2
$\begingroup$

I am reading the wikipedia article on ordered logit models. As I understand it, the model is specified by:

$\Pr(y \le k | \mathbf{x}) = \frac{1}{1 + e^{\mathbf{w} \cdot \mathbf{x} - \theta_k}}$

where $\theta_k$ are the thresholds $\theta_1 <\theta_2 < \dots < \theta_{K-1}$. (NB: The original wiki formula has $\theta_i$ appearing on the RHS. This does not seem meaningful to me so I changed it to $k$ instead.)

This implies that the probability that the observation $\mathbf{x}$ falls in the $k$-th class is

$\Pr(y = k | \mathbf{x}) = \Pr(y \le k | \mathbf{x}) - \Pr(y \le k - 1 | \mathbf{x})$.

The article then goes on to say that to fit the model you need to find the coefficients $\mathbf{w}$ and the thresholds $\theta_k$. So far so good, except it does not tell you how to implement a procedure to find these.

My question is, how do you determine the best set of coefficients and thresholds? Is it with respect to a maximum likelihood estimator for the predicted probability for the actual class of the response?

Note that I am not interested in packages/software that can fit the ordered logit model.

Improve this question
$\endgroup$
0

1 Answer 1

Reset to default
1
$\begingroup$

Yes, typically maximum likelihood estimation is used to find the coefficients and thresholds. Of course, there's no reason why a Bayesian approach wouldn't be a reasonable way to do it either.

Improve this answer
$\endgroup$
2
  • $\begingroup$ thanks, are you 1) able to elaborate a bit more on the Bayesian approach? 2) recommend me a reference on implementing the maximum likelihood estimator?. Also, was I correct in replacing the $i$ in the wikipedia formula with $k$? $\endgroup$
    – Alex
    Commented Jan 11, 2016 at 5:57
  • $\begingroup$ @Alex: honestly, it's a fairly straight forward problem so it's a little hard to know where to start, other than standard methods for glm's. $\endgroup$
    – Cliff AB
    Commented Jan 11, 2016 at 13:45

Your Answer

By clicking “Post 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.