I prefer to think of it in terms of the statistical inversion problem. That is, we have an event(s) that occur, which we may imperfectly understand. We take noi

I prefer to think of it in terms of the statistical inversion problem. That is, we have an event(s) that occur, which we may imperfectly understand. We take noisy measurements of that event. Clearly, the causal relationship is the events cause the measurements - a bad measurement does not cause the event to move. But, in practice all we have are measurements, and from that we want to find an optimal (or good) estimate for what the events were. Hence, inversion. Bayes formula expresses P(x|y) in terms of P(Y|x), so you can perform the inversion using bayes. In many fields establishing the prior is difficult, hence frequentist methods are popular. There are many techniques for the statistical inversion problem. Trying to track a ballistic object in a vacuum? Fit the measurements to a second order polynomial (parabola) and you are done (well, you have to decide least squares vs robust methods, but it is not such a hard problem in the scheme of things). Trying to track a manuevering jet, stock prices, or disease incidence rates. Now your model of the problem is much less clear. We model lack of information as random variables. It isn't "random" in the deterministic sense, just in the sense that we don't know. Establish a good probabilistic description of that lack of knowledge in your prior, and you are probably going to get good result: this jet fighter is probabilisticly either moving straight, performing a coordinated turn, or performing an uncoordinated turn. Use a Markov chain to model those likelihoods (e.g.), and you may end up with good results. But if your modeling of the prior is poor, well, good luck to you, your output is probably nonsense.