加载中…(个人笔记）Profile likelihood confidence interval
(2015-01-03 23:14:37)
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佛学 |
我们用来算LC的误差的，现在终于弄懂了。
profile likelihood就好像普通likelihood,只是有多个参数，当中很多是没有用的(nuisance parameters)。
x = events in the signal region
y = events in data sidebands (or from MC), measured with some uncertainty, statistical and systematic
z = measurement of the efficiency, measured with
some uncertainty, statistical and systematic
→ How do we set limits on the signal rate( = flux in LC)?
profile likelihood就好像普通likelihood,只是有多个参数，当中很多是没有用的(nuisance parameters)。
x
y = events in data sidebands (or from MC), measured with some uncertainty, statistical and systematic
z = measurement of the efficiency,
→ How do we set limits on the signal rate( = flux in LC)?
Profile Likelihood
x – number of events in signal region
y - number of events in data sidebands
τ – relative “size” of background region to signal region, so that y/ τ is estimate of background in signal region
m = number of MC events to test efficiency
z = number of MC events that survive the cuts
--> z/m = an estimate of the efficiency
Unknown Parameters:
μ – signal rate (what we want to know)
b – background rate in signal region
e – efficiency
x – number of events in signal region
y -
τ – relative “size” of background region to signal region, so that y/ τ is estimate of background in signal region
z = number of MC events that survive the cuts
--> z/m = an estimate of the efficiency
Unknown Parameters:
μ – signal rate (what we want to know)
b – background rate in signal region
e – efficiency
Probability Model:
X ~ Pois(eμ+b), Y ~ Pois(τb), Z ~ Binom(m,e)
Loglikelihood:
l(μ,b,e) = (-2)* (xlog(eμ+b)-log(x!)-(eμ+b) +
ylog(τb)-log(y!)-τb +
log(m!)-log(z!)-log((m-z)!)+zlog(e)+(m-z)log(1-e))
is a function of all parameters
Idea: for each μ find b and e which make the observations most likely – profile likelihood
X ~ Pois(eμ+b), Y ~ Pois(τb), Z ~ Binom(m,e)
Loglikelihood:
l(μ,b,e) = (-2)* (xlog(eμ+b)-log(x!)-(eμ+b) +
ylog(τb)-log(y!)-τb +
log(m!)-log(z!)-log((m-z)!)+zlog(e)+(m-z)log(1-e))
is a function of all parameters
Idea: for each μ find b and e which make the observations most likely – profile likelihood
Result: given the data (x,y,z,τ,m) the profile
likelihood is a function of μ alone
Since loglikehood follows chi-squared distribution, if you want to have a certain % confidence level, look for the value of the chi square. For the following, for a 95% confidence interval, from https://people.richland.edu/james/lecture/m170/tbl-chi.html, look for 1 d.o.f., 0.1, you get a value of 2.701, then you can the upper and lower boundaries.
For different combinations of b and e, the signal rate is different. You want to find out of n tests, 95% of the time that the true rate is in that region.
"A method is said to yield a 100(1-alpha)% confidence interval if, were the experiment tobe repeated many times, the resulting intervals would include ( or cover) the true parameter at least 100(1 - alpha)% of the time, no matter what the true parameter is." - Limits and Confidence Intervals in the Presence of Nuisance Parameters (W. Rolke， 2009）
![]( "(个人笔记）Profile <wbr>likelihood <wbr>confidence <wbr>interval")
参考
http://warnercnr.colostate.edu/~gwhite/mark/markhelp/profile_likelihood_confidence_intervals.htm
http://www.unc.edu/courses/2010fall/ecol/563/001/docs/lectures/lecture8.htm#profileci
http://www.math.umt.edu/patterson/ProfileLikelihoodCI.pdf
Since loglikehood follows chi-squared distribution, if you want to have a certain % confidence level, look for the value of the chi square. For the following, for a 95% confidence interval, from https://people.richland.edu/james/lecture/m170/tbl-chi.html, look for 1 d.o.f., 0.1, you get a value of 2.701, then you can the upper and lower boundaries.
For different combinations of b and e, the signal rate is different. You want to find out of n tests, 95% of the time that the true rate is in that region.
"A method is said to yield a 100(1-alpha)% confidence interval if, were the experiment tobe repeated many times, the resulting intervals would include ( or cover) the true parameter at least 100(1 - alpha)% of the time, no matter what the true parameter is." - Limits and Confidence Intervals in the Presence of Nuisance Parameters (W. Rolke， 2009）
参考
http://warnercnr.colostate.edu/~gwhite/mark/markhelp/profile_likelihood_confidence_intervals.htm
http://www.unc.edu/courses/2010fall/ecol/563/001/docs/lectures/lecture8.htm#profileci
http://www.math.umt.edu/patterson/ProfileLikelihoodCI.pdf
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