# On the parameter errors & contour plots - TMinuit

Two questions:

1. So, using TMinuit::Contour I gan get all relevant contour plots for two out of the NP parameters under minimization. Let’s Assume NP>2, so that for each contorur plot, we have at least one “hidden” dimension.
Say we have P1, P2, P3; and we get the plot of P1-vs-P2. Is this plots marginal wrt P3? Or is it maximized in the hidden dimension (P3 fixed at its best value)??

2. So, Minuit provides the parameter errors (and associated the contour plots) from the FCN as a function of the parameters around the minimum. In my guess, the uncertainty boundary are given by the parameter values which produce a change of the FCN value up to some value (in case of FCN=Chi2 and given n-sigma bounds). I wonder if this method is safe and may be applied regardless the NDoF of the problem (i.e. the number of data-points used to compute the FCN; this number is unknown to Minuit, but the ChiSquare statistical distribution depends on it).

…and question 3:
3) The 2D contour plots are made for N=1,2,3… sigmas of CL, just setting SetErrorDef(N*N), that will correspond to a given probability content P=34, 86.5, 99.9% etc. If I want make my plots for a given P instead N, is there a way to tell it to Minuit? I mean, setting P rather than N.
Or equivalently: Since N and P should be related by the cumulative Chi2 distribution, I wonder if such distribution function can be computed someheow under ROOT.

Hi,

First of all I have moved this topic in the Stat and Math Tools board.

1. The contour is obtained using the profile likelihood: for each value of your parameters of interest (P1,P2), the likelihood is maximized with respect to all other parameters.
The marginalization (integration) is a Bayesian method and can be done only when a prior distribution the other parameters is specified (if it is not known one could always assume an uniform distribution).

2. The size of the interval (e.g. at one sigma level) depends on the Fischer information matrix (i.e. inverse of the second derivative of your least square or likelihood function). More points you have automatically larger the information will be ( I is proportional to N) and therefore smaller the size of the 1-sigma interval.
The NDoF dependence is therefore already there. More points you have, the second derivative of your log-likelihood around the minimum will be larger: a steeper parabola or a smaller contour region for a fixed Delta log-likelihood value.

3)For making the plot on the probability content (it makes more sense for the 2-dimensional case), you can use this simple formula, given the desired value of prob: ( e.g. 0.68 or 0.95 ):

SetErrorDef ( ROOT::Math::chisquared_quantile( prob, 2 ) );

For the one dimensional interval instead use:

SetErrorDef ( ROOT::Math::chisquared_quantile( prob, 1 ) );

In case of log-likelihood functions you need to divide the values by 2. This comes from the fact that
the asymptotic distribution of your least square sum or 2 * log-likelihood function is a chi square.

Best Regards

Lorenzo

Thank you, Lorenzo,

Do you mean that the P3 value is different for each point of the (P1,P2) plane?
Is this choice more meaningful that other possible choices? For example, one may want to keep the same P3, fixed to its best-fit value: this should be equivalent to make a cross section of the P1-P2-P3 ellipsoid.

Is there a simple way to obtain marginalized plot (using uniform prior distributions) with Minuit?

Good to know!

Let’s say that I draw a 1-sigma contour of two parameters P1-P2 (it should be ~34%). Then, can I say that its one-dimensional X-projection (or Y-projection) corresponds to the 1-sigma error on the single parameters P1 (or P2)?
If the answer is NO, is there a way to plot the P1-P2 contour (see (1)) to have such meaning in its X-Y projections?

No, keepen the other parameter fixed to the best-fit value is not a good choice. Take for example the figure 1.1 of seal.web.cern.ch/seal/documents/ … nerror.pdf.
When making an interval of only P1 with P2 = best-fit value, the interval would be the intersection of the ellipsoid with the x- axis and not the interval A-B which is the real projection of the ellipsoid on the axis.

No, to marginalize you need to compute an integral which can be very complicated in case of many parameters. You can do yourself, using the ROOT integration classes, in case of few extra parameters, or you can use the RooStats classes, like the MCMCCalculator, for likelihood with many parameters.

[quote]
Let’s say that I draw a 1-sigma contour of two parameters P1-P2 (it should be ~34%). Then, can I say that its one-dimensional X-projection (or Y-projection) corresponds to the 1-sigma error on the single parameters P1 (or P2)? [/quote]

Yes, the projection of a 1-sigma contour in P1-P2 (the probability content is 39.3% , and it is obtained with SETErrorDef(1.) for a chi2) will give you the 1 -sigma error on P1. See again figure 1.1

Best Regards

Lorenzo