Wikipedia:Reference desk/Archives/Mathematics/2015 April 14

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

April 14

Unknown Prior Distributions?

I'm not sure there is a simple answer (of course I'm open) but I'm really looking to be pointed in the right direction. I'm using an analogy because I also want to know about more general answers.

Let's say there are pictures, and each pixel is either red or blue in reality. I cannot know this, as I have no access to the manufacturing process, but I do know that they are only made in two varieties - 100% blue, or 99% blue 1% red. I can look at the pictures (I can only look at a picture once - looking at it again is not an independent observation), but my eyes are a little broken. Sometimes my eyes see red as blue and vice versa. However, the inconsistency in my eyes is based on something that is reasonably random, like the exact air speed around them. I can estimate this, or divide it into subcategories, but I can never be sure how wrong my eyes are. I could take a lot of measurements of the wind speed and write down what I see a lot, and try to come up with a formula, but it would be imperfect. When I need it most, I never know the probability that they switch the color.

I have tested my eyes a lot, and written down the results I saw for a lot of pictures. Windspeed ~3mph, see 0.5% red. Windspeed x, see y% red. But i have no idea what the actual amount of red in those paintings was, so I don't know even how to calculate the error exactly in those cases.

If I have a new picture, how can I guess which kind it is? How can I correct for a prior error distribution I do not know, and cannot measure well? Any advice about what this is called or where to start is helpful.

Thanks, B4dA1r (talk) 02:26, 14 April 2015 (UTC)

Here is a little more info that might be relevant. The red->blue and blue->red switching rates are not the same, and both fluctuate, and they are not connected. Also, the error rate of my eyes is pretty significant - 1-2%, hedging a guess, though again I cannot know or estimate without getting good test cases. Also the proportion these pictures are made in is not known or consistent. B4dA1r (talk) 02:39, 14 April 2015 (UTC)

@B4dA1r: I found it really hard to follow your query because it was not clear what probability distributions were known and what were unknown; what were the observations; what was the "vanilla" version of the problem and what were the variants etc. Perhaps if you can present it more plainly, even in the form of a toy math problem (as in x is a binary random variable with unknown prior probability distribution $p_{X}(x)$; y is a binary rv with known conditional probability distribution etc), more responses will be forthcoming. Abecedare (talk) 19:38, 14 April 2015 (UTC)

Seems to me you are asking several different questions at once. So I'll begin by saying the standard way to approach this general problem is to decide on a family of models, for example, parametrized by a few real variables; this model will give the conditional probabilities of various events. Your model will include a parameter for the probability for the picture to be of the all-blue variety, and a value for $\mathrm {Pr} ({\textrm {seencolor}}|{\textrm {realcolor}},{\textrm {windspeed}})$ as a function of seen color, real color, wind speed and the model parameters. If you manage to estimate the specific distribution from the family, you can calculate any probability you please.

As for estimating the distribution - this can be done via Bayesian inference, using the data and a prior. Here comes your titular question of settling on specific priors for the model parameters. In cases you don't have any additional prior knowledge to shape your prior distribution, you usually can't go very wrong with choosing either a maximum entropy prior (which unfortunately is not robust to reparameterization), or a Jeffreys prior (which is generally improper). With sufficient data, your prior won't matter too much.

Note though, that if there's too much symmetry, you might never be able to glean useful information. -- Meni Rosenfeld (talk) 21:48, 14 April 2015 (UTC)

sum of reciprocals of prime number of primes in factorization? diverge/converge

Does the some of the reciprocals of the numbers that have an prime number of primes in their factorization converge or diverge? So 1/4 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/14 + 1/15 + 1/18 + 1/20 ... Naraht (talk) 10:36, 14 April 2015 (UTC)

Not the some but the sum. The answer is of course yes. What did you do to figure it out yourself? Bo Jacoby (talk) 11:12, 14 April 2015 (UTC).

Too early in the morning, of course "sum". I presume the yes for converge/diverge, it has to do one or the other. I wanted to ask here. It doesn't contain a subset that I recognize as converging...Naraht (talk) 11:43, 14 April 2015 (UTC)

Follows from the Divergence of the sum of the reciprocals of the primes.John Z (talk) 19:03, 14 April 2015 (UTC)

Yes. To spell it out: $\sum _{p{\text{ prime }}}{\frac {1}{p}}$ diverges $\Rightarrow \sum _{p{\text{ prime }}}{\frac {1}{2p}}$ diverges. And the latter is a sub-sequence of the sequence of Naraht's interest. Abecedare (talk) 19:24, 14 April 2015 (UTC)

Thanx.Naraht (talk) 20:18, 14 April 2015 (UTC)