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Intuition

How can we look at a simple table of counts—0 hits, 1 hit, 2 hits—and definitively say "Ah, the enemy was aiming at specific targets?"

It all comes down to the fundamental nature of randomness, and figuring out what the data should look like if the bombing was completely blind.

Pure Randomness vs. Clusters​

A single Poisson Distribution acts as the baseline for "purely random." If you close your eyes and randomly throw darts at a map of a city, the number of darts per city block will naturally follow a Poisson curve. Some blocks get zero, some get one or two, and a rare few might get three or four—just by pure luck.

A defining mathematical feature of "pure randomness" (a single Poisson) is that its Mean (the average hits per block) must equal its Variance (how spread out the counts are).

Let's look at the two cities:

London:

  • Mean ≈0.93\approx 0.93
  • Variance ≈0.93\approx 0.93
  • Observation: Because Mean = Variance, the data almost perfectly mirrors a single random process (K=1K=1). The bombs were flying blind and raining down evenly as a single, untargeted barrage.

Antwerp:

  • Mean ≈0.90\approx 0.90
  • Variance ≈1.74→\approx 1.74 \to High Variance!
  • Observation: This is called Overdispersion. The counts are stretched way too far apart to be random. There are too many 0's (325 areas) and dangerously many 5+'s (21 areas); the "middle" numbers are hollowed out.
pie title "What actually happened in Antwerp?"
    "Residential / Untargeted (Rate ~0.23)" : 66
    "Strategic Targets (Rate ~2.19)" : 34

The K=2K=2 Mixture Solution​

When we feed the Antwerp data into the EM algorithm from part (a) to look for a mixture of Poisson distributions, it easily finds two vastly different groups hiding inside the data.

It splits the map into:

  1. A "Safe" Zone: Makes up roughly 2/3 of the squares. These barely ever get hit (average 0.23 hits).
  2. A "Danger" Zone: Makes up exactly 1/3 of the squares. These are hammered constantly (average 2.19 hits).

This tells a clear story: The attacks on Antwerp weren't random at all. The attackers were trying very hard to hit specific targets (like the port of Antwerp), making "Danger Zones" out of 1/3 of the city, while the rest of the city mostly suffered from stray misses.