2
$\begingroup$

I asked a question about combining equally-likely observed PDFs: Combine three equally likely observed PDFs. I'm not sure how to combine them. I showed my best attempt so far. What more can I do?

Replying to the comment by @Glen_b over there: I do interpret densities to be the same as probabilities. I did explain how they arise—I spend individual observation sessions creating probability density functions of what I think the likelihood of $x$ being different values is and I consider each of these probability density observations to be equally likely.

| |
$\endgroup$
  • 3
    $\begingroup$ Did you see the comment under your question explaining why it was put on hold? It was posted 8 hours ago. $\endgroup$ – amoeba May 25 '17 at 14:45
  • $\begingroup$ yes. Here I respond to the comments. $\endgroup$ – kilojoules May 25 '17 at 14:55
  • 4
    $\begingroup$ IMHO it makes more sense to respond in the comments under your question on the main site. Here you ask why your Q was put on hold, and the answer to that is: see the comment explaining why. If you want to reply to the comment, please do it there and ping whoever posted that comment (in this case, Glen_b) by including @Glen_b into your comment. $\endgroup$ – amoeba May 25 '17 at 15:02
  • 2
    $\begingroup$ In addition to @amoeba's advice, you should incorporate the response addressing the close reason into your question via editing so that the current state of the question is clear, on-topic, etc. Then it may be reopenable. $\endgroup$ – gung - Reinstate Monica May 25 '17 at 15:25
  • $\begingroup$ @gung I think my question is clear and on topic. I don't understand why Glen_b didn't. I explain my problem, offer a reproducible example, and show what I tried so far. $\endgroup$ – kilojoules May 25 '17 at 15:37
  • 2
    $\begingroup$ Your question doesn't really make sense to me, but I'm not too sharp. I agree w/ @Glen_b's comment. Why not just average them? $\endgroup$ – gung - Reinstate Monica May 25 '17 at 16:02
  • $\begingroup$ @gung If I choose to average them, I have two options. I can divide the $x$ axis into small rectangles and find the average $f$ value for every $\Delta x$, or divide the $f$ axis into small rectangles and find the average $x$ for each $\Delta f$. I want to rigorously approach this problem, and it is not clear to me if one of these methods is better than the other, or if there is a better way. $\endgroup$ – kilojoules May 25 '17 at 16:05
  • 3
    $\begingroup$ The clarity that counts is in the eyes of potential answerers. Best thing you could do is explain the context and motivation in as simple terms as possible. $\endgroup$ – Scortchi - Reinstate Monica May 25 '17 at 16:08
  • $\begingroup$ FWIW, the question of how to average 3 densities by dividing "the x axis into small rectangles and find the average f value for every Δx, or divid[ing] the f axis into small rectangles and find the average x for each Δf" seems clear to me. $\endgroup$ – gung - Reinstate Monica May 25 '17 at 16:11
  • 1
    $\begingroup$ I voted to reopen. To be honest, I am not sure why it was closed, I think it was reasonably clear from the beginning; I suspect that everybody was confused by you not simply averaging the PDFs (which seems the most natural and the most reasonable thing to do; you just average three functions, there is only one way of doing it, not two). $\endgroup$ – amoeba May 25 '17 at 18:53
  • 1
    $\begingroup$ @Scortchi You should be able to see that in my example. I observe the same population three times by randomly sampling from it. This gives me three equally valid probability distributions. What about this doesn't make sense? $\endgroup$ – kilojoules May 25 '17 at 19:31
  • 1
    $\begingroup$ Another way to think about it is with a somewhat contrived example. Imagine I want to know the distribution of carrot sizes in my carrot farm. I take carrot samples in Fall, Summer, and Winter, only recording the probability density functions of carrot sizes found during each season. I want to combine these observed carrot size distributions to a distribution describing the probability of carrots I pick being different sizes in all seasons. Each observation of carrot size distributions used the same number of carrots, so they are each equally valid. $\endgroup$ – kilojoules May 25 '17 at 19:37
  • 1
    $\begingroup$ @kilojoules I should say that I appreciate that you took my initial comments seriously and made genuine attempts to improve the question (and indeed did improve it quite a bit), and also that you raised the issue here in a constructive manner. [I believe that there's what would ultimately be a very good question in there, and one that deserves good answers] $\endgroup$ – Glen_b May 25 '17 at 21:54
  • 1
    $\begingroup$ @kilojoule: I hadn't read the code - I rarely do read code unless it's already clear from the question text what it's supposed to do. (Trying to working out what code's supposed to do from what it in fact does is rather tiresome, especially if you're not familiar with the language; & assumes it does what it's supposed to do, which is often enough not the case.) I have had a look now you've trimmed it down & it seems you're just asking how to pool mean & standard deviation estimates from different samples for a common normal population/process. $\endgroup$ – Scortchi - Reinstate Monica May 26 '17 at 10:34
  • 1
    $\begingroup$ @Scortchi I don't want to be limited by gaussian distributions. I used gaussians in my example but I am looking for a general method to combine equally valid observed density functions. $\endgroup$ – kilojoules May 26 '17 at 15:24
7
$\begingroup$
  1. The general issue - Why questions that aren't really clear should be on hold:

    Questions that are not clear are supposed to be put on hold so that they can be made clear before people start posting answers that misunderstand the question, causing problems for the asker as well as later readers (who are the actual target of a good answer under the Stack Exchange model).

    Your feelings about how clear the question may be are not particularly relevant (except in so far as that makes it hard for you to see what to do). The central issue is how clear the average semi-knowledgeable reader is likely to find it. In the first place potential answerers need to clearly understand the question, but that alone isn't enough: by semi-knowledgeable readers I mean people who could read and benefit from an answer -- which is a much lower degree of understanding of statistics than required to make a good answer. If I (as a regular answerer with a reasonable level of understanding) find your question unclear -- and I don't seem to be alone, so it isn't just some odd quirk on my part -- the question would fall short of being clear enough.

  2. Specifics of what is unclear:

    I explained issues with the post in an extended comment under the post, but I think the main issues were:

    • the conflation of density with probability made it difficult to see what objects you were actually manipulating; it almost seems as if you were flipping between a set of probabilities and a density (specifying probabilities at values and then somehow making a density from that?)

    • the process by which you obtained and possibly manipulated these quantities was not sufficiently clear, making it hard to see how a model for that process might be constructed. Your comments here to gung indicate to me that this and the previous issue are still not clear enough. If you're taking information and then smoothing it before combining, why is that the best approach -- why not combine the original information instead ... and then smooth the result? If we don't know enough to answer that, we don't have a clear question. Is the quantity you're specifying probability for really discrete? If it is, don't make it into a density. If it's not discrete, what do the probabilities represent (what are they probabilities of?)

    • the part about convolution I initially found confusing because I couldn't see how it had anything to do with the problem -- if the previous issues had been clearer I think I'd have understood this earlier: I believe that's simply a misunderstanding that is best addressed in an answer. In effect it looks like you decided to try to solve a problem a particular way and asked about the problems you encountered with your attempted solution instead of asking how to solve your original problem. If I understand the situation correctly your solution is mistaken but we can address that as long as we have correctly understood what you have and what you want to get.

    • As a result I think the question would be much clearer if you removed almost all your information about your solution (apart from saying that using convolution was how you were thinking of solving it). I previously explained that I think all or almost almost all your code should be removed. Many of your plots (those about manipulating the convolution) add nothing to understanding your problem and I think they should be removed as well.

The question is definitely better than it was but some of my issues are not fully resolved.

In spite of these remaining misgivings I reopened on amoeba's request. If amoeba thinks it's answerable now I don't want to get in the way of that answer. I'd like it to made be clearer but an answer may indeed help clarify the question. However if an answer leaves the question as unclear as I still find it, I'll likely close it again -- I think there's a danger that not only might you end up with an answer that doesn't address the actual problem you have but one that may in turn confuse later readers with similar-sounding problems. Closure would at least flag that it needs to be treated with caution.

[I suspect the nub of an answer may be as simple as "Convolution is the wrong thing to do, you need to average the densities" (or probability functions or whatever these objects are), but even if that's right, details of the best way to do that would depend on information that I'm not sure is there yet.]

| |
$\endgroup$
  • $\begingroup$ +1 These are good points. $\endgroup$ – amoeba May 25 '17 at 21:34
  • 1
    $\begingroup$ What is the difference between probability and density? What other information can I provide? $\endgroup$ – kilojoules May 25 '17 at 21:58
  • 2
    $\begingroup$ The first question is answered multiple times on the main site. If you can't locate a good answer to it, comment again here and I'll find you one.. ... I'm not sure how much more explicit I can be about what I want but I'll attempt an edit of the answer above. $\endgroup$ – Glen_b May 25 '17 at 22:06
  • $\begingroup$ @kilojoules Actually, in case you don't spot the edits I'll post them here: Is the quantity you're specifying probability for really discrete? If it is, you probably shouldn't be making it into a density. If it's not discrete, what do the probabilities represent (what are they probabilities of?) $\endgroup$ – Glen_b May 25 '17 at 22:10
  • 1
    $\begingroup$ What's the ultimate use of what you get out? what's it to be used to do? I have misgivings about the entire process here. I imagine that if you address what you can address, it will be easier to explain more about what still needs to be clearer. $\endgroup$ – Glen_b May 25 '17 at 22:12
  • $\begingroup$ @Glen_b I am quantifying the model-form uncertainty of synthetic parameters in a reduced physics model. But, I think my question is more broad than that specific application. $\endgroup$ – kilojoules May 25 '17 at 22:13
  • 1
    $\begingroup$ I'm sure it is broader than your specific application; that's one reason why I am keen to make it a good question. However, there are a number of issues about the process by which the quantities you're asking about come into existence (the probabilities and then in turn the densities) that potentially affects suitable advice. We don't necessarily need the physics jargon but simple issues like understanding what the quantities are we're dealing with matter. For example I still don't understand the most basic issues like whether these quantities you're assigning probabilities to are discrete! $\endgroup$ – Glen_b May 25 '17 at 22:21
  • $\begingroup$ If that's anything but crystal clear (as well as several issues that follow form it), your question isn't close to being answerable. $\endgroup$ – Glen_b May 25 '17 at 22:22
  • 1
    $\begingroup$ @Glen_b nothing in my problem is discreet. My carrot lengths are continuously valued and can be any length. $\endgroup$ – kilojoules May 25 '17 at 22:42
  • 1
    $\begingroup$ @Glen_b I thought of another example that might represent my problem a little better. I notice a plume of smoke pass by my window once in a while. I trace it's path using binoculars and maps, and use available weather information to develop a probability distribution of where I think the smoke is coming from, taking uncertainty in my model inputs into account. Every time I observe the smoke, I develop a new probability distribution. After several observations, I want to know where I think the plume's origin is and where I think it is not, expressed as a probability function. $\endgroup$ – kilojoules May 25 '17 at 23:02
  • 1
    $\begingroup$ In both the carrot example and this new one, you jump over an essential part when you say "I develop a probability distribution". You're going from data (the object we should be dealing with) to an estimate (your fitted distribution) without any indication of what your data are like or how you got there. It's the data you should be working with, if at all possible. But if it's not possible it's best we understand how the distributions arise from the data because the actual best way to combine the information you're presenting as a density may not be via the density. $\endgroup$ – Glen_b May 25 '17 at 23:07
  • 1
    $\begingroup$ Even if you are doing the right things in going from data to density, your densities may not be equally precise at every value of $x$ (it would be astonishing to me if the uncertainty in the density estimate is a constant across the x-values), but we can't figure out what the relative uncertainties are if we don't know how they arise (how they relate to the data). Your analogies may be useful but they still need to convey the information we need to properly advise you. $\endgroup$ – Glen_b May 25 '17 at 23:17
  • $\begingroup$ @Glen_b Thanks very much for your patience and help. The probabilities are due to my uncertainty in modeling the turbulence. In this new example, I have a time-averaged differential equation that describes how the plume should behave, given initial conditions from the local weather service and a synthetic parameter, the "eddy viscosity." I chose a prior distribution of the synthetic parameter in the turbulence model and used bayesian calibration to find a distribution of the "eddy viscosity" for each observation, resulting in a probability distribution of where my model predicts the origin $\endgroup$ – kilojoules May 26 '17 at 5:31
  • $\begingroup$ That sounds fascinating ... is this distribution some function of parameters, or is it just a set of points with density values? $\endgroup$ – Glen_b May 26 '17 at 7:47

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .