Computing the Fisher Matrix numerically

In summary, The Fisher matrix can be computed for simple models with linear parameters, but becomes more complicated for models such as the power spectrum. The Fisher matrix is valid around a certain set of model parameters, but to deduce the covariances between parameters, it needs to be computed around different sets of parameters and then used to create a triangle plot. Useful resources for understanding the Fisher matrix include notes from CosmoCoffee and the MontePython paper.
  • #1
shahbaznihal
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TL;DR Summary
Computation of Fisher matrix for complicated models
Hi,

I have been studying the Fisher matrix to apply in a project. I understand how to compute a fisher matrix when you have a simple model for example which is linear in the model parameters (in that case the derivatives of the model with respect to the parameters are independent of the parameter). But I am unable to understand how to compute the Fisher matrix when the model is more complicated like the power spectrum which depends on the cosmological parameters in a complicated way. From the exercises that I have done, I understand the Fisher matrix computed around certain model parameters is valid around that set of model parameters but how do you go from varying parameters (and computing the Fisher matrix around each set of parameters) to deducing the covariances between the parameters in parameter space and making the triangle plot.

I have been unable to find the answer to this question all day. Thanks in advance for your help.
 
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  • #2
Hi,

I leave here the reply by Eric Linder from CosmoCoffee which had been useful for me,

You might find useful the notes at https://supernova.lbl.gov/~evlinder/scires.html, specifically https://supernova.lbl.gov/~evlinder/InfoMatrixNotes.pdf.

I also leave this link to the MontePython paper which is a pretty nice description of using the Fisher matrix approximation.

https://arxiv.org/pdf/1804.07261.pdf
 
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FAQ: Computing the Fisher Matrix numerically

1. What is the Fisher Matrix?

The Fisher Matrix is a mathematical tool used in statistics and data analysis to quantify the uncertainty in a set of parameters. It is commonly used in fields such as physics and engineering to estimate the precision of measurements and to determine the sensitivity of a system to changes in its parameters.

2. How is the Fisher Matrix computed numerically?

The Fisher Matrix can be computed numerically by using numerical differentiation techniques, such as finite difference methods, to approximate the derivatives of the model with respect to the parameters. These derivatives are then used to construct the Fisher Matrix, which is a matrix of second derivatives of the model.

3. What is the purpose of computing the Fisher Matrix numerically?

The purpose of computing the Fisher Matrix numerically is to estimate the uncertainty in the parameters of a model. This can be useful in a variety of applications, such as parameter estimation, model fitting, and optimization. It can also be used to determine the sensitivity of a system to changes in its parameters, which can aid in the design and optimization of experiments and systems.

4. What are the advantages and disadvantages of computing the Fisher Matrix numerically?

The main advantage of computing the Fisher Matrix numerically is that it can be applied to any model, regardless of its complexity or analytical form. This makes it a versatile tool for data analysis and parameter estimation. However, the numerical computation can be time-consuming and may introduce errors due to the use of approximations. Additionally, the accuracy of the results may depend on the choice of numerical differentiation method and step size.

5. Are there any alternative methods for computing the Fisher Matrix?

Yes, there are alternative methods for computing the Fisher Matrix, such as analytical methods and Monte Carlo simulations. Analytical methods involve deriving the Fisher Matrix directly from the model equations, which can be more accurate but may be limited to simple models. Monte Carlo simulations involve randomly sampling the parameter space and using the resulting data to estimate the Fisher Matrix, which can be computationally intensive but can provide more accurate results for complex models.

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