Free lesson · Statistics

Delta method: propagate estimator uncertainty through a smooth function

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Start here · the same idea in everyday words

Unpack a technical sentence

Technical wording: A first-order transformation scales sampling uncertainty by the local derivative.

Read it as: For a smooth calculation and small enough estimation error, multiply the input standard error by the absolute slope to estimate the output standard error.

A slower worked example · one step at a time

1. Use a positive rate estimate of 2 rate units, with standard error .1 rate units. Take its reciprocal to get .5 inverse-rate units.
2. At 2, the reciprocal’s slope is −1 divided by 2 squared, or −.25 inverse-rate units per rate unit.
3. Multiply the absolute slope .25 by input standard error .1. The approximate output standard error is .025 inverse-rate units.
4. An input increase from 2 to 2.1 predicts a reciprocal decrease of .025, giving .475. The exact reciprocal is about .476190.
5. The negative slope tells us direction. Its absolute value tells us the size of uncertainty. Standard errors are never negative. This approximation becomes unreliable when the denominator can get close to zero.

Study loop: predict the result, work through the arithmetic, run the Python, then explain which assumption would make the result unreliable.

• First estimate a parameter and its standard error. Linearise the transformation around the parameter; the first-order sampling variance multiplies by the square of the derivative.
• This is an asymptotic approximation for an asymptotically normal estimator and a differentiable transformation with a nonzero first derivative for the ordinary first-order result. It can be poor near boundaries or with large uncertainty.
• For a synthetic positive daily rate estimate θ̂=2 and standard error .1 in common rate units, transform to its reciprocal. The sensitivity has a negative sign, but a standard error uses its absolute magnitude.

An analogy to remember

A magnifying lens stretches small input errors by its local magnification. The derivative measures that magnification for an estimator.

How this becomes a building block

A transformed estimate can carry an approximate standard error through a smooth model interface instead of losing its uncertainty metadata.

Python · run and reuse

Architecture pseudocode

# Python 3.10+; standard library only unless NumPy is imported.
# Synthetic teaching inputs; no historical performance claim.
def reciprocal_delta(estimate, standard_error):
    # Positive rate estimate; SE in the same rate units.
    if estimate <= 0 or standard_error < 0: raise ValueError("Positive estimate and nonnegative SE required")
    return 1/estimate, standard_error/estimate**2

print(reciprocal_delta(2,.1))

# Replay the slower, synthetic example above. Units are stated in its steps.
def plain_language_example():
    from math import exp, e  # Standard library; no extra dependency.
    return {
        "estimate": 1/2,
        "se": .1/2**2,
        "prediction": .5-.25*.1,
        "actual": 1/2.1,
    }

print(plain_language_example())

# Python 3.10+; standard library only unless NumPy is imported.
# Synthetic teaching inputs; no historical performance claim.
def reciprocal_delta(estimate, standard_error):
    # Positive rate estimate; SE in the same rate units.
    if estimate <= 0 or standard_error < 0: raise ValueError("Positive estimate and nonnegative SE required")
    return 1/estimate, standard_error/estimate**2

print(reciprocal_delta(2,.1))

# Replay the slower, synthetic example above. Units are stated in its steps.
def plain_language_example():
    from math import exp, e  # Standard library; no extra dependency.
    return {
        "estimate": 1/2,
        "se": .1/2**2,
        "prediction": .5-.25*.1,
        "actual": 1/2.1,
    }

print(plain_language_example())