Testing Exchangeability

Abstract

The topic of this paper is testing exchangeability using e-values in the batch mode, with the Markov model as alternative. The null hypothesis of exchangeability is formalized as a Kolmogorov-type compression model, and the Bayes mixture of the Markov model w.r. to the uniform prior is taken as simple alternative hypothesis. Using e-values instead of p-values leads to a computationally efficient testing procedure. In the appendixes I explain connections with the algorithmic theory of randomness and with the traditional theory of testing statistical hypotheses. In the standard statistical terminology, this paper proposes a new permutation test. This test can also be interpreted as a poor man's version of Kolmogorov's deficiency of randomness.The version of this paper at http://alrw.net (Working Paper 38) is updated most often.

COVID‐19: A multiwave SIR‐based model for learning waves

“…In our experiments we have found s = 2.1 is a reasonable value for this parameter allowing the model to be sensitive to wave changes without overflagging. We use a threshold test in order to determine when M t has deviated significantly far away from M 0 (Ho, 2005;Vovk et al, 2003). Note that to ensure that the martingale is not too sensitive with respect to initial estimates of the parameters, whenever M t drops below 1, we reset the strangeness scores and martingale.…”

“…We use a threshold test in order to determine when $${M}_{t}$$ has deviated significantly far away from $${M}_{0}$$ (Ho, 2005; Vovk et al., 2003). Note that to ensure that the martingale is not too sensitive with respect to initial estimates of the parameters, whenever $${M}_{t}$$ drops below 1, we reset the strangeness scores and martingale.…”

“…We expect the error to go up because the infection rate has changed and we will be using the infection rate from wave $$n$$ to model what is happening in wave $$n+1$$. Thus, we can test whether a new wave has started by checking exchangeability, which consists largely of two steps (Vovk et al., 2003). First, we calculate a p ‐value measuring the proportion of previous instances that had errors equal to or bigger than the current data point.…”

“…Let $${p}_{t}$$ be defined as the following: $$$$\begin{equation} {p}_{t}=\frac{{\sum}_{u={T}_{n-1}}^{t}{\mathbb{1}}_{{Z}_{u}\ge {Z}_{t}}}{t-{T}_{n-1}}. \end{equation}$$$$As long as the underlying error remains exchangeable, then $${p}_{t}$$ is distributed uniformly in [0,1] (Vovk et al., 2003). However, when the next wave hits, we expect $${p}_{t}$$ to drop closer to 0 for a while, on account of larger estimation error.…”

“…Let be defined as the following: As long as the underlying error remains exchangeable, then is distributed uniformly in [0,1] (Vovk et al., 2003). However, when the next wave hits, we expect to drop closer to 0 for a while, on account of larger estimation error.…”