Statistics for Particle Physics

Lecture 1 Introduction

Statistics links physical theories with experiments

• Theory + response of measurement apparatus = model prediction
• Observations have uncertainty on many levels
• Statistics allows us to quantify this with probablility

Definition of probability

• Kolmogorov Axioms (1933)
• Defined in set theory
• Can be compacted to three axioms
  • For all A subset S, P(A) >= 0
  • P(S) = 1
  • if A^B = empty set, P(a or b) = P(A) + P(B)
• Also define conditional probability. This cannot be derived from the first axioms

Interpretations of Probability

• The axioms provide no interpretations of the elements of the sample space.
• Frequentist statistics treat probability the limiting frequency
  • A, B, … are outcomes of an experiment
  • That can be repeated an infinite number of times
  • Probability is limiting frequency
  • What does it mean to say whether a theory is favoured or disfavoured?
    • Preferred theories predict a high probability for the data that is ‘like’ the data observed
• Bayesian or Subjective statistics treats probability as a degree of belief
  • A, B, … are hypotheses
  • P(A) = degree of belief that A is true
  • S is sometimes called the hypothesis space
  • As opposed to frequentist interpretation, bayes theroem says “if your priors wehre p, then it says how these probabilities should change in light of the data.”
  • There is NO recipe for finding the prior probabilities. This is the subjective nature of bayesian statistics.
    • You can’t enumerate all the hypotheses! (denominator)
• Both these interpretations are consistent with the Kolmogorov axioms
• Then it also satisfies Bayes theorem

Bayes theorem

• Relates P(AgivenB) to P(BgivenA)
• link the essay
• Both interpretations of statistics is consistent with bayes theorem
  • add these things

Law of total probability

• We express P(B) as a sum of P(BgivenA_i)P(A_i)
• This is often used in bayes theorem

Probability density and mass fn

• The first case is continuous, the second is discrete

Cumulative Distribution fn

• We can integrate the density fn
• Or differentiate the cumulative distribution to get the denisty fn

Expectation values

• can be used to summarise a complicated pdf
• E(X) = mu “centre of gravity” of the pdf

Lecture 2: Parameter estimation

Outcomes to probabilities - Frequentist! There are also bayesian

Hypotheses and likelihoods

• A rules that assigns a probability to each data outcome
  • P(x given H), “what’s the probability of x under the assumption of some hypothesis”
  • this is likelihood function $L(\theta )$
    • We fix x in L, so x is hidden.
  • $L$ is not a pdf for the parameter. It’s a pdf for the data!

Parameters

• of a pdf are any constants that characterise it.
• we want a function, or estimator, to estimate the parameters.
  • theta hat of x

Properties of estimators

• we want small bias
  • E(theta hat) - theta
• we want small variance
  • V(theta hat)
• These are conflicting criteria, optimising for both is difficult.

Maximum likelihood estimators

• Find theta such that theta hat is maximised.
• Equivalent to maximising log likelihood
• MLEs are not guaranteed to have optimal properties.
  • ??

Properties of MLE

• the bias is 0
• the variance is theta squared/n

Monte carlo variance

• In most cases, calculating the variance of estimators is not easy. One way is to simulate the experiment many times
• The distribution of estimates is guassian for ML in large sample limit
  • Central limit theorem

Variance of estimators from information inequality

• Sets lower bound on variance for any estimator.
• For small bias, the inequality becomes an equality.
• ⇒ MLE is ‘efficient’

Lecture 3: Hypothesis testing & confidence intervals

Suppose a measurement produces some data $x$, consider hypotheses $H_0,H_1$. We can reject or accept H0.

Setup a critical region such that the probability of finding the data there, under the hypothesis, is less than or equal to (discrete) a small probability alpha.

If $x$ is observed in the critical region, reject $H_0$.

The alternative hypothesis motivates the placement of the critical region

Test significance / goodness of fit

’Discovery’ at 5 sigma in particle physics

Lecture 4: Machine Learning

Curve fitting