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Comment by antioedipus | Hacker News Reader

* You can think of it as a factor graph with linear residuals and Gaussian noise functions in factors that connect a chain of variables, with all but the most recent variable marginalized. It’s a well known fact that linear, Gaussian factors result in a closed-form expression that gives the optimal maximum a posteriori estimate. The Kalman filter exploits this very special case. You can also write a LQR down with a factor graph (as the parent commented, the KF and LQR are duals).

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That's the same as the first point in the parent comment's list: a factor graph is a visualisation of the conditional probability distribution. But yes it is very helpful to draw out the factor graph (or Bayesian graph) for the Kalman filter, probably more useful than just writing out the equations.

By the way (as if my original comment above isn't already nitpicky enough, this is even worse...):

It bugs me when people use the word "optimal" in the Gaussian / Bayesian formulation. As the top-level comment above says, if you assume the various prior and conditional distributions are Gaussian then the posterior distribution is Gaussian too. This is not optimal, it's exact, just like you wouldn't say x=2 is optimal solution to x+1=3.

It is the optimal solution in the quadratic optimisation formulation, as the top-level comment also correctly said.

I'm not a mathematician at all (mechanical engineer), but to me, "exact" sounds like "deterministic" as an opposition to stochastic.

I though optimal conveyed the idea of "literally the best possible solution but you're still in the presence of a fully random system here".

Which might be the wrong interpretation, but hopefully it explains why some people (who aren't necessarily familiar with rigorous mathematics) use optimal.

I do see your point. But if you're talking about a probability or probability distribution, it can still be an exact solution to a model. For example, if I throw two standard dice, what is the probability of throwing two sixes? The answer is 1/36. To me, it sounds odd to describe 1/36 as the "optimal" solution to that problem, even though it's stochastic. Even "exact" solution is a bit odd, I'll concede, but a lot less so. "The solution" or "the answer", with no more qualification needed, sounds best to me.

1/36 is indeed the most optimal estimator of the expected frequency of two sixes, it's not odd at all.

1/36 is an estimate, it is not an estimator at all. An estimator is a formula based on data from the rolls.

It's an estimator in this case. A fixed number is an estimator too, it's just not going to be desirable in most cases. But single numbers are absolutely and unquestionably also valid estimators.

In any case, what I'm trying to get at there is that in estimator theory there is a concept of optimality for an estimator over a distribution.

Sure, a single number is a trivial example of a procedure/formula.

But an estimator estimates an unknown parameter from data (or in such a trivial estimator possibly without data) - and I believe this is central to the confusion.

That's a known probability distribution. The optimal is about being the minimum variance unbiased estimator for the unknown probability distribution.

Oh yes, of course.

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