I've been learning Bayesian statistical analysis on my spare time using textbooks, videos on YT, etc. I'm slowly going up that mountain. Please correct me if my wording below is poor or ask for clarification if it doesn't make sense.
I understand that conceptually you can use an analytically closed form if you "know" (somehow) that you have conjugacy - that is, when the posterior distribution is in the same family as the prior distribution. If we do not know, or if we know that the prior and posterior have different families, we would need to rely on MCMC techniques.
I'm a practitioner though, so this conceptual explanation is unhelpful to me. Concretely, how does one evaluate whether we have conjugacy or not? Or does a practitioner just avoid making that decision nowadays, and ends up routinely using MCMC techniques regardless?