Fisher Forecasting

9. Fisher Forecasting#

9.1. Precision vs Accuracy#

Definitions.

Examples of systematic errors (reduced accuracy):

  • Using a stretched measuring tape

  • A scale that is not properly zeroed

  • reading an indicator line from a poor angle

Examples of random errors (reduced precision):

  • Measuring timing that depends on reaction time

  • Multiple people taking turns making the reading

  • Rounding numbers

These types of errors affect our estimations and model fitting in different ways. For a Gaussian, systematic errors shift the mean while random errors inflate the standard deviation.

9.2. Fisher Matrixes#

Read https://wittman.physics.ucdavis.edu/Fisher-matrix-guide.pdf.

Here is an updated code block since the example in the text is for Python 2 & NumPy < 2.0.

"""fisher.py

Calculates fisher matrix and covariance for a model of a line.
"""

import numpy as np

npar = 2
xvals = (0, 1)
sigmavals = [0.1] * len(xvals)

F = np.zeros([npar, npar])

for x, sigma in zip(xvals, sigmavals):
    for i in range(npar):
        # dfdp is calculated outside program.
        if i == 0:
            dfdp_i = x
        else:
            dfdp_i = 1

        for j in range(npar):
            if j == 0:
                dfdp_j = x
            else:
                dfdp_j = 1

            F[i, j] += sigma ** (-2) * dfdp_i * dfdp_j


print(f"Observatoins at {xvals}:\n")
print("Fisher matrix:")
print(F)
print("\nCovarance (m, b):")
print(np.asmatrix(F).I)  # numpy v2 needs np.asmatrix() not np.mat()

9.3. Suggested Reading#

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