Chapter 6 — Calculus II

Lesson map

Lesson 06 published. Map, example, EX016.

flowchart TD
  N0["Learning outcomes"]
  N1["6.1 Accumulated exposure"]
  N2["6.2 Decay and staleness"]
  N3["6.3 Series and accumulation"]
  N4["6.4 Approximation"]
  N5["Synthesis"]
  N0 --> N1 --> N2 --> N3 --> N4 --> N5
  N5 --> EX["Worked example"]
  EX --> Q["Exercise EX016"]

Logical drawing

flowchart LR
  N0["Learning outcomes"]
  N1["6.1 Accumulated exposure"]
  N2["6.2 Decay and staleness"]
  N3["6.3 Series and accumulation"]
  N4["6.4 Approximation"]
  N5["Synthesis"]
  N0 --> N1 --> N2 --> N3 --> N4 --> N5 --> Q["EX016"]

[Learning outcomes]
    |
    v
[6.1 Accumulated exposure]
    |
    v
[6.2 Decay and staleness]
    |
    v
[6.3 Series and accumulation]
    |
    v
[6.4 Approximation]
    |
    v
[Synthesis]
    |
    v
[Worked example] --> [Exercise EX016]

Learning outcomes

You will (1) integrate exposure over time; (2) model exponential decay of risk; (3) use series to accumulate privacy loss.

6.1 Accumulated exposure

Zara models Pellmore's exposure as the number of patient records stored multiplied by the time they are stored. Integrating storage over a year gives patient-days of exposure. A retention period of thirty days instead of indefinite storage for video recordings reduces the integral dramatically, even though the daily inflow is unchanged. She shows the calculation to Pellmore's engineering lead as an area under two curves. The visual makes the effect of retention immediate, and it gives a single number that can be tracked as a key indicator.

6.2 Decay and staleness

Some risks decay. An old address is less useful to an attacker than a current one, and the usefulness of contact data might be modelled as decaying exponentially with time. Zara warns against over-using the model: health information does not lose sensitivity with age, and a diagnosis from ten years ago can still harm someone. She concludes that decay models can guide retention for low-sensitivity data, but special categories need explicit rules rather than an assumption that risk fades away naturally over time.

6.3 Series and accumulation

In later work Zara will publish repeated statistics. Calculus II introduces series, sums of many terms. If each release leaks a small amount of information, the total after many releases is the sum. Zara writes the sum for daily releases and sees that it grows without bound. This foreshadows the composition property of differential privacy described by Dwork and Roth (2014), where privacy loss adds up across queries, and it makes her sceptical of any claim that frequent releases are safe because each one is individually harmless.

6.4 Approximation

Taylor approximations let engineers replace a complex function with a simple one near a point. Zara uses the idea to estimate how a small change in retention affects exposure: near the current policy, exposure changes roughly in proportion to the retention period. She uses this to explain that halving retention roughly halves exposure for that category. Prof. Holm reminds her that approximations hold only near the point where they were made, which is a useful caution whenever engineers extrapolate from small changes to large redesigns.

Worked example

Zara integrates storage to get patient-days of exposure, limits decay models to low-sensitivity data, sums repeated releases to foreshadow composition and uses a linear approximation for retention changes.

Synthetic case. Exposure figures are synthetic; Dwork and Roth (2014) is real.

Errors and limits

Patient-days ignores sensitivity differences between records; weight categories when needed.

Exercises

• EX016 — Patient-days. What does integrating storage over time measure?
  Show marking criteria

  Rubric: 3 = accumulated exposure in patient-days;; 2 = mentions area;; 1 = storage size;; 0 = none.
• EX017 — Decay caution. Why are decay models unsuitable for health data?
  Show marking criteria

  Rubric: 3 = health information keeps its sensitivity over time;; 2 = mentions sensitivity;; 1 = accepts decay;; 0 = none.
• EX018 — Summed releases. What does summing daily leakage show?
  Show marking criteria

  Rubric: 3 = total privacy loss grows without bound as releases continue;; 2 = mentions growth;; 1 = says it stays small;; 0 = none.

Synthesis

Chapter 6 accumulates exposure and loss. Chapter 7 uses probability to collect sensitive data safely.

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