From what I have gathered so far, an AI has some prior (stored in the form of some probability distribution), and, based on experiences/data, changes the distribution (via Bayes rule) accordingly. This idea seems intuitively correct, as humans do something similar: we have some prejudice about certain things and refine it further based on additional observations.
I am wondering if there is a different (possibly, non-probabilistic) setting for designing an AI.
Yes, there is symbolic AI. This was the 'original' approach to AI, at a time when there was very little data and/or processing power available. The focus was on logic and calculus, not on machine learning, which was just in its infancy.
A lot of natural language processing was developed using grammar rules (which only later were learned from data).
There still is a lot of this around, but often it's now hybrid, where human-authored rules correct the systematic mistakes of statistical approaches.
Update: as an example the Teneo system for building conversational agents (aka chatbots) uses both pattern matching rules and machine learning for intent recognition. The (human-created) patterns are more precise, but sometimes lack breadth of coverage, which the ML part provides, which works as a fallback.
What you're looking for are Expert systems and Knowledge Based Systems. Really similar to each other, they encompass all systems built upon experts knowledge, from which analytic rules are derived in order to allow their implementation in computer programs.
A trivial example could be a self driving system based on proximity sensors, a camera and a set of predefined rules (basically a set of explicit if-else statements) like:
