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Exercise 4: Compute the test statistic

For each data-set we can compute the Likelihood ratio test statistic, defined as

\(X = −2{\rm Log}(Q)\), with \(Q = \frac{L(µ = 1)}{L(µ = 0)}\)

, where the likelihood is the same as we have used in previoous exercises:

\({\rm -2 \cdot Log(Likelihood)} = -2\sum_{\rm bins} {\rm Log(Poisson(n |   \alpha \cdot f_{\rm SM}) + \mu \cdot f_{\rm Higgs} )}\).

Note that in this exercise we will not scale the background, so \(\alpha=1.00\). Also note that we do not compute the likelihood directly, but -2Log(Likelihood) and it is also convenient to use the fact that log(a)/log(b) = log(a)-log(b).

Code you could use form the skeleton code:

  • Get_TestStatistic()

Exercise 4a)

Write a function that computes the likelihood ratio test-statistic for a given data-set (h_mass_dataset) from the expected ’template’ distributions for the background and the signal, also histograms. You can start with the routing Get_TestStatistic() in the skeleton code.

Note: we will use this routine extensively in Exercise 5, so ask the function to return the value of the test statistic.

Exercise 4b)

Use the function in exericse 4a) to compute:

  • Compute \(X_{\rm data}\) , the value of the likelihood ratio test-statistic for the data
  • Compute \(X_{\rm data}\) , the value of the likelihood ratio test-statistic for the SM hypothesis histogram
  • Compute \(X_{\rm data}\) , the value of the likelihood ratio test-statistic for the SM+Higgs hypothesis histogram