Re: [julia-users] Deducing probability density functions from model equations
There's tons of WIP-code that implements arithmetic on Distributions. No one has the time to finish that code or volunteer to maintain it, so it's just sitting in limbo. OP: What you're asking for sounds like it's largely equivalent to probabilistic programming. There are a ton of ways you could implement it, but it's almost surely a mistake to implement this sort of thing on your own. I would encourage you to learn Stan (which you can call from Julia), which is a formal language for specifying joint probability distributions. A Stan encoding of your probability model will allow you to draw samples from arbitrary conditional distributions by conditioning on observed data. On Tuesday, July 21, 2015 at 6:43:05 AM UTC-7, Tamas Papp wrote: In general, f(x,theta) does not necessarily belong to any frequently used distribution family, even if theta does -- it is easy to come up with examples. Some distribution families are closed under certain operations (eg addition, and multiplication by scalars for the normal), and some distributions arise as transformations of other distributions (eg chi square from normal, etc). AFAIK Distributions.jl does not support arithmetic on distributions, but you can always try programming the relevant generic functions, eg + for (Real, Distributions.Normal) if the results of your transformations are distributions in Distribution.jl. Best, Tamas On Tue, Jul 21 2015, amik...@gmail.com wrote: Dear all, I don't know if that's the best place to ask such a question but I'll give it a try: I need to code stochastic models: xnplus1 = f(xn, theta) where xn is the state of my system at time n and theta is a set of parameters for this model, constant through time, and possibly containing noise. As a matter of example, let's consider the following model: function transition(n, xn, theta) e = rand(Normal(theta.mu, theta.sigma)) xnplus1 = xn + e return xnplus1 end My goal is to write such equations and to deduce automatically the transition probability density function: p(xnplus1 | xn, theta). I intended to parse the code of the model and look for the rand keyword, the name of the law used to generate this random variable and the values given to it. In the previous case, I could deduce that: p(xnplus1 | xn, theta) = normal_pdf(xnplus1, xn + theta.mu, theta.sigma) where normal_pdf(x, mu, sigma) = 1 / (sigma * sqrt(2 * pi)) * exp( - 1 / 2 * (x - mu) ^ 2 / sigma ^ 2) (rather) easily I think by specifying that whenever a constant is added to a normal variable, the mean of the new variable (left value in the equation) is normal and of mean increased by this constant and of the same standard deviation. But that's the simplest case of all and it requires me to specify some rules about the normal distribution. I therefore have the following questions: - is what I'm trying to do understandable? - is it doable at all? and in Julia? - is specifying rules for each distribution (normal, uniform, Poisson; correlated, uncorrelated) the way to go or should I think of something even more generic? - do you have any other suggestions to solve this problem? Thank you very much,
[julia-users] Deducing probability density functions from model equations
Dear all, I don't know if that's the best place to ask such a question but I'll give it a try: I need to code stochastic models: xnplus1 = f(xn, theta) where xn is the state of my system at time n and theta is a set of parameters for this model, constant through time, and possibly containing noise. As a matter of example, let's consider the following model: function transition(n, xn, theta) e = rand(Normal(theta.mu, theta.sigma)) xnplus1 = xn + e return xnplus1 end My goal is to write such equations and to deduce automatically the transition probability density function: p(xnplus1 | xn, theta). I intended to parse the code of the model and look for the rand keyword, the name of the law used to generate this random variable and the values given to it. In the previous case, I could deduce that: p(xnplus1 | xn, theta) = normal_pdf(xnplus1, xn + theta.mu, theta.sigma) where normal_pdf(x, mu, sigma) = 1 / (sigma * sqrt(2 * pi)) * exp( - 1 / 2 * (x - mu) ^ 2 / sigma ^ 2) (rather) easily I think by specifying that whenever a constant is added to a normal variable, the mean of the new variable (left value in the equation) is normal and of mean increased by this constant and of the same standard deviation. But that's the simplest case of all and it requires me to specify some rules about the normal distribution. I therefore have the following questions: - is what I'm trying to do understandable? - is it doable at all? and in Julia? - is specifying rules for each distribution (normal, uniform, Poisson; correlated, uncorrelated) the way to go or should I think of something even more generic? - do you have any other suggestions to solve this problem? Thank you very much,
Re: [julia-users] Deducing probability density functions from model equations
In general, f(x,theta) does not necessarily belong to any frequently used distribution family, even if theta does -- it is easy to come up with examples. Some distribution families are closed under certain operations (eg addition, and multiplication by scalars for the normal), and some distributions arise as transformations of other distributions (eg chi square from normal, etc). AFAIK Distributions.jl does not support arithmetic on distributions, but you can always try programming the relevant generic functions, eg + for (Real, Distributions.Normal) if the results of your transformations are distributions in Distribution.jl. Best, Tamas On Tue, Jul 21 2015, amik...@gmail.com wrote: Dear all, I don't know if that's the best place to ask such a question but I'll give it a try: I need to code stochastic models: xnplus1 = f(xn, theta) where xn is the state of my system at time n and theta is a set of parameters for this model, constant through time, and possibly containing noise. As a matter of example, let's consider the following model: function transition(n, xn, theta) e = rand(Normal(theta.mu, theta.sigma)) xnplus1 = xn + e return xnplus1 end My goal is to write such equations and to deduce automatically the transition probability density function: p(xnplus1 | xn, theta). I intended to parse the code of the model and look for the rand keyword, the name of the law used to generate this random variable and the values given to it. In the previous case, I could deduce that: p(xnplus1 | xn, theta) = normal_pdf(xnplus1, xn + theta.mu, theta.sigma) where normal_pdf(x, mu, sigma) = 1 / (sigma * sqrt(2 * pi)) * exp( - 1 / 2 * (x - mu) ^ 2 / sigma ^ 2) (rather) easily I think by specifying that whenever a constant is added to a normal variable, the mean of the new variable (left value in the equation) is normal and of mean increased by this constant and of the same standard deviation. But that's the simplest case of all and it requires me to specify some rules about the normal distribution. I therefore have the following questions: - is what I'm trying to do understandable? - is it doable at all? and in Julia? - is specifying rules for each distribution (normal, uniform, Poisson; correlated, uncorrelated) the way to go or should I think of something even more generic? - do you have any other suggestions to solve this problem? Thank you very much,
