1
$\begingroup$

Consider random variables $X$ and $Y$. $Z = f_a(X, Y)$ where $f(\cdot, \cdot)$ is a deterministic, not random function $f_a: \mathbb{R}^2 \to \mathbb{R}$ depending on a deterministic real parameter $a$. For example, let $f_a(X,Y)=aX+(1-a)Y$.

I've read a lot about Bayesian modelling and fitting with the help of pymc3 Python module. But how can I model $Z$ in a frequentist's manner? I am to calculate quantiles, optimize $a$ over setting $\mathbb{E}Z$ to $0$ etc.

What is the most convinient way to do this? I haven't found solution neither in scipy.optimize nor scipy.stats.

$\endgroup$
9
  • $\begingroup$ Would you specify what the deterministic function is, please? It might give the question a broader appeal if more of us understand it. $\endgroup$
    – Carl
    Commented Sep 13, 2016 at 16:17
  • $\begingroup$ actually, any function. I try to find a general approach. Let, e.g. $f_a(X, Y) = aX + (1-a) Y$. $\endgroup$ Commented Sep 13, 2016 at 16:19
  • $\begingroup$ OK, in that case the Bayesian priors would likely be independent for $X$ and $Y$ because there is no $XY$ covariance term and iff $0<a<1$, $aX+(1-a)Y$ is a mixture distribution, for which two independent processes are adding to the density. In other words, there are two distributions with no convolution chaining or cross-talk that are mixed with superpositioning to make final density function. With a different form, for example with an $XY$ term, that would not necessarily be the case, such that I am guessing that it may be necessary to divide up the problem into special cases. $\endgroup$
    – Carl
    Commented Sep 13, 2016 at 16:47
  • $\begingroup$ @Carl my question is mainly about Python modeling. I don't need to fit my model, I need to optimize distribution parameter $a$ $\endgroup$ Commented Sep 13, 2016 at 18:07
  • 1
    $\begingroup$ @Carl could you write it in detail? $\endgroup$ Commented Sep 13, 2016 at 18:30

1 Answer 1

1
$\begingroup$

OK, in that case the Bayesian priors would likely be independent for 𝑋 and π‘Œ because there is no π‘‹π‘Œ covariance term and iff 0<π‘Ž<1, π‘Žπ‘‹+(1βˆ’π‘Ž)π‘Œ is a mixture distribution, for which two independent processes are adding to the density. In other words, there are two distributions with no convolution chaining or cross-talk that are mixed with superpositioning to make final density function. With a different form, for example with an π‘‹π‘Œ term, that would not necessarily be the case, such that I am guessing that it may be necessary to divide up the problem into special cases.

I would use Tikhonov regularization error propagation of π‘Ž adaptively minimized for the error, or relative error of π‘Ž.

$\endgroup$

Your Answer

By clicking β€œPost Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.