Inverse-Variance Fusion

Ada re-derives this chapter’s own numbers step by step, at full precision

ADA · CALCULATION AUDIT

Inverse-Variance Fusion

The chapter fuses three obstacle-range readings — camera 44.8 m, lidar 45.1 m, radar 45.5 m — each with its own uncertainty, into a single value with a published band of 0.25 m, tighter than any one sensor. That tightness is the whole appeal of inverse-variance fusion, and also its trap. This audit re-runs the weights and the 45.18 m fused range to show when that confident 0.25 m is earned and when it is not.

Companion to the chapter Sensor Fusion Applications — every number here comes from that chapter.

Ada: A fused number is trustworthy only when its weights and its published uncertainty come from the same arithmetic. Let me re-run the obstacle-range example above using only its stated values, carry full precision, and round at the end.

The three measurements are camera 44.8 m (sigma 1.0 m), lidar 45.1 m (sigma 0.3 m), and radar 45.5 m (sigma 0.5 m). Precision is the inverse of variance:

• Camera precision: 1 / 1.0^2 = 1.000000
• Lidar precision: 1 / 0.3^2 = 1 / 0.09 = 11.111111
• Radar precision: 1 / 0.5^2 = 1 / 0.25 = 4.000000
• Total precision: 1.000000 + 11.111111 + 4.000000 = 16.111111, which the chapter rounds to 16.11.

Each normalized weight is a precision divided by that total:

• Camera: 1.000000 / 16.111111 = 0.062069, rounded to 0.062.
• Lidar: 11.111111 / 16.111111 = 0.689655, rounded to 0.690.
• Radar: 4.000000 / 16.111111 = 0.248276, rounded to 0.248.

The fused range is the weighted sum:

• 44.8 x 0.062069 + 45.1 x 0.689655 + 45.5 x 0.248276
• = 2.780690 + 31.103448 + 11.296552 = 45.180690 m, rounded to 45.18 m.

The published uncertainty is the standard deviation implied by the total precision:

• sqrt(1 / 16.111111) = sqrt(0.062069) = 0.249136 m, rounded to 0.25 m.

That 0.25 m is tighter than any single sensor’s sigma, which is exactly why the fused value looks confident. But the same arithmetic assumes the three errors are independent and current, and the freshness check in this section shows the risk: a camera frame 120 ms old on an object moving 1.6 m/s has already slipped 1.6 x 0.120 = 0.192000 m, past a 0.10 m stale budget, while the 40 ms lidar frame has slipped only 1.6 x 0.040 = 0.064000 m. The audit’s lesson is narrow: publish the 0.25 m band only after proving the inputs were independent and fresh, because otherwise the tightest number in the record is the least honest.

Every number above is taken from the chapter’s own material and re-derived step by step.