Merge pull request #994 from JuliaRobotics/23Q4/docs/schur

Schur complicament example in docs

docs/src/dev/internal_fncs.md

@@ -1,5 +1,52 @@
+## Good to know
+
+# Conditional Multivariate Normals
+
+```julia
+using Distributions
+using LinearAlgebra
+
+##
+
+# P(A|B)
+
+Σab = 0.2*randn(3,3)
+Σab += Σab'
+Σab += diagm([1.0;1.0;1.0])
+
+μ_ab = [10.0;0.0;-1.0]
+μ_1 = μ_ab[1:1]
+μ_2 = μ_ab[2:3]
+
+Σ_11 = Σab[1:1,1:1]
+Σ_12 = Σab[1:1,2:3]
+Σ_21 = Σab[2:3,1:1]
+Σ_22 = Σab[2:3,2:3]
+
+##
+
+# P(A|B) = P(A,B) / P(B)
+P_AB = MvNormal(μ_ab, Σab)                       # likelihood
+P_B = MvNormal([-0.5;0.75], [0.75 0.3; 0.3 2.0]) # evidence
+
+# Schur compliment
+μ_(b) = μ_1 + Σ_12*Σ_22^(-1)*(b-μ_2)
+Σ_ = Σ_11 + Σ_12*Σ_22^(-1)*Σ_21
+
+P_AB_B(a,b) = pdf(P_AB, [a;b]) / pdf(P_B, b)
+P_A_B(a,b; mv = MvNormal(μ_(b), Σ_)) = pdf(mv, a) 
+
+##
+
+# probability density: p(a) = P(A=a | B=b)
+@show P_A_B([1.;],[0.;0.])
+@show P_AB_B([1.;],[0.;0.])
+
+P(A|B=B(.))
+```
+
 # Various Internal Function Docs
 
 ```@docs
 _solveCCWNumeric!
-```
+```