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Let be $y_i$ some observed values of a given time-series. We denote $\hat{y}_i$ the corresponding predicted values. We also assume to have a prediction interval $p_i$ such that $\hat{y}_i \in p_i = [\hat{y}_i^l , \hat{y}_i^u]$ with $1-\alpha = 0.95$ average coverage.

I am wondering whether one can extract the probability that $y_{i+1}$ is higher than $y_i$ like that : $$ P(y_i \leq y_{i+1} | \alpha) = \frac{\hat{y}_{i+1} - \hat{y}_{i+1}^l}{\hat{y}_{i+1}^u - \hat{y}_{i+1}^l} $$

Can someone explain whether that idea makes sense or not ? I would really appreciate any help, adding some sources would be great :) Many thanks :)

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