Let $\textbf{X} = (X_1, \ldots, X_n)$ be a vector of responses, where $X_i = (p_1, \ldots, p_k)$ is itself a vector of probabilities.
What method does one use to analyze such data? I want the logic/steps/ideas/concepts behind the method outlined in a short summary, so that I know what's going on before I study it in depth.
To be more specific: I am particularly interested in how one would go about developing the analogue of a linear model: we have some covariates for each response, and we may be interested in whether the mean of the probabilities is influenced by some linear function of the covariates, and then we want to find these parameters, test them, do confidence intervals, and predictions.
What's the "general" idea here? One way could be to just straight-up assume that $X_i$ follows a Dirichlet distribution, but I am more interested in the standard "log-ratio analysis" approach, and what the underlying plan is.