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Are there any accepted methods of interpolating between two discrete probability distributions that vary in time?

Say we have some data at time $t=0$ with pmf $p_1$ and some data at time $t=10$ with another with some other pmf $p_{10}$. Are there any methods to interpolate between two two time-varying data and construct some $p_{5'}$ that would be an interpolated pmf at $t=5$?

Thanks!

Stephan Kolassa
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mightykitten
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    Do you have particular probability distributions you want to interpolate/a particular data set? The usual way of doing this is to introduce a regression-type term into the model, but the details of this might depend on the distributions you are fitting – jcken May 10 '21 at 07:08
  • I'm considering data that arise from simulations so there's no prior knowledge of what distribution the data might take. In essence, I was trying to see if there was a probabilistic method of interpolating time-varying data other than linearly interpolating across spatial locations – mightykitten May 10 '21 at 19:05
  • Quantile kriging maybe? https://www.jstor.org/stable/24587033?seq=7#metadata_info_tab_contents – jcken May 11 '21 at 07:52
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    @jcken Kriging over one dimension is usually not useful, especially for time series. Since abstractly this is a standard interpolation problem -- a distribution with $d$ supports is a point in the $d-1$ simplex in $\mathbb R^d$ and a varying distribution is just a curve in that space -- you can apply virtually any interpolation method you like. As a simple (yet very flexible) example, you could apply the ILR, interpolate that (linearly or otherwise), and invert the transformation. – whuber Dec 29 '23 at 14:29

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In a simple case with discrete distributions coming from the same family, e.g. two Poisson distributions with parameters $\lambda_0$ and $\lambda_{10}$ or two negative binomial distributions with parameter pairs $(\mu_0,o_0)$ and $(\mu_{10},o_{10})$, you could simply linearly interpolate between the parameters and use the same distribution family.

If you want to interpolate between distributions from different families, or perhaps between distributions that are only known based on their PMFs (perhaps they arise from simulations), you can interpolate between the probability masses.

Specifically, assume that we have PMFs $(p_{i,t})_{i\geq 0}$ for two times $t_0$ and $t_1$. That they are PMFs means that $\sum_{i=0}^\infty p_{i,t}=1$ for $t\in\{t_0,t_1\}$. Now, define a new set of probability masses by linear interpolation for each separate $i$:

$$ \hat p_{i,t} := \frac{t-t_0}{t_1-t_0}(p_{i,t_1}-p_{i,t_0}) + p_{i,t_0}. $$

Then $\hat p_{i,t_0}=p_{t,t_0}$ and $\hat p_{i,t_1}=p_{i,t_1}$ for all $i$, so $\hat p$ indeed interpolates between the two original PMFs. And $\sum_{i=0}^\infty\hat p_{i,t}=1$ for all $t$, since the summation commutes with the linear scaling, so $\hat p$ is a true PMF.

Note that the second approach (linearly interpolating probability masses separately) will not give the same result as the first one (linearly interpolating distribution parameters) in cases where both make sense.

Stephan Kolassa
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