Can someone please provide a definition of the ordered probit model so that I can calculate the value of $\Pr(y \le k | \mathbf{x})$, i.e. probability of observation falling in the $k$-th class or worse given the observations $x$.
I am trying to follow the definition of the ordered logit in this wikipedia article, which I have reproduced here.
However, in the probit case, I cannot find out what the function $\sigma$ should be even after following the links in the article.
In this article $\sigma$ is any function which maps an individual's score, which can typically be any number along $(- \infty, \infty)$, to the interval $(0, 1)$. The most commonly-used function is the sigmoid (the inverse logit) which looks like
$$ \frac{e^{\beta^T x}}{e^{\beta^T x} + 1} $$
and this is mentioned in the article. The function $\sigma$ will usually be a distribution function (that is, it will express the probability that some random variable is less than or equal to the argument of the function) because these have the properties of mapping to the unit interval while being monotone increasing, the two main characteristics we want inverse link functions to have. In the probit case $\sigma(x) = \Phi(x)$ where $\Phi$ is the standard normal distribution function. That is, if $Z \sim$ normal$(0, 1)$ then $\Phi(x) = P(Z \leq x) = \int_{-\infty}^{x} e^{-t^2 / 2} / \sqrt{2 \pi} \, dt$.
