Added the likelihood equations/form from the discrete bayes
chapter to better tie in that form of reasoning. then I converted
the 1d equations to the orthogonal projection form to show how
the Kalman gain is computed and where the residual comes from
computationally. This should make the full KF equations much more
approachable.
All my code in this chapter hard coded the computation of the
likelihood inside the update() function, where it had no business.
Also, my treatment of the likelihood was rather hand wavey. By
pulling it out of update() and maing it explicit I have created
a firm foundation for the rest of the book.
It rendered fine in the notebook, but \text{foo_bar} was causing
an error in the pdf generation. Switching to \mathtt{foo\_bar}
mostly fixes it (though in the notebook it renders as -, not _.
Oye.
I was using a bunch of variable names that weren't consistent
with the rest of the book (but perhaps are more consistent with
the literature). It just made everything more challenging than
it needed to be, so instead of \mu and \sigma (e.g.) I use
\bar x and \bar P.
I also am in the middle of rewriting some sections for clarity,
but that work is not completed.
I added titles so you can see where in the processing the
graph is.
This surfaced a bug in matplotlib 1.5 (#5399) so I added
some code to work around that.
* integrals should have "du", etc. (right??)
* $\mathit{VAR}$ gives better spacing than $VAR$
* I find "within $1\sigma$" to be ambiguous; "within $\pm1sigma$" much more clear
* Changed dashes in text from hyphens to em-dashes
* Changed (some) straight quotation marks to curly ones (and removed single quotes in favour of double ones)
* Improved "°C" to just use a text symbol
* Assorted random typos, etc., but avoided rewording anything except for one particularly difficult sentence
I was conflating thematrix exponential with Taylor series, making
the whole thing quite confusing.
There was also significant duplication of content, such as presenting
the calculus form of Newton's equations multiple times in slightly
different ways. I've made it all hang together now.
