4 ms·

In term 1, you say P(z_k | x_k) is a probability (which should be a scalar value between 0.0 and 1.0 inclusive). How did we get from the vector pdfs of measure

In term 1, you say P(z_k | x_k) is a probability (which should be a scalar value between 0.0 and 1.0 inclusive). How did we get from the vector pdfs of measurement and estimate to a scalar probability? Maybe you meant evaluating a pdf at an arbitrary point z_k, getting its density (scalar non-negative value, between 0.0 and infinity) at that point? I have the same confusion on your other enumerated terms. If they are probabilities, I'm only familiar with getting those by integrating the pdfs over some set of bounds or range. I've seen the k subscripts plenty before, and some people use + and - to indicate pre and post parts of an operation, others put conditional vertical bars in the subscripts, but your definition for the pdf of z_k is a bit different than I'm used to. If we focus on the current instant of time (dropping the subscripts), ignore the motion model part, and only deal with a single measurement, I would say the estimate and measurement are defined like this: estimate ~ Normal(mean=x, covar=P) measurement ~ Normal(mean=z, covar=R) Then there is an elegant formula looking at it as an information filter, using Matlab-ish notation: Y = inv(P) + H'*inv(R)*H y = inv(P)*x + H'*inv(R)*z update ~ Normal(mean=inv(Y)*y, covar=inv(Y)) Ignoring the information filter stuff, and getting back to your list, Term 1 looks like a pdf with the dimensionality of z, but the result in Term 4 is a pdf with the dimensionality of x. In otherwords, the resultant Term 4 is a pdf (function) which takes a vector argument in the x space, and I don't see where to pick any given z for the pdf in Term 1. It's getting late, and I'm getting spacey, but it's tempting to try and write the type signatures of these functions in some strongly typed pseudocode so I can show you my confusion.

abstrakraft 9y ago
Concerning the difference between our definitions of measurement, I was referring to the prior probability of making a measurement given the true state, which is P(z|x), or, if you prefer, f(z,x) = P(Z=z|X=x). This PDF is Normal(Hx, R). The distribution you refer to is that of the image of the true state in measurement space (Hx), which is Normal(z, R). Concerning the general question of dimensionality and numbers vs. functions, I'm afraid I still don't understand the fundamental issue. I agree that probability notation isn't the best, but each of those terms can be replaced with a Gaussian (or in the case of the denominator, an integral of the product of two Gaussians). You end up with a function in terms of x_k and the sequence of z's. You can then choose which variables are free or bound and evaluate the function as your application requires. If all variables are bound (the case that you know the measurement z_k, and are evaluating the posterior probability of the state at a given point), all the terms are numbers (yes, densities), and there are no dimensional issues. If you leave x_k free, you end up with two functions in the numerator, and a number in the denominator. Those two functions are Mahalanobis distances (which are scalars) in an exponent, and again, no dimensional issues. In general, such operations with PDFs represent various combinations and manipulations of event spaces that need not be commensurate. Computing the PDF of a coin flip given the distribution of all of the positions and velocities of the molecules of air through which the coin flies requires comparing a two-state PMF to a virtually infinite dimensional PDF. Notationally speaking, no problem.
- civility 9y ago
  > I agree that probability notation isn't the best [...] There is so much hidden context in mapping Bayes' theorem to Kalman filters that I still have to cringe. First we've got the sub-expression "p(x_k)" in the numerator, and you know that is a multivariate PDF for the state estimate, or possibly the density value of that PDF at x_k. However, we've got a very similar looking expression "p(z_k)" in the denominator, and in this case it means "an integral of the product of two PDFs". Those are wildly different substitutions for nearly the same syntax. That same confusion applies to p(x_k|z_k) and p(z_k|x_k). The first seems to indicate "the PDF of my estimate given this measurement", but the second really says something like "evaluate the PDF in the measurement space at the location which corresponds to my estimate". Essentially, p_Z(h(x_k)). If you don't already know the solution, this notation is not prescriptive for getting there.

abstrakraft 9y ago

Concerning the difference between our definitions of measurement, I was referring to the prior probability of making a measurement given the true state, which is P(z|x), or, if you prefer, f(z,x) = P(Z=z|X=x). This PDF is Normal(Hx, R). The distribution you refer to is that of the image of the true state in measurement space (Hx), which is Normal(z, R). Concerning the general question of dimensionality and numbers vs. functions, I'm afraid I still don't understand the fundamental issue. I agree that probability notation isn't the best, but each of those terms can be replaced with a Gaussian (or in the case of the denominator, an integral of the product of two Gaussians). You end up with a function in terms of x_k and the sequence of z's. You can then choose which variables are free or bound and evaluate the function as your application requires. If all variables are bound (the case that you know the measurement z_k, and are evaluating the posterior probability of the state at a given point), all the terms are numbers (yes, densities), and there are no dimensional issues. If you leave x_k free, you end up with two functions in the numerator, and a number in the denominator. Those two functions are Mahalanobis distances (which are scalars) in an exponent, and again, no dimensional issues. In general, such operations with PDFs represent various combinations and manipulations of event spaces that need not be commensurate. Computing the PDF of a coin flip given the distribution of all of the positions and velocities of the molecules of air through which the coin flies requires comparing a two-state PMF to a virtually infinite dimensional PDF. Notationally speaking, no problem.

civility 9y ago
> I agree that probability notation isn't the best [...] There is so much hidden context in mapping Bayes' theorem to Kalman filters that I still have to cringe. First we've got the sub-expression "p(x_k)" in the numerator, and you know that is a multivariate PDF for the state estimate, or possibly the density value of that PDF at x_k. However, we've got a very similar looking expression "p(z_k)" in the denominator, and in this case it means "an integral of the product of two PDFs". Those are wildly different substitutions for nearly the same syntax. That same confusion applies to p(x_k|z_k) and p(z_k|x_k). The first seems to indicate "the PDF of my estimate given this measurement", but the second really says something like "evaluate the PDF in the measurement space at the location which corresponds to my estimate". Essentially, p_Z(h(x_k)). If you don't already know the solution, this notation is not prescriptive for getting there.