into account the uncertainty over the parameters so you can sample a number of parameters from the chain which will then simulate a range of choices and RTs expected given the uncertainty Michael

…

On Tue, Jan 14, 2025 at 11:03 AM JoeSu112 ***@***.***> wrote: Thank you so much for your response! I came across the sample_posterior_predictive function, but I'm a bit confused about the "chain" dimension in the sampling result. I had initially expected there to be one rt and response result for each trial in the data. Could you help me understand how to interpret this "chain" dimension and its role in the output? Thank you again for your help! — Reply to this email directly, view it on GitHub <#615 (reply in thread)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AAG7TFE4UYKDK4QML45D6E32KUYNZAVCNFSM6AAAAABUF7C3FGVHI2DSMVQWIX3LMV43URDJONRXK43TNFXW4Q3PNVWWK3TUHMYTCOBTGMZTGNI> . You are receiving this because you commented.Message ID: ***@***.***>