In my application (spaced-repetition system for quizzes), I have an unknown half-life (in units of time), $h>0$. This half-life governs a Bernoulli trial $X$, such that $P(X=1) = 2^{-t / h}$, for some known, deterministic time $t>0$ (same units as $h$, this corresponds to when the Bernoulli test was conducted).

I’d like to know what prior to place on $h$ such that the posterior $P(h | x)$ is conjugate, i.e.,

  • $h \sim \, ??? $ such that
  • $π = 2^{-t/h}$, a deterministic function, and
  • $x \sim Bernoulli(π)$, so that
  • the posterior $p(h | x) \propto p(x | h) \cdot p(h)$ has same structure as $p(h)$, i.e., conjugate.

I’d be happy to give $h$ a Gamma prior, because that’s conjugate to the Exponential distribution, which $π$ appears to have, up to a constant factor, but my experimental result, $x$, is binary-valued, not a count as expected by an Exponential random variable.

I’d also be happy to give $π$ a Beta prior, since that’s the conjugate prior to the Bernoulli distribution, but I can’t find a neat way to convert a Beta-distributed $π$ into any kind neat distribution on $h$.

I’d like everything to be conjugate to allow for quick updates for online updates (i.e., successive $x$ will come irregularly as a student learns a fact). If I must do MCMC, that’ll greatly expand the memory/storage requirements between updates.

  • $\begingroup$ Why do you need $\pi=2^{-t/h}$? A Beta-Bernoulli model followed by sampling from the posterior of $h$ (given $\pi$) seems much easier. $\endgroup$ – nth Apr 10 '17 at 1:40
  • $\begingroup$ @jth Thanks for the note. I need some way to capture the Bernoulli probability on the passage of time $t$, and in the application, exponential decay is the standard one. (This matters because if we model each fact a student is learning as exponentially evaporating from memory, this model has to tell us which fact is closest to being forgotten right now.) I wonder if there’s some way to normalize time with $h$ and use Beta-Bernoulli… I will see! $\endgroup$ – Ahmed Fasih Apr 10 '17 at 1:58
  • $\begingroup$ @jth I think I see what you mean. I can draw samples of $h$, convert to $π$ and fit a Beta distribution, update the posterior conjugatedly, convert the resulting $π$ back to $h$ and fit a Gamma to it again. I like it! $\endgroup$ – Ahmed Fasih Apr 10 '17 at 2:12

After some Monte Carlo simulation and analysis, I realized that the half-life $h$ in this context is totally a nuisance parameter and that I can deal solely with priors on $π$.

I wrote up the math to accompany a little (Python) library I wrote: https://fasiha.github.io/ebisu/ but in a nutshell, rather than a prior on $h$, I just put a Beta prior on $π$ at some time, and then propagate that distribution through time via the deterministic equation $π = 2^{-t/h}$ (which makes the new prior non-Beta), then apply the Bernoulli likelihood analytically, before fitting the final posterior to a new Beta distribution to function as the prior for the next round.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.