Unbiased and Consistent

Given a gaussian estimate of our entire path (from anything solving NLNG Problem Statement or LG Problem Statement)

$$\mathcal{N}({\hat{\mathbf{x}}}_k,{\hat{P}}_k)\text{ for }k=1\dots K$$

And a ground truth, $x_{true,k}$ we can compute a straightforward error

$${\hat{e}}_k={\hat{x}}_k-x_{true,k}$$

A estimate is considered unbiased if ${\hat{e}}_k$ hovers around zero and consistent if the error stays within $3{\sigma }_k$ where ${\sigma }_k=\sqrt{{\hat{P}}_k}$ 99.7% of the time.

We usually measure error over a single trial throughout all of its timesteps, instead of having multiple trials. This is because we assume that the average of the error overtime is the same as the average of the error over many trials (known as ergodic hypothesis)

Unbias

We require that

$$E[{\hat{e}}_k]=0$$

Consistent

We require that

$$E\left[\frac{{\hat{e}}_k^2}{{\hat{P}}_k}\right]=1$$