4
$\begingroup$

Suppose that data X have a Normal distribution with some mean $\mu$ and some variance $\sigma^2$. However, you don't get to see X. Instead, you see $Y = g(X)$ where $g$ is a known function. Assume that $g$ is complicated and is not invertible.

How do you write the likelihood function for $Y$? Or how do you use some numerical method to approximate a likelihood in a way that allows inference about $\mu$ or $\sigma^2$?

Seems like a basic question that has totally stumped me.

Improve this question
$\endgroup$
7
  • 2
    $\begingroup$ It depends on how smooth $g$ is and whether it is one-to-one. That landscape can be subdivided into cases where (a) $g$ is differentiable, (b) $g$ is continuous, and (c) $g$ is discontinuous, because the calculations of the likelihoods differ fundamentally among those cases. Any of our threads on transforming PDFs will apply to (a) when $g$ is one-to-one. $\endgroup$
    – whuber ♦
    Commented Jun 10, 2014 at 19:34
  • 1
    $\begingroup$ @whuber The question states that the function g is not invertible. This means that g is not one-to-one. $\endgroup$
    – tom_0
    Commented Jun 10, 2014 at 20:06
  • $\begingroup$ This can be very tricky because it can cause complex patterns of non-identifiability in the parameters $\mu$ and $\sigma$. A very simple example is $g(X)=X^2$, for which the sign of $\mu$ cannot be determined. Because of this it is rare to formulate models using non-invertible $g$. Do you have a particular $g$ in mind? (Thanks for clearing up your meaning of "non-invertible": to many readers, noting the context mentions "complicated" and "numerical methods," that would only have meant that computing $g^{-1}$ is impracticable.) $\endgroup$
    – whuber ♦
    Commented Jun 10, 2014 at 20:09
  • 2
    $\begingroup$ The question is inspired by observed data that come from a known differential equation with unknown parameters. Inference about the parameters is of interest. These unknown parameters can be given a known probability distribution with unspecified hyperparameters. Likelihood inference for the hyperparameters is of interest. Essentially, $Y = g(X)$, $X \sim N(\theta, \sigma^2)$, all you see is $Y$, find good estimates for the parameters $\theta$ and $\sigma^2$. Perhaps likelihood methods are not the best approach (but that seems to rule out Bayesian as well). $\endgroup$
    – tom_0
    Commented Jun 10, 2014 at 20:20
  • 1
    $\begingroup$ That's pretty cool--thanks for the explanation. $\endgroup$
    – whuber ♦
    Commented Jun 10, 2014 at 21:10

0

Reset to default

Your Answer

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