The Blackstone Ratio: Type I vs Type II at Maximum Asymmetry
~12 min · blackstone, asymmetry, type-i, type-ii, courtroom
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"'Better that ten guilty persons escape than that one innocent suffer' is a moral and legal maxim, not a calibrated 10:1 loss function."
The Quote
The 18th-century English jurist William Blackstone wrote in his Commentaries on the Laws of England that it is better for ten guilty persons to escape than for one innocent person to suffer. Related formulations predate him, and his wording became the best-known expression of a priority against wrongful conviction.
What the Number Does—and Does Not—Mean
The number ten is rhetorical. It is not an empirical estimate of wrongful convictions, a literal exchange rate between two errors, or a formula from which an optimal α can be calculated. Blackstone's maxim does not by itself specify every rule of criminal procedure.
A statistical error-cost analogy can make the priority vivid: wrongful conviction receives special weight. But a legal system also protects rights, constrains state power, defines admissible proof, and embodies constitutional and moral judgments that are not consequences of one cost ratio.
Across Systems and Over Time
Legal traditions differ in institutions, procedures, and safeguards. They cannot be ranked by an invented "effective Blackstone ratio" without evidence and a defined measure. Emergency laws or political pressure may weaken protections, but the change should be described through the actual doctrine and observed outcomes.
The Principle to Keep
Keep the maxim as a warning against treating wrongful conviction as an ordinary cost. Then evaluate proposed reforms by their legal mechanism, evidence, and effects rather than pretending that ten is a statistical parameter.
Code
Blackstone ratio → optimal threshold → wrongful-conviction rate·python
import numpy as np
rng = np.random.default_rng(170)
# Compute the 'optimal' threshold for different Blackstone ratios.
# Assume 5% of accused are innocent, evidence distributions as before.
N = 100_000
p_innocent = 0.05
is_innocent = rng.random(N) < p_innocent
evidence = np.where(
is_innocent,
rng.normal(0.4, 0.2, N),
rng.normal(0.85, 0.15, N),
).clip(0, 1)
print("Different cost weights → different optimal thresholds")
print(" (cost of Type I) : (cost of Type II) → best threshold → resulting Type I rate (% of convictions)")
print("-" * 90)
for ratio in (1, 2, 5, 10, 20, 100):
# Brute-force search for the threshold that minimizes weighted total cost.
best_cost = float('inf')
best_thr = None
for thr in np.linspace(0, 1, 101):
convicted = evidence > thr
type_I = (convicted & is_innocent).sum()
type_II = (~convicted & ~is_innocent).sum()
cost = ratio * type_I + type_II
if cost < best_cost:
best_cost, best_thr = cost, thr
convicted = evidence > best_thr
type_I = (convicted & is_innocent).sum()
rate = type_I / max(convicted.sum(), 1) * 100
print(f" {ratio:>3}:1 → thr={best_thr:.2f} → {rate:5.2f}% wrongful conviction rate")
# As the Blackstone ratio rises (Type I treated as more costly),
# the optimal threshold for conviction rises, and the wrongful-conviction
# rate falls. The 10:1 ratio is roughly what common law has chosen as
# the operating point.
Name a domain (other than criminal law) where you implicitly apply a Blackstone-like ratio: how much more averse are you to one error type than the other? Examples: (a) hiring — false positive vs false negative; (b) medical treatment — over-treating vs under-treating; (c) parenting decisions — over-protecting vs under-protecting. Make the ratio explicit, then ask whether it matches the actual costs in your situation.
Hint
Most domains have implicit Blackstone ratios that nobody has ever articulated. Articulating them is the first step to calibrating them against actual costs rather than gut feel.
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