Singapore School Mathematics Operating Manual · Chapter 33
Mathematics often asks us to reason about things we cannot see directly.
We may observe a total but not its parts, a graph but not its generating parameters, several measurements but not the hidden state that produced them, an output but not the input, or a pattern of consequences but not the unknown quantity underneath.
Two questions then become essential. Observability asks whether the available observations contain enough information to determine the hidden state. Identifiability asks whether only one hidden state can explain those observations under the chosen model.
These are BTT operating-manual terms for a family of ideas already present across school Mathematics: simultaneous equations, loci, inverse functions, data sufficiency, graph reconstruction, probability, parameter fitting and model checking.
Observability · Identifiability · Equations · Geometry · Data · Practice · Worked answers
1. A hidden state is a quantity or configuration not directly supplied
If a problem gives x+y=10 but asks for x and y separately, the individual values are hidden states.
The observed information is the total.
The key question is whether that total alone reveals both parts.
It does not: (1,9), (4,6), (10,0) and infinitely many other real pairs work.
2. Observability depends on the measurements available
Add the second observation x−y=2.
Now the pair is determined: adding gives 2x=12, so x=6 and y=4.
The state that was unobservable from one measurement becomes observable from two independent measurements.
3. More measurements help only if they add new information
If the second equation were 2x+2y=20, it would merely repeat x+y=10.
The number of equations increased, but the information did not.
Observability depends on independent constraints, not sentence count.
4. Observable does not always mean uniquely identifiable
Suppose x²=25.
The magnitude |x| is observable: it must be 5.
The sign is not identifiable over the full real domain because x=5 and x=−5 produce the same squared observation.
A measurement can reveal part of a hidden state while leaving another component ambiguous.
5. Domain information can restore identifiability
If the same problem states x>0, the positive branch is selected and x=5 becomes identifiable.
The additional condition does not create a new numerical observation. It narrows the admissible state space.
6. A forward map explains what information is visible
If y=f(x), the forward map turns hidden input x into observable output y.
If distinct x-values can give the same y, the inverse state is not uniquely identifiable from y alone.
The horizontal-line test for functions is therefore an identifiability test.
7. One-to-one functions preserve enough information for unique inversion
For f(x)=3x+2, every real output corresponds to exactly one real input.
Knowing y therefore identifies x uniquely.
For f(x)=x² over all real x, it does not.
8. Restricting a function domain can make the inverse identifiable
x² is one-to-one on x≥0.
Within that restricted state space, y uniquely identifies x=√y.
This is why inverse functions often require domain restrictions.
9. Simultaneous equations are observability systems
Each independent equation cuts down the feasible set.
One line in two unknowns leaves infinitely many points.
Two non-parallel lines leave one intersection.
Two parallel distinct lines leave none.
Coincident lines still leave infinitely many.
The geometry visualises observability, uniqueness and inconsistency.
10. Rank is the advanced language behind independent information
In later linear algebra, the rank of a system records how many independent constraints it contains.
School students need not use formal matrix rank to use the principle: repeated equations do not improve identifiability.
11. Hidden variables can be only partially identified
If x+y=10, the total is known exactly even though x and y are not.
Some functions of the hidden state are identifiable while the full state is not.
This is a powerful distinction: lack of complete reconstruction does not mean nothing can be concluded.
12. Bounds can provide partial observability
If x and y are positive and x+y=10, then 0<x<10 and 0<y<10.
The exact values are unidentified, but useful bounds are observable from the constraints.
13. Loci turn hidden positions into observable constraints
If point P is equidistant from A and B, P lies on the perpendicular bisector of AB.
One distance-equality observation gives a whole line of possible positions.
A second independent locus can intersect it and identify P.
14. Two circles can create two candidate hidden positions
Knowing distances from P to two fixed centres often gives two possible intersections.
The position is observable up to a mirror ambiguity.
A third suitable observation can resolve the ambiguity.
15. Symmetry commonly causes non-identifiability
If a model is unchanged when two hidden states are swapped or reflected, observations may be unable to distinguish them.
The Symmetry and Case Reduction chapter explains the preserved structure. Here symmetry is viewed as a source of hidden-state ambiguity.
16. Measurement direction matters
Knowing only a vector’s magnitude does not identify its direction.
Knowing both horizontal and vertical components does.
Different measurement designs expose different parts of the state.
17. Summary statistics do not identify the original data set
A mean of 50 can arise from 50,50,50 or from 0,50,100.
The mean reveals one feature while hiding spread and individual observations.
A compressed summary is therefore not a full-state observation.
18. Mean plus range still may not identify all data
Even several summaries can leave many compatible data sets.
Whether the original observations are identifiable depends on how many degrees of freedom remain.
19. A histogram hides within-bin positions
If ten values fall in 20≤x<30, the histogram does not reveal their exact values.
The data are observable only at class-interval resolution.
20. Sampling can make different underlying processes look identical
If a signal is observed too sparsely, distinct hidden behaviours can generate the same samples.
This is the aliasing idea developed in Sampling, Resolution and Aliasing.
21. Model assumptions affect identifiability
Two points uniquely identify a straight line, provided we assume the model is linear and the x-values differ.
Without the linear-model assumption, infinitely many curves pass through the same two points.
Identifiability belongs to data plus model, not data alone.
22. Sequence reconstruction depends on the assumed family
The terms 2,5,8,11 suggest an arithmetic sequence with common difference 3.
Within the class of arithmetic sequences, the rule is identifiable.
Among all imaginable sequences, it is not.
23. Parameter identification needs enough independent observations
For y=ax+b, one point gives one equation in a and b and does not identify both.
Two points with distinct x-values identify the line.
Three points can provide a consistency check for the linear model.
24. Noise changes exact identification into estimation
If three measured points almost but not exactly lie on a line, no exact pair a,b may satisfy all equations.
The task becomes estimating parameters under a best-fit criterion.
Identifiability and estimation precision are different questions.
25. Unique does not automatically mean stable
A hidden state may be uniquely identifiable but highly sensitive to measurement error.
Nearly parallel lines have one intersection, yet tiny gradient changes can move it greatly.
This is the distinction between identifiability and conditioning.
Use Conditioning, Ill-Posedness and Sensitivity for stability.
26. Observability can be designed
If the current measurements cannot distinguish two candidates, ask what additional observation would separate them.
This is a powerful planning question in geometry, experiments, data collection and algebraic modelling.
27. Good extra information targets the ambiguity directly
If x²=25 leaves a sign ambiguity, learning x>0 resolves it.
If two geometric positions are mirror images, one orientation or side-of-line condition may distinguish them.
If two model parameters trade off against each other, a new independent measurement may separate them.
28. Redundant observations are still useful for verification
Once the hidden state is identified, an extra observation can test whether the reconstructed state reproduces all measurements.
Redundancy may not improve uniqueness, but it can improve confidence and error detection.
29. A candidate hidden state should be run forward
After reconstruction, calculate the observations that candidate would produce.
If they do not match the actual observations within the required tolerance, reject the candidate or revisit the model.
This forward check closes the inference loop.
30. An observability audit
Ask:
What is hidden? What is actually observed? Which parts of the hidden state affect each observation? Are the observations independent? Can two different states produce the same observations? What domain or model assumptions are active? What additional observation would separate the remaining candidates? Is the unique reconstruction stable?
31. Independent practice
1. Does x+y=12 identify x and y uniquely over the reals?
2. Add x−y=4 and solve.
3. Why do x+y=12 and 2x+2y=24 not identify the pair uniquely?
4. Does x²=36 identify x over all reals?
5. What extra condition makes the answer unique?
6. Two points with distinct x-values are known to lie on a straight line. Are its gradient and intercept identifiable?
7. Why does a mean not identify the original data set?
8. A point is known to be equidistant from A and B. What geometric set contains every possible position?
9. Why can a unique intersection of nearly parallel lines still be unreliable?
10. What is the best general check after identifying a hidden state?
32. Worked answers
1. No. Infinitely many real pairs sum to 12.
2. Adding gives 2x=16, so x=8 and y=4.
3. The second equation is only a multiple of the first and adds no independent information.
4. No. x=6 and x=−6 both work.
5. For example x>0 selects x=6.
6. Yes. Two distinct points determine one non-vertical or vertical line uniquely; gradient-intercept parameters apply to the non-vertical case.
7. Many different data sets share the same arithmetic mean.
8. The perpendicular bisector of AB.
9. Small changes in the measured gradients can move the intersection by a large amount.
10. Run it through the original forward model and check every observation and constraint.
33. Continue through Batch 09
Use Compression, Sufficient Summaries and Information Loss to ask what a summary preserves, Mathematical Provenance and Result Lineage to trace the observations and assumptions supporting a reconstruction, and Error Budgets and Cumulative Uncertainty when observations are approximate.
Return to the BTT Mathematics Hub for Batch 09.