Bayesian Inversion

definition theorem example

In a Markov Category, let $p:I\to X$ be a state (a prior) and $f:X\to Y$ a kernel (a likelihood, or channel). A Bayesian inverse of $f$ at $p$ is a kernel $f_p^{†}:Y\to X$ such that the joint distribution can be produced in either direction:

$$p;{\mathrm{copy}}_X;(1_X\otimes f)=(f_{\ast }p);{\mathrm{copy}}_Y;(f_p^{†}\otimes 1_Y):I\to X\otimes Y,$$

“sample $x\sim p$, then $y\sim f(\cdot \mid x)$” equals “sample $y$ from the pushforward $f_{\ast }p$, then $x\sim f_p^{†}(\cdot \mid y)$“. In $\mathbf{FinStoch}$ this is Bayes’ law,

$$f_p^{†}(x\mid y)=\frac{f(y\mid x)p(x)}{{\sum }_{x^{\prime }}f(y\mid x^{\prime })p(x^{\prime })}\text{wherever }(f_{\ast }p)(y)>0.$$

Bayesian inversion is a special disintegration: integrate $p$ and $f$ into a joint state, then disintegrate it the other way (Cho & Jacobs §3).

Sources: Cho & Jacobs arXiv:1709.00322 (notes) §3 (Eq. (5), Examples 3.8–3.9, Proposition 3.10), §7; Fritz arXiv:1908.07021 (notes) Proposition 11.17, Remark 13.10; Braithwaite, Hedges & St Clere Smithe arXiv:2305.06112 (notes) Definition 1, Proposition 11, Theorem 20; St Clere Smithe arXiv:2006.01631 (notes) §2, Theorem 5.2; St Clere Smithe & Perin arXiv:2503.18608 (notes) §3, Theorem 13; Clerc, Danos, Dahlqvist & Garnier (2017), Pointless learning.

The chain rule

Inverting a composite is composing the inverses in reverse order, with the pushed-forward prior for the second one:

$$(f;g{)}_p^{†}=g_{f_{\ast }p}^{†};f_p^{†}(\text{almost surely}).$$

Compare the reverse-mode chain rule ${\mathrm{d}}_x(g\circ f)={\mathrm{d}}_{x}f\circ {\mathrm{d}}_{f(x)}g$:

autodiff	Bayes
point $x$	prior $p$
pushforward point $f(x)$	pushforward prior $f_{\ast }p$
${\mathrm{d}}_{x}f$ (transpose)	$f_p^{†}$
composite reverses order	composite reverses order

The prior plays the role of the linearisation point. The chain rule makes Bayesian inversion a functor into Bayesian lenses — exactly, or up to almost-sure equality (Braithwaite et al. Proposition 11; St Clere Smithe Theorem 5.2), and on the nose once inverses are restricted to supports (Braithwaite et al. Theorem 20). AutoBayes (Theorem 13) proves the analogue for open models, where it holds without integrating out intermediate variables.

Inversion as a dagger

On the category $\mathbf{ProbStoch}(\mathcal{C})$ of probability spaces $(X,p)$ and measure-preserving kernels (modulo a.s. equality), Bayesian inversion is a symmetric monoidal dagger (Fritz Remark 13.10; Clerc et al.): $(gf{)}^{†}=f^{†}{g}^{†}$, $f^{††}=f$. The dagger is “run the channel backwards, given where you started”.

Why it is the hard part of inference

Computing $f_p^{†}$ requires the normaliser $(f_{\ast }p)(y)=\int f(y\mid x)p(dx)$ — a marginalisation, generally intractable. Every approximate-inference method replaces the exact inverse by a chosen kernel of the same type — amortised encoder, mean-field Gaussian, particle set, solver fixed point — and measures the error with a divergence; the categorical bookkeeping for that is the Bayesian Lens and the Statistical Game.

Lenticulum.jl

A Lenticulum factor-to-variable message is a Bayesian inversion $c_{\pi }^{\prime }$ of the factor at the belief of the other variables. See Messages are Inversions.

JuliaLeanHaskell

Docs: Theories (Catlab): copy/delete — ThMonoidalCategoryWithDiagonals

# Bayes' law in FinStoch and the chain rule (f;g)†_p = g†_{f*p} ; f†_p.
pushforward(p, f) = vec(p' * f)
function invert(p, f)                                   # f†_p : Y → X, rows = y
    q = pushforward(p, f)
    [q[y] > 0 ? f[x, y] * p[x] / q[y] : 1 / length(p) for y in axes(f, 2), x in eachindex(p)]
end
p = [0.2, 0.5, 0.3]                                     # prior on X
f = [0.9 0.1; 0.4 0.6; 0.1 0.9]                         # X → Y
g = [0.7 0.2 0.1; 0.1 0.3 0.6]                          # Y → Z
lhs = invert(p, f * g)                                   # (f;g)†_p : Z → X
rhs = invert(pushforward(p, f), g) * invert(p, f)        # g†_{f*p} ; f†_p
lhs ≈ rhs                                                # true
# the defining equation: both ways of sampling give the same joint on X × Y
q = pushforward(p, f); fd = invert(p, f)
[p[x] * f[x, y] for x in 1:3, y in 1:2] ≈ [q[y] * fd[y, x] for x in 1:3, y in 1:2]   # true

import Mathlib
open MeasureTheory ProbabilityTheory
-- Mathlib: the posterior kernel is the Bayesian inverse of κ at the prior μ,
-- characterised by μ ⊗ₘ κ = (κ ∘ₘ μ) ⊗ₘ κ†μ up to swapping the factors.
#check @posterior
-- (see the lemmas in Mathlib.Probability.Kernel.Posterior)

-- Bayes' law for finite channels
type Channel x y = x -> [(y, Double)]
 
pushforward :: Eq y => [(x, Double)] -> Channel x y -> [y] -> [(y, Double)]
pushforward p f ys = [ (y, sum [ px * w | (x, px) <- p, (y', w) <- f x, y' == y ]) | y <- ys ]
 
invert :: Eq y => [(x, Double)] -> Channel x y -> [y] -> Channel y x
invert p f ys y = [ (x, px * like x / py) | (x, px) <- p, py > 0 ]
  where like x = sum [ w | (y', w) <- f x, y' == y ]
        py = sum [ px * like x | (x, px) <- p ]