Timeline for Approximate Posterior Predictive Quantiles with Numerical Methods
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 1, 2022 at 17:52 | vote | accept | Closed Limelike Curves | ||
| Jan 1, 2022 at 17:52 | vote | accept | Closed Limelike Curves | ||
| Jan 1, 2022 at 17:52 | |||||
| Dec 31, 2021 at 23:17 | answer | added | jbowman | timeline score: 1 | |
| Dec 31, 2021 at 12:00 | history | tweeted | twitter.com/StackStats/status/1476885879025188877 | ||
| Dec 31, 2021 at 4:03 | comment | added | Closed Limelike Curves | @jbowman That makes sense, thanks! I was a bit confused at first since I expect the bulk of the processing time to be likelihood evaluations, and I thought you were saying those could be precalculated. | |
| Dec 30, 2021 at 16:48 | comment | added | jbowman | I'll write this up as an answer later today, but the general idea is that you can integrate the GPD with respect to the bivariate Normal posterior using Gaussian quadrature, which can be rewritten to be a weighted sum of GPD distributions. The pre-calculation involves selecting the points from the bivariate Normal and calculating the density values at those points, so you basically have a pre-selected $n \times 2$ matrix of (shape, scale) pairs and a pre-computed vector of length $n$ of weights, $n$ being chosen to hit an accuracy target. | |
| Dec 30, 2021 at 16:42 | comment | added | Closed Limelike Curves | @jbowman Gaussian quadrature sounds like a great idea, thanks! What do you suggest pre-computing? I’m not 100% sure what you mean by that. | |
| Dec 30, 2021 at 2:26 | comment | added | jbowman | I should think a 2-dimensional Gaussian quadrature, e.g., jaeckel.org/ANoteOnMultivariateGaussHermiteQuadrature.pdf would work quite well. You could pre-compute almost everything, and the final calculation of the posterior predictive dist'n value at $x$ would just be a weighted average of some number, say 50 or so, generalized Pareto distribution values at $x$. | |
| Dec 29, 2021 at 22:52 | comment | added | user225256 | I think this is standard, but I have no reference for it. If you want a generalized Pareto distribution instead, then you can choose the parameters which make that cdf fit the data best, and the quantile function will be the inverse of that cdf. | |
| Dec 29, 2021 at 22:21 | comment | added | Closed Limelike Curves | @MattF. interesting -- does the CDF have to be Gaussian? The Generalized Pareto is very far from a Gaussian, but I could probably start by using the quantile function for a generalized Pareto distribution where the posterior mode is the plug-in estimator, then apply the correction you suggested. Do you have a source/paper for your method? | |
| Dec 29, 2021 at 22:13 | comment | added | Closed Limelike Curves | @Tim The model is very simple -- it's just X ~ GeneralizedPareto(σ, ξ) with unknown σ, ξ. I'm using a Jeffreys prior. There's no analytic solution for the normalizing constant or the posterior predictive, but the posterior itself is easy to approximate using Laplace. I want to calculate the quantile function of X, though. | |
| Dec 29, 2021 at 22:08 | comment | added | Closed Limelike Curves | @jbowman my posterior has 2 parameters, the scale and shape; the location parameter is known to be 0. | |
| Dec 29, 2021 at 20:54 | comment | added | user225256 | If you have $N$ ordered samples $x_i$, you can use a transformation with the Gaussian cdf $\Phi$: Fit the function $f(p)$ to the datapoints $(\frac{i}{N+1}, \Phi(x_i))$, and then approximate the quantile function $Q(p)$ as $\Phi^{-1}(f(p))$. For instance, you might fit to functions of the form $f(p)=p+ap(1-p)$ with $-1<a<1$, since any such $f$ leads to a quantile function with $Q(0)=-\infty$, $Q(1)=+\infty$, and $Q(p)<Q(p')$ when $p<p'$. | |
| Dec 29, 2021 at 20:11 | comment | added | Tim | Could you describe the actual model you are using? The generic answer would be to sample, but there may be problem specific solutions. | |
| Dec 29, 2021 at 19:05 | comment | added | jbowman | The generalized Pareto distribution has three parameters, but your posterior at first glance would appear to only be on one of them (otherwise you'd have more than two parameters.) Is this correct? If so, how are you dealing with the other two parameters, and which ones are which? | |
| S Dec 29, 2021 at 16:36 | review | First questions | |||
| Dec 29, 2021 at 16:50 | |||||
| S Dec 29, 2021 at 16:36 | history | asked | Closed Limelike Curves | CC BY-SA 4.0 |