As a scientist, what should I do when I encounter a seemingly fundamental problem that also seems strangely unfamiliar? Is it unfamiliar because I am up to something really new, or am I re-discovering something that has been around for centuries, and I have just missed it?

This is a short story about an exploration that began with such a problem, and led to this manuscript. It began one day when I was pondering the idea of probability:

Scientists often estimate probability (P) of an event, where the event can be a disease transmission, species extinction, volcano eruption, particle detection, you name it. If the P is

, there has to be someestimatedabout the estimate -- every estimate is imperfect, otherwise it is not an estimate. I felt that the uncertainty about P has to be bounded: I couldn't imagine a high estimate of P, say P=0.99, associated with high uncertainty about it. The reason is that an increased uncertainty about P means broader distribution of probability density of P, which means that the mean (expected value) of the probability density shifts towards P=0.5. Inversely, as P approaches 0 or 1, the maximum possible uncertainty about P must decrease. Is there a way to calculate the bounds of uncertainty exactly? Have anybody already calculated this?uncertainty

First, I did some research. Wikipedia: nothing. I hit the books: Jaynes's Probability theory, Johnson *et al*'s Continuous univariate distributions, statistical textbooks, all the usual suspects, nothing. Web of Science: A bunch of seemingly related papers, but no exact hit. ArXiv: More seemingly related semi-legible papers by physicists and statisticians, but not exactly what I was looking for. After some time I put the search on hold.

Then one day, for a completely different reason, I was skimming through John Harte's Maximum Entropy Theory and Ecology, and it struck me. Maybe there is a MaxEnt function that can give the maximum uncertainty about P, given that we only have the value of P and nothing else. Back to the web and, finally, I found something useful by UConn mathematician Keith Conrad: a MaxEnt probability density function on a bounded interval with known mean.

I adjusted the function for my problem, and I started to implement it in R, drafting some documentation on side, and I realized that I have actually started working on a paper!

Then the doubts came. I discussed the problem with my colleagues, but I hadn't learnt much -- many well-intentioned suggestions but no advice hit the core of the problem, which is: Is the idea any good? And anyway, as a wee ecologists with little formal mathematical education, can I actually dare to enter the realm of probability theory where the demi-gods have spoken pure math for centuries?

Then I took the courage and sent my draft to two sharp minds that I admire for the depth of their formal thinking (and for their long beards): Arnost Sizling the mathematician, and Bob O'Hara the biostatistician. For a week nothing happened, then I received two quite different and stimulating opinions, a semi-sceptical review from Arnost, and a semi-optimistic email from Bob. I tried to incorporate their comments, but I am still unsure about the whole thing.

I don't have a clue whether and where such thing should be published. It feels too long and complex for a blog post, yet somehow trivial for a full-length research paper; it is perhaps too simple for a statistical journal, yet too technical for a biological journal.

And so I've decided to pre-print it on **PeerJ** for everybody to see. I guess that it could be the perfect place for it. If it is really new then it is citable and I can claim my primacy. If it is all bullshit, then people will tell me (hopefully), or they will perhaps collaborate and help me to improve the paper, becoming co-authors. It is an experiment, and I am curious to see what happens next.

The usual way that a frequentist reasoner would go when faced with this is to reason from a repeated sample of events (and obtain the one-sample confidence interval for a proportion), or reason from a model like a logistic regression if the probability depends on covariates, again obtaining a confidence interval for the probability, conditional on values of the covariates.

A Bayesian would (or ought to) obtain a posterior distribution for the probability, and then summarize it in some way (eg. a credible interval) that depicts the uncertainty as well as the best estimate.

I didn't study the details of your paper, but I didn't immediately see how it adds to what I describe above. I am happy to be enlightened, however!

Hi Ken,

My study is NOT about how to estimate the uncertainty (using either confidence or credible intervals). That is easy and well known.

My study is about how to get the maximum possible value of uncertainty for a given P. In other words: I have found that you cannot have very high or very low (say 0.9 or 0.1) estimates of P, and at the same have high uncertainty (span of the confidence or credible intervals) about the estimates. If your mean P=0.9, then you inevitably must have very narrow posterior -- there is no way to have broad posterior (i.e. high uncertainty), and high expected value (mean of the posterior of P) at the same time.

Thanks for the comment though -- it shows that my main message may not be entirely clear.

Love the idea of this Petr. I don't have anything to contribute though I do have a gut feeling what you are trying to achieve is a good thing - I just can't help also feeling (form my A-level stats) that this already exists, somewhere...!

I think what happens when you get close to 0% or 100% and you have high uncertainty, the distribution is no longer Gaussian. Instead you get a different distribution, for example Poisson. The distribution is lopsided and this keeps it in the 0% to 100% region.

Look at a graph called 'probability paper'.

Also you can take the log of the measurement on the 0% side to move the graph away from zero. Then you get a realistic result: 99% -> 90% to 99.9%.

Let's take some examples: The probability of drawing an ace from a deck of cards is 4/52 and there is no uncertainty about that probability.

Now what's the probability that another tsunami will hit Fukushima within the next 5 years? Suppose that the estimate is 1/10 000 000 000. The heart of the problem is: "How in the world do we get such an estimate?"

Apart from the trivial theoretical bounds (i.e. a probability is a real number between 0 and 1) you will have to get into the nitty-gritty of your model to refine your predictions, with an optimistic scenario that will give you a lower bound and a pessimistic scenario that will give you an upper bound. There is no universal technique (similar to bootstrap for samples) to extract information for such non-frequentist events that will give you a universal confidence interval.

PS: well, did I say that a probability was always in [0;1] ? Apparently there is more to the story, according to Feynman ðŸ™‚

http://cds.cern.ch/record/154856/files/pre-27827.pdf

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