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Some has given me a positive integer (5). It’s actually a forecast of the mean number of hurricanes for this summer, so we can think of it as an estimate of the mean of a Poisson distribution. But I know they rounded it, so I know the actual number was [4.5,5.5). So I’m going to sample randomly from that interval, by way of incorporating the uncertainty that they removed by rounding it. But with what distribution should I sample? One might say uniform, but that doesn’t seem right. For instance, if someone gives me 1 it’s surely more likely to have come from below 1 than above 1. Perhaps I should use a log-scale somehow. That’s often the answer to these things. I vaguely remember something about how more numbers have 1 at the start than any other digit, which seems like a similar thing. Any thoughts anyone? Thanks

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    $\begingroup$ There's not enough information. If you could provide details of how the forecast was made, using what data, then perhaps something definite could be said. $\endgroup$
    – whuber
    Mar 7, 2023 at 16:31
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    $\begingroup$ I don't see how this is valuable given the information you've provided, but you can sample n times from a truncated Poisson distribution (on [x-0.5,x+0.5]) for each of the integers x which appear in your sample n times. $\endgroup$ Mar 7, 2023 at 16:35
  • $\begingroup$ @whuber: there isn't any other information (the forecasts come from a giant numerical model). Ideally I'd go back and get the unrounded forecast, but that's not feasible. Given the lack of information, it's a bit like the question of trying to determine an uninformative prior in Bayesian statistics. $\endgroup$ Mar 7, 2023 at 17:08
  • $\begingroup$ @Floyd Everest: an argument for why it's valuable might be: what if what really matters is the probability that there will be 10 hurricanes. That probability will be a bit higher if I include this uncertainty around the 5 than if not. But in general it's just about trying to not eliminate uncertainty from the forecast. I'm not sure sampling from the Poisson is right, because this is the mean of a Poisson, not a random variable from a Poisson. $\endgroup$ Mar 7, 2023 at 17:12
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    $\begingroup$ Anything you do will be just a more or less indirect way of imposing unfounded assumptions on the results. But since you appear to be looking at conducting a form of sensitivity analysis, why adopt any distributional assumption? Just explore the consequences of varying the prediction between 4.5 and 5.5. $\endgroup$
    – whuber
    Mar 7, 2023 at 17:46

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