Direct Taylor linearisation of the likelihood

[SamDuffield]

I'd really like to support a linearisation method that simply uses a 2nd order Taylor expansion in $x$ without touching $y$.

That is linearizes a more general potential $G(x)$ rather than conditional distribution $p(y \mid x)$.

This is used in abile here https://github.com/SamDuffield/abile/blob/a10fb8c328fdb088fdbafd3f82bb537d89d67bb6/abile/models/extended_kalman.py#L101
where it is particularly useful when $y$ is categorical.

The problem is with just the gradient $g = \nabla G(x)$ and Hessian $H = \nabla^2 G(x)$ I don't see how to get it into the $(H, d, L)$ form required for generalized-kalman.

[SamDuffield]

Ok I figured a workaround by allowing generalised_kalman to additionally support a PotentialParams for the update step which only requires an unconditional potential function G(x) rather than p(y | x) (and thus just uses dummy $H$ and $y$ for the actual Kalman update calculation)

[SamDuffield]

PotentialParams and linearize_taylor added in #26