Combining Measurements With Different Reliability

Two thermometers disagree. Their known error variances tell us how strongly each reading should influence a shared estimate.

13 sections · 4 questions · 2 coding challenges

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Following a Quantity Through Noisy Measurements

BOTIE

Kalman’s 1960 paper recast linear filtering as a recursive estimation problem. An estimate and its uncertainty could be carried forward and corrected as new measurements arrived.

Our thermometers isolate the measurement-combination step. The full Kalman filter also predicts how the state and its uncertainty change over time; averaging simultaneous readings alone is not that complete algorithm.

Sources for this history

Kalman: A New Approach to Linear Filtering and Prediction Problems (1960)

Which Thermometer Should Botie Trust More?

Two thermometers read the same room at the same time. One is much noisier than the other. Simply averaging them gives both the same influence, even though they do not carry the same information.

How much more influence should the quieter thermometer have? We can derive a weighting rule if the errors are independent and Gaussian, have no bias, and have known positive variances. Under those assumptions, a smaller error variance gives a reading more weight. Its reciprocal is called precision. We will use it to work out the contribution of each thermometer.

The measurement variances are supplied and the instrument errors are assumed independent.

Step 1 of 4

Read both instruments

The example variances describe measurement errors, not the temperatures.

An invented example for this lesson.
Instrument	Readings	Variances
Instrument A	20	1
Instrument B	24	4

Instrument A
Readings
20
Variances
1
Instrument B
Readings
24
Variances
4

Step 2 of 4

Convert variance to precision

A larger error variance gives a smaller weight.

An invented example for this lesson.
Instrument	Precisions	Normalised weights
Instrument A	1	0.8
Instrument B	0.25	0.2

Instrument A
Precisions
1
Normalised weights
0.8
Instrument B
Precisions
0.25
Normalised weights
0.2

Step 3 of 4

Combine evidence

The more precise instrument pulls the estimate closer to its reading.

Estimate = (0.8 × 20) + (0.2 × 24) = 20.8
Combined variance = 1 / (1 + 0.25) = 0.8

The more precise instrument contributes more to the combined estimate, under the independent Gaussian measurement assumptions.

Step 4 of 4

Two readings can share the same error

If both instruments respond to the same disturbance, their errors are related. Treating them as independent would count some evidence twice. We need that dependence before interpreting the reduced variance as a reliable uncertainty estimate.

The independence assumption concerns errors, not the number of instruments.
Measurement situation	What the second reading adds
Independent instrument errors	Another independent piece of evidence
A shared disturbance	Some information already present in the first reading

Independent instrument errors
What the second reading adds
Another independent piece of evidence
A shared disturbance
What the second reading adds
Some information already present in the first reading

The independence assumption concerns errors, not the number of instruments.

Precision expresses how much independent evidence a reading supplies

Independent measurements contribute in proportion to their precision.

Which thermometer should influence the estimate more?

Two thermometers disagree, but one may be much noisier. Change an instrument's variance and watch its influence on the combined temperature.

BOTIE

Botie needs one estimate from independently noisy thermometers.

Choose an example

Temperature readings

Positive error variances

Precision-weighted temperature estimate

20.8

Follow the operations

1. Temperature readings
1. 020
2. 124
2. Positive error variances
1. 01
2. 14
3. Take positive reciprocals
1. 01
2. 10.25
4. Dot product
```
26
```
5. Add a list
```
1.25
```
6. Divide
```
20.8
```
7. Result
```
20.8
```

Requires common units, positive known variances, unbiased readings, and independent errors.

1. Separate Error Size From Measurement Value

A reading is an observed temperature. Its error variance describes the spread of repeated measurement errors around the true temperature. A high temperature is not automatically a noisy temperature.

We assume both thermometers measure the same quantity in the same units and have zero average error. Calibration bias is not repaired by pretending it is independent random noise.

The known-variance assumption lets us study weighting before estimating reliability from data.

3. Carry the Remaining Uncertainty

The variance of the combined estimate is the reciprocal of total precision for independent Gaussian errors. It is smaller than either individual variance, because both readings supply independent information.

Equal variances give the ordinary mean. A very noisy instrument contributes little, while a zero variance would claim perfect information and needs separate treatment rather than division by zero.

Combined variance = 1 / (1 + 0.25) = 0.8

This is uncertainty about the common temperature under the stated error model.

Start with two equally reliable thermometers

The instruments report different temperatures but have the same known variance. Calculate their precision weights before combining the readings.

Python

import numpy as np
readings = np.array([18., 22.])
variances = np.array([4., 4.])
precision = 1.0 / variances
weights = precision / precision.sum()
print(weights.tolist())
print(float(weights @ readings))
print(float(1.0 / precision.sum()))

Output

[0.5, 0.5]
20.0
2.0

Equal variances give equal influence. Under independent errors, the combined variance is smaller than either instrument's variance. The challenge then gives the instruments different reliabilities.

4. Ask Whether the Errors Are Independent

Two instruments can share a power supply problem or calibration mistake. Their agreement then does not provide two independent confirmations. A full covariance model is needed when errors are correlated.

Check residuals against a reference instrument, inspect bias, and avoid fitting error variances using the same tiny sample you want to evaluate. An impressively narrow interval can be misleading if the model assumptions are wrong.

Botie’s workshop exposes each precision so the weighting remains visible.

5. Independent Evidence Earns Additional Precision

Two independent noisy readings of the same quantity can reduce uncertainty. A reading with smaller variance contributes more precision and therefore more weight. The combined variance is smaller because we have added independent information.

The variances must describe uncertainty in comparable units. A sensor's short-term repeatability is not necessarily its total measurement error if it also has an unknown calibration offset.

Fuse a precise reading with a noisy one

Both thermometers are assumed unbiased and independent for this calculation.

Python

import numpy as np
reading = np.array([20., 24.])
variance = np.array([1., 9.])
precision = 1/variance
mean = (precision@reading)/precision.sum()
combined_variance = 1/precision.sum()
print(round(float(mean), 3))
print(round(float(combined_variance), 3))

Output

20.4
0.9

The noisier reading moves the estimate only modestly. We give it less influence because its variance says that it is a less precise measurement.

6. Copying One Thermometer Does Not Give Us Two Thermometers

If both displayed readings come from the same instrument, their errors can move together. Treating them as independent would count the same information twice and report unjustified precision.

For equal-variance readings, the variance of their average includes their covariance. As correlation approaches one, averaging offers almost no reduction. Independence is therefore a substantive assumption about the sensors, not a convenient detail of the formula.

Var((X + Y)/2) = (Var(X) + Var(Y) + 2 Cov(X,Y)) / 4
Equal variance σ² and correlation ρ:
Var(average) = σ²(1 + ρ)/2

7. Shared Bias Survives Averaging

Two thermometers can agree closely because both were calibrated against the same incorrect reference. Averaging reduces certain random errors; it does not automatically remove a shared offset.

Compare against an independent reference when possible, retain calibration records and inspect disagreement over time. If the readings differ far more than their stated uncertainties predict, investigate the assumptions before producing a confident combined estimate.