Cowan - Aachen14 - 1 - Statistical Methods For Particle Physics
Cowan - Aachen14 - 1 - Statistical Methods For Particle Physics
Cowan - Aachen14 - 1 - Statistical Methods For Particle Physics
Graduierten-Kolleg
RWTH Aachen
10-14 February 2014
Glen Cowan
Physics Department
Royal Holloway, University of London
g.cowan@rhul.ac.uk
www.pp.rhul.ac.uk/~cowan
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Outline
1 Probability
Definition, Bayes theorem, probability densities
and their properties, catalogue of pdfs, Monte Carlo
2 Statistical tests
general concepts, test statistics, multivariate methods,
goodness-of-fit tests
3 Parameter estimation
general concepts, maximum likelihood, variance of
estimators, least squares
4 Hypothesis tests for discovery and exclusion
discovery significance, sensitivity, setting limits
5 Further topics
systematic errors, Bayesian methods, MCMC
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A definition of probability
Consider a set S with subsets A, B, ...
Kolmogorov
axioms (1933)
From these axioms we can derive further properties, e.g.
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Interpretation of probability
I. Relative frequency
A, B, ... are outcomes of a repeatable experiment
Bayes theorem
From the definition of conditional probability we have,
and
but
, so
Bayes theorem
Ai
B Ai
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posterior probability
i.e. youre probably OK!
Your viewpoint: my degree of belief that I have AIDS is 3.2%
Your doctors viewpoint: 3.2% of people like this will have AIDS
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).
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Expectation values
Consider continuous r.v. x with pdf f (x).
Define expectation (mean) value as
Notation (often):
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If x, y, independent, i.e.,
, then
x and y, uncorrelated
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Correlation (cont.)
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Error propagation
Suppose we measure a set of values
and we have the covariances
which quantify the measurement errors in the xi.
Now consider a function
What is the variance of
The hard way: use joint pdf
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or in matrix notation
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where
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y(x)
y
x
y(x)
?
N.B. We have said nothing about the exact pdf of the xi,
e.g., it doesnt have to be Gaussian.
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with
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Some distributions
Distribution/pdf
Binomial
Multinomial
Poisson
Uniform
Exponential
Gaussian
Chi-square
Cauchy
Landau
Beta
Gamma
Students t
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Binomial distribution
Consider N independent experiments (Bernoulli trials):
outcome of each is success or failure,
probability of success on any given trial is p.
Define discrete r.v. n = number of successes (0 n N).
Probability of a specific outcome (in order), e.g. ssfsf is
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random
variable
parameters
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Multinomial distribution
Like binomial but now m outcomes instead of two, probabilities are
nm of outcome m.
This is the multinomial distribution for
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Example:
represents a histogram
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Poisson distribution
Consider binomial n in the limit
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Uniform distribution
Consider a continuous r.v. x with - < x < . Uniform pdf is:
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Exponential distribution
The exponential pdf for the continuous r.v. x is defined by:
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Gaussian distribution
The Gaussian (normal) pdf for a continuous r.v. x is defined by:
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For n = 2 this is
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Landau distribution
For a charged particle with = v /c traversing a layer of matter
of thickness d, the energy loss follows the Landau pdf:
+ - + -
- + - +
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Beta distribution
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Gamma distribution
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Student's t distribution
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sequence repeats
Choose a, m to obtain long period (maximum = m - 1); m usually
close to the largest integer that can represented in the computer.
Only use a subset of a single period of the sequence.
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Require:
i.e.
That is,
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set
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A simulated event
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Next time...
Today we have focused on probabilities, how they are
defined, interpreted, quantified, manipulated, etc.
In the following lecture we will begin talking about statistics,
i.e., how to make inferences about probabilities (e.g., probabilistic
models or hypotheses) given a sample of data.
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Extra slides
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Multivariate distributions
Outcome of experiment characterized by several values, e.g. an
n-component vector, (x1, ... xn)
joint pdf
Normalization:
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Marginal pdf
Sometimes we want only pdf of
some (or one) of the components:
marginal pdf
x1, x2 independent if
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Marginal pdf ~
projection of joint pdf
onto individual axes.
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Conditional pdf
Sometimes we want to consider some components of joint pdf as
constant. Recall conditional probability:
conditional pdfs:
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Basically treat some of the r.v.s as constant, then divide the joint
pdf by the marginal pdf of those variables being held constant so
that what is left has correct normalization, e.g.,
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Example:
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and a function
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(Mellin convolution)
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exist.
For e.g.
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integrate
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