I am trying to implement a model to predict the next time a user will login to some system. The only data I have is the user ID and the login time.

The distribution for the time between each users' logins (in seconds) is exponential. I'm struggling to implement a model that one could use to predict the next n $\Delta_t$.

There was a very similar question previously asked, and The answer was to model the data as such:

$ (\Delta_t)i∼$Exponential$(\lambda_i)$ $i=1…N$


Where the Exponential distribution is paramatarised by $\lambda$ and the $P$ is the distribution for $\lambda_i$ which are gamma distributed.

I'm strugging to understand how I would implement this in Python using numpy or scipy for example. I'm very familiar with training machine learning models using scikit-Learn or PyTorch, but how would I implement the model above?

Given the data below, where do I go next?

time_between_logins = scipy.stats.expon.rvs(450, 80000, 1000)

enter image description here

  • $\begingroup$ This looks like something the PyMC package would readily handle, but coding questions are not on topic here. $\endgroup$
    – Galen
    Sep 21, 2022 at 3:34
  • $\begingroup$ @Galen Should I move this question to stackoverflow instead? $\endgroup$
    – jwnz
    Sep 21, 2022 at 3:49
  • $\begingroup$ The resulting distribution from the inter-login times $\Delta_{t,i}$ is a Pareto distribution, see e.g. math.stackexchange.com/questions/646852/…. Therefore you can use any function that allows you to fit a Pareto distribution, or you can do it yourself by maximum likelihood. $\endgroup$ Sep 23, 2022 at 5:12

1 Answer 1


I have managed to get an implementation that works for my purposes.

Below is a snippet of how I implemented just the model definition in NumPyro. You can also find a Gist of the full working and runnable code here.

def model(user_ids, time_deltas=None):

    n_users = len(set(user_ids))

    with numpyro.plate("user_plate", n_users):
        gamma_dist = dist.Gamma(concentration, gamma_rate)
        scaledGamma = dist.TransformedDistribution(
        _r = numpyro.sample("rate", scaledGamma)

    rate = _r[user_ids]

    with numpyro.plate("data", len(time_deltas)):
        exp = dist.Exponential(1 / rate)
        numpyro.sample("obs", exp, obs=time_deltas)

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