mathematics//statistics//estimate

What one decides to believe about a quantity nobody sees directly, kept apart from the quantity itself and from what the instrument says. Three objects are involved, and they are not interchangeable. The **truth** \(x\) exists but is never seen directly; one only surrounds it (the room is at 21.3 °C). The **measurement** \(z\) is what the sensor says (the thermometer reads 22.1 °C). The **estimate** \(\hat x\) is what one decides to believe after thinking; the little hat means *this is an opinion of mine about \(x\)* (I believe it is at 20 °C, \(\hat x=20\)).

What one decides to believe about a quantity nobody sees directly, kept apart from the quantity itself and from what the instrument says. Three objects are involved, and they are not interchangeable. The truth xxx exists but is never seen directly; one only surrounds it (the room is at 21.3 °C). The measurement zzz is what the sensor says (the thermometer reads 22.1 °C). The estimate x^\hat xx^ is what one decides to believe after thinking; the little hat means this is an opinion of mine about xxx (I believe it is at 20 °C, x^=20\hat x=20x^=20).

Just as x≠zx\neq zx=z, also x≠x^x\neq\hat xx=x^. Neither of them is the truth, and two different errors follow:

v=z−x⏟measurement noisee=x^−x⏟estimation error\underbrace{v=z-x}_{\text{measurement noise}}\qquad\qquad \underbrace{e=\hat x-x}_{\text{estimation error}}measurement noisev=z−x​​estimation errore=x^−x​​

Both errors have a sign. zzz and x^\hat xx^ can fall above or below xxx, so the errors live on both sides of zero, and repeating the experiment many times draws a distribution, very often a bell (normal distribution). Whether that bell is a fair assumption is decided with data (goodness of fit).

The two errors belong to different owners. The measurement noise is a property of the instrument and is characterized once, against a known truth (sensor calibration). The estimation error belongs to the reasoning that produced x^\hat xx^, and a good estimator makes it smaller than the noise of any single measurement (inverse-variance weighting).

Neither error can be computed in operation, because computing it needs xxx. What can be known is its size on average, a variance, and that is what a filter carries forward (covariance propagation).

When time enters, the estimate splits in two, before and after the sensor speaks: a priori and a posteriori.