Two independent measurements of the same quantity always carry more information than one. By combining them with the right weights you produce an estimate with smaller variance than either input. Picking only the best sensor throws away the information in the other.
\(\hat{x} = w_1 z_1 + w_2 z_2\) subject to \(w_1 + w_2 = 1\). The constraint keeps the estimate unbiased when both measurements are unbiased.
An unweighted average gives every sensor equal influence. If \(\sigma_1 \ll \sigma_2\), letting the noisy sensor pull on the estimate as hard as the quiet one inflates the fused variance unnecessarily. Optimal weights respect the relative quality of the measurements.
\(w_1 = \dfrac{1/\sigma_1^2}{1/\sigma_1^2 + 1/\sigma_2^2}\) and \(w_2 = \dfrac{1/\sigma_2^2}{1/\sigma_1^2 + 1/\sigma_2^2}\). Lower-variance sensors get larger weights.
\(\hat{x}_{\text{opt}} = \dfrac{z_1/\sigma_1^2 + z_2/\sigma_2^2}{1/\sigma_1^2 + 1/\sigma_2^2}\). Multiply each measurement by its information, sum, and normalize by the total information.
\(\sigma_{\text{opt}}^2 = \left(\dfrac{1}{\sigma_1^2} + \dfrac{1}{\sigma_2^2}\right)^{-1}\). The fused variance is always strictly smaller than the smaller of the two input variances.
Information is the inverse of variance: \(I = 1/\sigma^2\). Independent measurements add information: \(I_{\text{total}} = I_1 + I_2 + \cdots + I_n\). The fused variance is the inverse of the total information.
\(I_1 = 0.25,\ I_2 = 1/36 \approx 0.0278,\ I_{\text{total}} \approx 0.2778\). \(\hat{x}_{\text{opt}} = (100\cdot 0.25 + 110\cdot 0.0278)/0.2778 \approx 101.0\). The estimate is pulled toward the lower-variance sensor.
\(\sigma_{\text{opt}} = 1/\sqrt{I_{\text{total}}} \approx 1/\sqrt{0.2778} \approx 1.897\). Smaller than \(\sigma_1 = 2\), so adding the noisy sensor still bought you a small improvement.
Correlation reduces the effective information in the second measurement. The optimal fusion rule still exists, but the inverse-variance form is replaced by a more general inverse-covariance form using \(\boldsymbol{\Sigma}^{-1}\). We will see this exact pattern in the multi-state Kalman filter in Block 5.
\(\hat{x}_k^+ = \hat{x}_k^- + K_k (z_k - \hat{x}_k^-)\) with the Kalman gain \(K_k = P_k^- / (P_k^- + R_k)\). The innovation \(z_k - \hat{x}_k^-\) is the new information; \(K_k\) is the optimal-fusion weight on it.
If \(R_k \gg P_k^-\), then \(K_k \to 0\): the filter trusts the prediction and largely ignores the measurement. If \(P_k^- \gg R_k\), then \(K_k \to 1\): the filter snaps the estimate onto the measurement. Everywhere in between, it blends the two using the optimal trust factor.
Because for independent measurements, information adds. Adding two measurements simultaneously and adding them one-after-the-other produce the same total information and therefore the same fused variance. Recursion is just an implementation that uses constant memory, which is what makes the Kalman filter practical for real-time navigation.
Information ratio: \(I_R / I_B = (\sigma_B/\sigma_R)^2 = 25\). So \(w_R = 25/26 \approx 0.9615\) and \(w_B = 1/26 \approx 0.0385\). The radar dominates by a factor of 25 to 1.
\(\sigma_{\text{opt}}^2 = 1/(1/4 + 1/100) \approx 3.846\), so \(\sigma_{\text{opt}} \approx 1.961\) m. The radar alone is \(\sigma = 2\) m, so fusion buys about 0.04 m of additional precision. When one sensor is much better than the other, the marginal value of the worse sensor is small but non-zero.