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* 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 r
by antioedipus 4y ago
* 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).
- alex_sf 4y agoOh yes, of course.
- quietbritishjim 4y agoThat'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.
- quietbritishjim 4y agoBy 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.
- auxym 4y agoI'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.
- quietbritishjim 4y agoI 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.
- sudosysgen 4y ago1/36 is indeed the most optimal estimator of the expected frequency of two sixes, it's not odd at all.
- jmalicki 4y ago1/36 is an estimate, it is not an estimator at all. An estimator is a formula based on data from the rolls.
- sudosysgen 4y agoIt'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.
- jmalicki 4y agoSure, 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.
- jmalicki 4y ago