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| Original file line number | Diff line number | Diff line change |
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| [](https://travis-ci.com/MorganAskins/watchfish) | ||
| [](https://morganaskins.github.io/watchfish) | ||
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| Given $M$ components, each with an estimated rate $\vec{\beta}$ determined by a | ||
| normal distribution with uncertainty $\vec{\sigma}$, calculate the confidence | ||
| itervals and perform a hypothesis tests for each parameter $b$. | ||
| Given <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/fb97d38bcc19230b0acd442e17db879c.svg?invert_in_darkmode" align=middle width=17.73973739999999pt height=22.465723500000017pt/> components, each with an estimated rate <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/c90119f20c10a72dd5dccbfc89cd0785.svg?invert_in_darkmode" align=middle width=11.826559799999991pt height=32.16441360000002pt/> determined by a | ||
| normal distribution with uncertainty <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/3f9f71491501df368c7b0ef70db38d54.svg?invert_in_darkmode" align=middle width=10.747741949999991pt height=23.488575000000026pt/>, calculate the confidence | ||
| itervals and perform a hypothesis tests for each parameter <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/4bdc8d9bcfb35e1c9bfb51fc69687dfc.svg?invert_in_darkmode" align=middle width=7.054796099999991pt height=22.831056599999986pt/>. | ||
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| Nominally each event corresponds to a set of observables $\vec{x}$ of $N$ | ||
| Nominally each event corresponds to a set of observables <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/19e3f7018228f8a8c6559d0ea5500aa2.svg?invert_in_darkmode" align=middle width=10.747741949999991pt height=23.488575000000026pt/> of <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/f9c4988898e7f532b9f826a75014ed3c.svg?invert_in_darkmode" align=middle width=14.99998994999999pt height=22.465723500000017pt/> | ||
| measurements, for any given measurement, the probability for that particular | ||
| measurement to come from a particular components is given by | ||
|
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| $$ P_i(\vec{x}) $$ | ||
| <p align="center"><img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/21122488bdced611e7ed8bffcd42b543.svg?invert_in_darkmode" align=middle width=38.20684395pt height=16.438356pt/></p> | ||
|
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||
| The prior probability is then formed through a combination of these components | ||
| such that the total probability is | ||
|
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| $$ \mathbf{P} = \sum_i^M P_i(\vec{x}) $$ | ||
| <p align="center"><img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/33f2760534db4fe806e49280c548fc68.svg?invert_in_darkmode" align=middle width=99.53078024999999pt height=47.806078649999996pt/></p> | ||
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| The likelihood for a full data set of $N$ measurements is the product of each | ||
| The likelihood for a full data set of <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/f9c4988898e7f532b9f826a75014ed3c.svg?invert_in_darkmode" align=middle width=14.99998994999999pt height=22.465723500000017pt/> measurements is the product of each | ||
| event total probability | ||
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| $$\mathcal{L}(\vec{x}) = \prod_j^N \left( \sum_i^M b_iP_i(\vec{x}) \right) / \sum_i^Mb_i $$ | ||
| <p align="center"><img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/a364c165d0dd83eefa02e86048508140.svg?invert_in_darkmode" align=middle width=234.3143352pt height=50.399845649999996pt/></p> | ||
|
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||
| We can extend the likelihood by proclaiming that each components as well as the | ||
| sum of components are simply a stochastic process, produces the extended | ||
| likelihood: | ||
|
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||
| $$\mathcal{L}(\vec{x}) = \frac{\text{e}^{-\sum_i^Mb_i}}{N!} \prod_j^N \left( \sum_i^M b_iP_i(\vec{x}) \right) $$ | ||
| <p align="center"><img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/1624ea1a01ed7e584701d845dd89b4d7.svg?invert_in_darkmode" align=middle width=249.07579454999998pt height=50.399845649999996pt/></p> | ||
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||
| Finally, we can claim that we have _a priori_ knowledge of the parameters, | ||
| whether it be through side-band analysis or external constraints, by including | ||
|
|
@@ -37,27 +37,26 @@ the shape of that prior, we will consider the information we receive on the | |
| variables to be normally distributed and multiply the likelihood by those | ||
| constraints | ||
|
|
||
| $$\mathcal{L}(\vec{x}) = \frac{\text{e}^{-\sum_i^Mb_i}}{N!} \prod_j^N \left( \sum_i^M b_iP_i(\vec{x}) \right) \frac{1}{\sqrt{2\pi \sigma_j^2}}\text{exp}\left({\frac{-(\beta_i-b_i)^2}{2\sigma_i^2}}\right)$$ | ||
| <p align="center"><img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/3a59d3e79684a56191cc0718fda97fe6.svg?invert_in_darkmode" align=middle width=444.07050929999997pt height=57.205834949999996pt/></p> | ||
|
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||
| A few definitions to simplify things: | ||
| $$ \lambda := \sum_i^Mb_i $$ | ||
| <p align="center"><img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/5dbbdf06403eb766a9eaf09d34a85148.svg?invert_in_darkmode" align=middle width=74.26263239999999pt height=47.806078649999996pt/></p> | ||
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| Then then our objective function $\mathcal{O} = -\text{Ln}\mathcal{L}$ | ||
| Then then our objective function <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/cffa95e5b84679edc10428c782fe8e2f.svg?invert_in_darkmode" align=middle width=78.99089384999999pt height=22.465723500000017pt/> | ||
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| $$\mathcal{O} = \lambda + \text{Ln}N! -\sum_j^N\text{Ln}\left( \sum_i^M b_iP_i(\vec{x}) \right) + \sum_i^M \left( \frac{(\beta_i-b_i)^2}{2\sigma_i^2} + \text{Ln}\sqrt{2\pi \sigma_i} \right)$$ | ||
| <p align="center"><img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/4d56d8302518412d75f39e43357f5a75.svg?invert_in_darkmode" align=middle width=504.59342834999995pt height=50.399845649999996pt/></p> | ||
|
|
||
| Finally, for a counting analysis we assume that an optimal set of cuts has been | ||
| applied which optimizes the sensitivity to a particular parameter, which | ||
| simplifies the likelihood such that | ||
| $$ P_i(\vec{x}) := 1 $$ | ||
| <p align="center"><img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/a2654de0d1534690d918217913385c8e.svg?invert_in_darkmode" align=middle width=72.90990795pt height=16.438356pt/></p> | ||
| Also, because the shape of the likelihood space is independent of constant | ||
| parameters, we can drop the $\text{Ln}\sqrt{2\pi \sigma_i}$ terms. We could | ||
| also remove the $\text{Ln}N!$ term as well, but for numerical stability we will | ||
| keep it around, but use Sterling's approximation: $\text{Ln}N! \approx | ||
| N\text{Ln}N - N$. The remaining objective function we will thus use is: | ||
| parameters, we can drop the <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/96a94d4478eee20a423dcae01dfa99ba.svg?invert_in_darkmode" align=middle width=66.15028695pt height=26.045612999999992pt/> terms. We could | ||
| also remove the <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/be53185ed16aa9438e74a8f287753791.svg?invert_in_darkmode" align=middle width=38.97263864999999pt height=22.831056599999986pt/> term as well, but for numerical stability we will | ||
| keep it around, but use Sterling's approximation: <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/83b15a890ca32ebd305a780c615eec74.svg?invert_in_darkmode" align=middle width=145.38783435pt height=22.831056599999986pt/>. The remaining objective function we will thus use is: | ||
|
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||
| $$\mathcal{O} = \lambda - N\text{Ln}\lambda + N\text{Ln}N - N + \sum_i^M \left( \frac{(\beta_i-b_i)^2}{2\sigma_i^2} \right)$$ | ||
| <p align="center"><img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/fadac3b98225fbb22296866ea07908f8.svg?invert_in_darkmode" align=middle width=355.9963737pt height=47.806078649999996pt/></p> | ||
|
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||
| _Note: If the different values of $\beta$ differ by orders of magnitude, it | ||
| _Note: If the different values of <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/8217ed3c32a785f0b5aad4055f432ad8.svg?invert_in_darkmode" align=middle width=10.16555099999999pt height=22.831056599999986pt/> differ by orders of magnitude, it | ||
| might be worth forming an affine invariant form of the likelihood, otherwise | ||
| the $\text{Ln}\sqrt{2\pi \sigma_i}$ term should not matter_ | ||
| the <img src="https://rawgit.com/in [email protected]:MorganAskins/watchfish/master/svgs/96a94d4478eee20a423dcae01dfa99ba.svg?invert_in_darkmode" align=middle width=66.15028695pt height=26.045612999999992pt/> term should not matter_ | ||
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,63 @@ | ||
| # Watchfish | ||
|
|
||
| [](https://mybinder.org/v2/gh/morganaskins/watchfish/master) | ||
| [](https://travis-ci.com/MorganAskins/watchfish) | ||
| [](https://morganaskins.github.io/watchfish) | ||
|
|
||
| Given $M$ components, each with an estimated rate $\vec{\beta}$ determined by a | ||
| normal distribution with uncertainty $\vec{\sigma}$, calculate the confidence | ||
| itervals and perform a hypothesis tests for each parameter $b$. | ||
|
|
||
| Nominally each event corresponds to a set of observables $\vec{x}$ of $N$ | ||
| measurements, for any given measurement, the probability for that particular | ||
| measurement to come from a particular components is given by | ||
|
|
||
| $$ P_i(\vec{x}) $$ | ||
|
|
||
| The prior probability is then formed through a combination of these components | ||
| such that the total probability is | ||
|
|
||
| $$ \mathbf{P} = \sum_i^M P_i(\vec{x}) $$ | ||
|
|
||
| The likelihood for a full data set of $N$ measurements is the product of each | ||
| event total probability | ||
|
|
||
| $$\mathcal{L}(\vec{x}) = \prod_j^N \left( \sum_i^M b_iP_i(\vec{x}) \right) / \sum_i^Mb_i $$ | ||
|
|
||
| We can extend the likelihood by proclaiming that each components as well as the | ||
| sum of components are simply a stochastic process, produces the extended | ||
| likelihood: | ||
|
|
||
| $$\mathcal{L}(\vec{x}) = \frac{\text{e}^{-\sum_i^Mb_i}}{N!} \prod_j^N \left( \sum_i^M b_iP_i(\vec{x}) \right) $$ | ||
|
|
||
| Finally, we can claim that we have _a priori_ knowledge of the parameters, | ||
| whether it be through side-band analysis or external constraints, by including | ||
| those constraints via some prior probability. Given no specific knowledge of | ||
| the shape of that prior, we will consider the information we receive on the | ||
| variables to be normally distributed and multiply the likelihood by those | ||
| constraints | ||
|
|
||
| $$\mathcal{L}(\vec{x}) = \frac{\text{e}^{-\sum_i^Mb_i}}{N!} \prod_j^N \left( \sum_i^M b_iP_i(\vec{x}) \right) \frac{1}{\sqrt{2\pi \sigma_j^2}}\text{exp}\left({\frac{-(\beta_i-b_i)^2}{2\sigma_i^2}}\right)$$ | ||
|
|
||
| A few definitions to simplify things: | ||
| $$ \lambda := \sum_i^Mb_i $$ | ||
|
|
||
| Then then our objective function $\mathcal{O} = -\text{Ln}\mathcal{L}$ | ||
|
|
||
| $$\mathcal{O} = \lambda + \text{Ln}N! -\sum_j^N\text{Ln}\left( \sum_i^M b_iP_i(\vec{x}) \right) + \sum_i^M \left( \frac{(\beta_i-b_i)^2}{2\sigma_i^2} + \text{Ln}\sqrt{2\pi \sigma_i} \right)$$ | ||
|
|
||
| Finally, for a counting analysis we assume that an optimal set of cuts has been | ||
| applied which optimizes the sensitivity to a particular parameter, which | ||
| simplifies the likelihood such that | ||
| $$ P_i(\vec{x}) := 1 $$ | ||
| Also, because the shape of the likelihood space is independent of constant | ||
| parameters, we can drop the $\text{Ln}\sqrt{2\pi \sigma_i}$ terms. We could | ||
| also remove the $\text{Ln}N!$ term as well, but for numerical stability we will | ||
| keep it around, but use Sterling's approximation: $\text{Ln}N! \approx | ||
| N\text{Ln}N - N$. The remaining objective function we will thus use is: | ||
|
|
||
| $$\mathcal{O} = \lambda - N\text{Ln}\lambda + N\text{Ln}N - N + \sum_i^M \left( \frac{(\beta_i-b_i)^2}{2\sigma_i^2} \right)$$ | ||
|
|
||
| _Note: If the different values of $\beta$ differ by orders of magnitude, it | ||
| might be worth forming an affine invariant form of the likelihood, otherwise | ||
| the $\text{Ln}\sqrt{2\pi \sigma_i}$ term should not matter_ |