meters: variance=0.040000, full NLL=-0.565499, density=1.760327 per meter
centimeters: variance=400.000000, full NLL=4.039671, density=0.017603 per centimeter
NLL difference=4.605170; log(100)=4.605170
interval probability: [-0.1,0.1] m=0.382925; [-10,10] cm=0.382925
constant omitted NLL=-1.484438; difference=0.918939
incorrectly passing sigma as variance: NLL=0.139220
exact-fit NLL as sigma shrinks: 0.918939, -1.383647, -3.686232
checks passed: Jacobian scaling, interval invariance, constant, variance boundary
