Your probability changes with $t$ but as Michael said, you don't know how. linearly or not ? It looks like a model selection problem where your probablity $p$ :
$p=\Phi(g(t,\theta))$ may depend on a highly non linear $g(t,\theta)$ function. $\Phi$ is just a bounding function that guarantees between 0 and 1 probabilities.
A simple exploratory approach would be to try several probits for $\Phi$ with different non linear $g()$ and to perform a $g()$ model selection based on standard Information Criterias.