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bapowell
Science Advisor
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The \chi^2 depends on the values obtained by the data and the values predicted by the model.
bapowell said:The best-fit model minimizes the chi^2; it maximizes the likelihood. The Fisher matrix is the matrix of second derivatives of the likelihood function about some fiducial (reference) point. The Fisher matrix tells you nothing about what the best-fit parameter values are, rather, it tells you what their theoretical variances are. For this reason, the Fisher matrix is typically used as a quick and easy way of doing error forecasting -- you simply pick a fiducial model and calculate the Fisher matrix at that point. The resulting errors constitute an accurate projection only if the true parameter distributions are uncorrelated Gaussians.
So, in summary, the maximum likelihood is not related to the Fisher matrix -- the Fisher matrix is the 2nd derivative of the likelihood function about some reference point. For Gaussian distributions, the likelihood is related to the chi^2 as
[tex]\mathcal{L} \sim {\rm exp}(-\chi^2/2)[/tex]
Okay, that's good. You're almost there. Typically what is done is to write down the probability distribution as a Gaussian, and then draw contours that enclose 68% and 95% of the probability (these are the "one sigma" and "two sigma" contours). With a two-dimensional Gaussian probability distribution, these contours are ellipses.humanist rho said:I've completed upto maximum likelihood estimation.
The likelihood is obtained as gaussian.(not sure whether it is true).
But donno how to set desired confidence levels to draw the contours to estimate the parameters.Can u give any hint?
I can't get the ellipses.Chalnoth said:Okay, that's good. You're almost there. Typically what is done is to write down the probability distribution as a Gaussian, and then draw contours that enclose 68% and 95% of the probability (these are the "one sigma" and "two sigma" contours). With a two-dimensional Gaussian probability distribution, these contours are ellipses.
There are a few ways you could figure out what the ellipses are for your distribution. You could do it analytically by first figuring out what circle encloses 68% and 95% of the probability for a two-dimensional Gaussian with two independent, unit variance variables, and then performing a transformation on that to get what it looks like for your data.
Or you could do it numerically by computing the normalized values of your probability distribution in a grid, and then figuring out what level of probability makes it so that the total probability for values above that level encloses 68% and 95% of the probability, respectively. The boundary between values below and above this level makes your contour. Just bear in mind that you have to be sure to have a grid that is large enough to capture the whole distribution.
Um, okay. Maybe you can describe in more detail what you're trying to do.humanist rho said:I can't get the ellipses.
My probability distribution is in the form Exp(chi2-chimin2)/2
I've tried to draw them with varrying the two parameters omega(m) and omega(lamda).
ToyDistribution :TPDF.bmpChalnoth said:P.S. I'd start figuring out how to draw a circle from a toy probability distribution that has unit variance in two independent parameters (that is, [itex]P(x,y) \propto e^{-(x^2 + y^2)/2}[/itex]).