Section 1.

[x] L30: sensitivity ... 40% etc. 40% in what? Higgs mass? Width?

[ ] L35: likelihood function of arbitrary normalization etc. This is confusing - the likelihood function is not normalized,

Likelihood function = L(x) = p(data|x) and \int L(x) dx != 1 
Sampling distribution = f(data) = p(data|x) and \int f(data) d(data) = 1

We want to say that in past analyses, the sampling distribution was missing a normalization factor that was a function of x, such that

\int p_unnormed(data|x) * norm(x) d(data) = 1

This means that the likelihood function was missing a factor

L(x) = L_unnormed(x) * norm(x)

In the likelihood function, this is not a normalization factor, but just a missing factor that affects the shape of the function. i.e. past analyses used the wrong (shape of the) likelihood function.

[x] L45: Are we sure MadGraph cannot handle tau-decays?

Section 2.

[ ] The beginning of section 2. should mention the words maximum likelihood estimator (MLE) because hat's the name of what you are describing

[x] L70: maximizes the probability should read maximizes the probability density

[x] L90, Eq. (2): The meaning of M_\tau should be made clearer. I'd prefer e.g. M_{\tau\to\cdots} to make it clear it was a tau-decay, not just a tau leg on a Feynman diagram.

[ ] L105, Fig. (2): The plots actually show 1/\Gamma d\Gamma/dz not d\Gamma/dz I believe? The caption and/or axis-label should say this.

[ ] L120: I found the bracket enclosing the components of a four-vector confusing in the middle of a sentence, especially as the first component has an equals, like this (E=PT, P_x, P_y, 0).

[x] L125, Eq (15): What is \tilde p? I can guess, but make it explicit.

[x] L150, Eq (21). The = cases are missing from the conditions

[x] Fig (3): The units of the y-axis are presumably GeV^-1? Can this be marked? Or if the unit is arbitrary as shape is all that matters, say that in the caption.

[ ] Eq. (27): Why introduce \rho at all? Why not just put the covariance on the off-diagonals, cov(x,y), since \rho sigma(x) sigma(y) = cov(x,y)? This is much more reader-friendly.

[x] L175: Missing closing bracket f(x_b

[ ] L180: Discussion about number of dof required to parametrize the phase-spaces is confusing. For a two-body final-state one needs 2*3 - 4 = 2. For a three-body decay, 3*3 - 4 = 5. Why does the text say 3 for the latter?

[ ] L185: Eq. (44) and (64) are forward referenced. This is not reader-friendly. It should say e.g. in parenetheses (for more details see Eq. (44) and (64) of Appendix...)

[ ] L205: We have verified that Eq. (35) is correct. Since the verification is quite trivial, I don't think we need to say this.

[ ] L200, Eq. (36), (37): Eq. (36) and (37) should be removed. It's quite clear that you never need to think about transfer functions etc if you want to calculate the cross-section.

[ ] L210: I'd like more information about the numerical integration. E.g. is VEGAS a Monte-Carlo integrator? or e.g. Simpson's rule with adaptive step-sizes?

[ ] L225, Eq. (39): Why are the steps in log-space? i.e. log(m_i) = log(m_{i-1}) + const? Given that we're probably only interested in Higgs masses of with one order of magnitude, i.e. about 100 GeV, I don't see why this is necessary. Can the text just say equally spaced in log-space? It's much clearer than saying defined by a recursive relation etc.

[ ] L230: There is too much detail about the numerical methods used to maximise a function. Furthermore, I don't much like the method described. Even if you don't have analytic derivatives, a Newton's method or methods based on bisection with numerical derivatives seem preferable to scanning in mass (twice) then fitting a polynomial. Why was this chosen?

Section 3.

[ ] L310: How exactly are these values of kappa being chosen?

[x] Table 1: Extra space in [GeV ]

[ ] Table 1: Why does it show RMS/mean? Wouldn't standard deviation be much more reader-friendly?

[ ] Table 1: Where can I see the 40% number mentioned in the abstract and conclusions?

Hi Andrew,

Section 1.

L30: |sensitivity ... 40%| etc. 40% in what? Higgs mass? Width?

I will add "increases the sensitivity of the analysis for measuring the signal rate..."

L35: |likelihood function of arbitrary normalization| etc. This is
confusing - the likelihood function is *not* normalized,

|Likelihood function = L(x) = p(data|x) and \int L(x) dx != 1
Sampling distribution = f(data) = p(data|x) and \int f(data)
d(data) = 1 |

We want to say that in past analyses, the sampling distribution
was missing a normalization factor that was a function of |x|,
such that

|\int p_unnormed(data|x) * norm(x) d(data) = 1 |

This means that the likelihood function was missing a factor

|L(x) = L_unnormed(x) * norm(x) |

In the likelihood function, this is *not* a normalization factor,
but just a missing factor that affects the shape of the function.
i.e. past analyses used the wrong (shape of the) likelihood function.

hmh, I take your point, but I don't think it is really worth explaining this point in detail. It would be good if you could think about a concrete suggestion for how to rephrase the text.

L45: Are we sure MadGraph cannot handle tau-decays?

yes, I am sure.

Section 2.

The beginning of section 2. should mention the words |maximum
likelihood estimator (MLE)| because hat's the name of what you are
describing

I prefer not to use the term "likelihood" in this context, in order to avoid confusion with the likelihood based version of the SVfit algorithm (the one without proper normalization) that we used in the past.

L70: |maximizes the probability| should read |maximizes the
probability density|

|done.|

L90, Eq. (2): The meaning of |M_\tau| should be made clearer. I'd
prefer e.g. |M_{\tau\to\cdots}| to make it clear it was a
tau-decay, not just a tau leg on a Feynman diagram.

ok, I will replace |M\tau| by |M{\tau\to\cdots} everywhere.|

L105, Fig. (2): The plots actually show |1/\Gamma d\Gamma/dz|
*not* |d\Gamma/dz| I believe? The caption and/or axis-label should
say this.

yes, it should be |1/\Gamma d\Gamma/dz. I will update the y-axis label accordingly.|

L120: I found the bracket enclosing the components of a
four-vector confusing in the middle of a sentence, especially as
the first component has an equals, like this |(E=PT, P_x, P_y, 0)|.

hmh, ok. Do you have an alternative proposal for the notation, which you find more clear.

L125, Eq (15): What is |\tilde p|? Also see my longer comment.

The tilde refers to the true value of the momentum, in the frame in which the Higgs boson has zero pT. This is explained in the sentence L125+1.

L150, Eq (21). The |=| cases are missing from the conditions

fixed.

*
Fig (3): The units of the y-axis are presumably |GeV^-1|? Can this
be marked? Or if the unit is arbitrary as shape is all that
matters, say that in the caption.
yes, the units are GeV^-1. I will fix this.

*
Eq. (27): Why introduce |\rho| at all? Why not just put the
covariance on the off-diagonals, |cov(x,y)|, since |\rho sigma(x)
sigma(y) = cov(x,y)|? This is much more reader-friendly.
hmh, this is probably a matter of preference. My personal preference is to keep the current notation, as it makes it more clear that who=0 corresponds to the case that the resolutions sigmaX and sigmaY in x- and y-direction are uncorrelated.

*
L175: Missing closing bracket |f(x_b|

|fixed.|

*
L180: Discussion about number of dof required to parametrize the
phase-spaces is confusing. For a two-body final-state one needs
|2*3 - 4 = 2|. For a three-body decay, |3*3 - 4 = 5|. Why does the
text say |3| for the latter?
the 3 DoF refer to the parametrization that we explain in section 5.2 of the appendix. The point is that the momenta of the 2 neutrinos produced in tau -> mu nu nu and tau -> e nu nu decays cannot be measured individually, which reduces the number of DoF.

— Reply to this email directly or view it on GitHub https://github.com/veelken/SVfitMEM_paper/issues/3.

veelken / SVfitMEM_paper

Typos/minor corrections #3

Section 1.

Section 2.

Section 3.