Il corso sarà di 40 ore, con inizio a
20 gennaio 2015.
Programma
Modalità di esame
Visto il numero di ore e tenendo conto di una specifica richiesta
di Roma 3, l'esame consiste in due verifiche:
- una verifica scritta su una sottoparte del corso
(valida dal 26.mo al 29.mo ciclo e soggetta a cambiamento):
/dott-prob_26/programma_scritto.html)
- una presentazione sotto forma seminariale su tema concordato,
che prevedano possibilmente, ma non necessariamente, sviluppo/utilizzo
di programmi per risolvere problemi pratici o basati su toy model.
Further details
- Students who claim a credit of 40 hours have to pass
both tests (written and 'seminarial');
- Students who claim a credit of 20 hours have the choice
of the test to pass.
Note: in the case of low mark in the written exam,
some additional work will be required (typically
an extra topic to present), obviously depending
somehow on the mark (but the is no strict rule).
Date esami
- First written exam: 29 April, 16:30, Aula 2:
→ please register soon sending an email.
- First session of oral presentation: sometimes in May.
- Second written exam:
- Second session of oral presentation:
Orario
Dettaglio degli argomenti delle lezioni
- Lezione 1 (20/1/15)
- Introduction to the course:
- Issues concerning timetable, organization and exams.
- This is not a `statistics' course, but about probability
and uncertainty focused on inferential issues.
(Not a collection of formulae, or of tests
“with russian names”
)
- What is “Statistics”?
→ Lecture
at CERN (Lecture 1, sl. 5-13).
- “Claims of discoveries based on sigmas”
- First intro to R.
A foundamental question (if you do not try to answer
yourself you will not
understand the issues behind it!)
- rnorm(1, sample(1:2)[1], 0.5)
Question: from which of the two 'μ' does
the resulting number come from?
The histogram resulting from the following command can help
(or most likely confuse!) you:
- n=100000; hist(c(rnorm(n,1,0.5), rnorm(n,2,0.5)), nc=100)
- variant just to remark that statisticians have
strange ideas about
what “prob” is!
n=100000; hist(c(rnorm(n,1,0.5), rnorm(n,2,0.5)), prob=TRUE, nc=100)
- Lezione 2 (21/1/15)
- Introduction (continued)
- Which generator has produced the observed number?
(Question from Lecture 1 with R code):
- An Android game based on that problem:
- Continuing “Claims of discoveries based on sigmas”
- HASCO
Summer School, sl. 32-51.
- Again on descriptive statistics Vs Inferential statistics
(and 'interference': the question on n-1 in the
formula of the variance from a sample of observations).
- Mechanical analogy of mean and variance of distributions
(of experimental samples or of probability distributions).
- “From Observations to Hypotheses: Probabilistic Reasoning Versus Falsificationism and its Statistical Variations”,
physics/0412148v2.
Further readings ... and more (first week)
- Lezione 3 (27/1/15)
- Continuing the critical introduction.
- Deep reason of uncertainty: cause ↔ effect.
- “The essential problem of the experimental method”.
- Uncertainty about physics quantities Vs
“data uncertainty”(??)
- Uncertainty and probability (Vs frequentistic ideology).
- Further examples of misleading conclusions from p-values.
- Probability based on combinatorial evaluations and on
past frequencies (+Models!).
- Probability of observation Vs Probability of true values.
- Sources of uncertainties ('decalogo'), uncertainty, error
and true value according to ISO/NIST.
- Usual handling of uncertainties.
(We shall come back to the issue after we have learned
how to do using probability theory).
- About the meaning of “average +- sigma/sqrt(n)”.
- Some R commands.
- Generating random walks with just one (long) command line:
for(i in 1:50) {if(i>1) par(new=TRUE); plot( cumsum(rbinom(1000,1,0.5)-1/2 ), ty='l', ylim=c(-50,50), col='blue')}
(the proposed exercise
is to understand the command, and to get a feeling with randomicity).
Reference, further readings ... and more
- GdA, “Bayesian Reasoning in Data Analysis”, Chapter 1
(Based on first chapter of
CERN Yellow Report)
- GdA, “From Observations to Hypotheses: Probabilistic Reasoning Versus Falsificationism and its Statistical Variations”,
physics/0412148v2.
- Wikipedia, P-Value
[expecially 'Frequent misunderstandings' and the article
"Historical
background on the widespread confusion of the p-value:
a major educational failure" di R. Hubbard e J.S. Armstrong,
therein cited
(local copy)].
- R. Hubbard and M.J. Bayarri,
Confusion over measures of evidence (p's) versus errors (α's)
in classical statistical testing,
American Statistician, 57 (2003) 171-178:
Alternative in the case of difficulty to download the article:
- A joke from “Plato and a platypus
walk into a bar... − Understanding philosophy
through jokes”
(recensione su Scienza per Tutti):
The
scientist, his wife
and the shorn sheep (equivalent to that about the philosopher.
the physicist, the matematician... and the cow).
- Guide to the Expression of Uncertainty in Measurement (GUM),
by ISO: see here;
(Browsable version at
iso.org:
the 'decalogo' is in section
3.3.2)
- Lezione 4 (2/2/15)
- Comments about "Fisichetta" methods − restarting.
- Theory of maximum bounds and its byproducts
(according to usual teaching in Rome).
- About 'classical' confidence intervals:
- → /dott-prob_22/CI/ (not linked!)
References, further readings... and more
- GdA, Errori e incertezze di misura - Rassegna critica
e proposte per l'insegnamento
(vedi qui),
pp. 7-23 e 82-84.
- J. D. Mollon, A. J. Perkins, Errors of
judgement at Greenwich in 1796,
Nature 380, 101 - 102 (14 Mar 1996)
(for the pdf seehere, or local copy)
- Ole Rømer and his determination of the speed of light
(interessante sito in italiano)
- Dava Sobel, Longitude.
- G. Cowan, "Statistical data analysis", Chapter 9,
on "Statistical errors, confidence intervals and limits"
("/dott-prob_22/CI/"): for
self study
(we shall come back to it later).
- Do check on your preferred books and lecture notes.
- Lezione 5 (4/2/15)
- Probability of causes, probability of effects: A toy experiment.
- Still on fake claims of discoveries:
- the invented practical
joke of 1000 scientists to a journal;
- the “HERA events” and an
anti-publication.
- The Six Boxes toy experiment: what (and where) is probability?
- Probability vs Chance
('propensity' − “physical 'probability'”):
Probability, propensity and probability of propensity values.
- Ellsberg
paradox ("The Most Dangerous Man in America").
- Cognitive biases (just mentioned) and “Systems of
the Mind” (“System 1” and “System 2”).
- About the usual text book “definitions” of probability.
References, further readings... and more
- GdA, “Bayesian Reasoning
versus Conventional
Statistics in High Energy Physics”
- Tacchino induttivista
[See also the example of the inductivist turkey
(but be carefull to the last statement, “Science progresses
through falsification of old theories rather than
proving of new ones.”, on which we have already commented in the
first lecture!)]
- David Hume, An Enquiry Concerning Human Understanding
(also available at Librivox):
- Fiction Vs belief: par. 39 (search for "Nothing is more free than the imagination of man");
- On probability: par. 46 (search for "Though there be no such thing as Chance in the world").
- Daniel Kahneman, Thinking, Fast and Slow.
- Dan Ariely, A Beginner's Guide to Irrational Behavior (Coursera).
- Scott E. Page,Model Thinking (Coursera −
Just restarted this week!)
- R code to simulate n extractions from a 'box' extracted
"at random" out of six (0/1 → black/white):
n=10; rbinom(n, 1, sample(0:5)[1]/5)
(but you will never know the 'truth').
- Variation, close to the physics situation in which
you will not know the true box, but you might be interested
in a next extraction:
n=11; e<-rbinom(n, 1, sample(0:5)[1]/5); e[1:(n-1)]
Then, to see the result of the n-th extraction:
e[n]
- Two lines of R code to solve our main question
after having observed 5 times a black ball from the box:
nw=0; N=5; ( m.o <- outer(0:5, 0:5, function(x,y) dbinom(nw, N, x/5)/dbinom(nw, N, y/5) ) )
1/m.o[1,] / sum( 1/m.o[1,] )
The exercise is to
understand the code and the result
[Note: you will obtain the same result if you
replace the secondo line by "1/m.o[2,] / sum( 1/m.o[2,] )",
etc. (but not "1/m.o[6,] / sum( 1/m.o[6,] )").
Why?]
- Other examples of use of outer():
- outer(1:10, 1:10)
- outer(1:10, 1:10, '+')
- outer(1:10, 1:10, '-')
- outer(1:5, 1:5, '^')
- round( outer(1:5, 1:5, function(x,y) x^y/(x^2+y^2)), 2)
- Lezione 6 (6/2/15)
- Beliefs, probabilities and formal probability theory.
- Still on frequentistic
“Confidence Intervals” and
“Confidence Levels”
they do not imply levels of confidence
(in the human sense!):
Uncertainty and probability:
- False, true, and probable (de Finetti scheme)
- Dependence on the status of information: → probability
always conditional probability
→ subjective probability.
→ Three box problems.
- Bets, coherents bets, and basic rules of probability.
- Main rules of probability.
- Combinatorial evaluation of probability based on
equiprobable elementary events.
- Probabilistic dependence/independence.
- Uncertain numbers (not 'random' variables!)
- difference between "sample(0:9)[1]" (in R),
and the n-th digit of π?
Try for example with
Wolfram Alpha
(also app at the cost of a few Euros, but it needs
internet connection).
(Note the sloppy sintax of the command, accepted by Alpha!
Try for example with
integral of x^2 from 2 to 5: !!)
- Uncertainty on numerical quantities. Probability distributions of discrete
uncertain numbers: f(x), F(x),
expected value and (standard) uncertainty.
- Intro to MC methods.
- Example in R: intro_MC_cumsum.R
To execute the file from R console
- source("intro_MC_cumsum.R")
→ try different functions;
→ try to implement a binary search.
References, further readings... and more
- Lezione 7 (9/2/15)
- Rules of probabilities, probability distributions.
- From the last lecture:
- About the (second variant of the) three boxes problems:
- Remarks about randomness.
- Monte Carlo sampling in R:
- The chords 'experiment':
→ write a program
(in R, or your preferred programming language, or even
an app) to choose at random
the chords. using the algorithm you prefer.
- Uncertain numbers (CERN Lecture nr 3, from p. 33):
- Bernoulli trials and related probability distributions:
geometrical, binomial, Pascal.
- Summaries of probability distributions (and of frequency
distributions − pay attention!)
- Observations about combinatorial evaluation of probabilities
(Chapter 3 of lecture notes in Italian −
“può essere saltato”!)
- Coherence and basic rules of probability. In particular,
meaning of a conditional events 'E|H' and of the relation between
P(E&H), P(E|H) and P(H) [Remember that they should be under the same
background condition I:
P(E&H|I),
P(E|H,I)
and P(H|I)].
(Chapter 4 of lecture notes in Italian.)
References, further readings... and more
- R. Scozzafava, Incertezza e probabilità (Zanichelli),
1.1-1.16.
- GdA, Probabilità e incertezze di misura: Chapter 3, 4 and 6.
- GdA, Role and meaning of subjective probability: some
comments on common misconceptions, MaxEnt2000,
physics/0010064.
- Stanford Encyclopedia of Philosophy:
Interpretation
of probability.
- D. Lewis, A Subjectivist's Guide to Objective Chance
(local copy),
Published in Richard C. Jeffrey (ed.), Studies in Inductive Logic and Probability, Vol. II.
Berkeley: University of Berkeley Press, 263-293.
Reprinted with Postscripts in David Lewis (1986), Philosophical Papers. Vol. II.
Oxford: Oxford University Press, 83-132.
- A very nice page on uncertainty and probability:
Marguerite Yourcenar, Memoirs of Hadrian →
Look Inside!: p. 5.
- Lezione 8 (11/2/15)
- Probability distributions (+ comments on basic issues of probability)
- Issues from past lectures:
- Standard deviation Vs r.m.s. (expecially to particle
physics students using Root).
- Three prisoners problem.
- Bertand problem ('paradox'):
- Poisson process and related distributions: uniform Vs exponential.
Absence of memory of the geometrical and of the exponential
distributions.
- About the meaning of lifetimes ('tau'). Radiactive decays:
probabilistic decays and "continuous" variations of
the remaining nr of nuclides. Lifetimes Vs exponential
time constant of exponential low of remaining nuclides.
How to measure lifetimes.
- Probability density function. Expected value, standard uncertainty;
variation coefficient (v) to quantify the relative uncertinty.
- Standard deviation and probability intervals. Marcov theorem
and Cebicev theorem.
- Uniform distribution, triangular distributions, Gaussian distribution.
- Random number generator inverting F(x) applied to
distributions of continuous continuous variables. Examples:
- Exponential distribution (with further details on
visualizing and summarizing
the result):
- n<-100000; tau<-7; t <- -tau*log(runif(n))
- hist(t, nc=100, xlim=c(0, 5*tau), col='cyan')
- mean(t)
- sd(t)
- f(x) = k*x in the range [0,1]:
- f(θ) = k*sin(θ) in the range [0, π]:
- th <- acos( 1 - 2*runif(n) )
(It is then interesting to look at the distribution of
sin(th), cos(th), etc., an issue related with
the Bertrand 'paradox'.)
- Lezione 9 (13/2/15)
- Probability distributions. Important theorems.
- Bar plot of a probability distribution
of a discrete variable, e.g.
- n=10; p=0.4; x=0:n; fx = dbinom(x, n, p)
- barplot(fx, names=x)
- title = sprintf("Binomial(%d, %.1f)", n, p)
- barplot(fx, names=x, col='cyan', xlab='x', ylab='f(x)', main=title)
(I did not remember 'names' during the lecture...)
- Hit/miss method for sampling and integrating.
Example: MC_hit-miss.R
- Estimate of cumulative distribution from the sample (on request):
- h <- hist(xr, nc=200, col='red')
- fx.ext <- h$density
- Fx.ext <- cumsum(h$density) * (h$breaks[2] - h$breaks[1])
- barplot(fx.ext, names=h$mids)
- plot(h$mids, Fx.ext, ty='l')
- Some remarks on R language
(commands_13feb.R).
- Summary on distributions related to the Bernoulli process
(the Gamma will be derived in a future lecture).
- Expected value and variance of linear combination
of uncertain numbers.
- Expectations about future relative frequencies: Bernoulli theorem.
- Propagation of uncertainties (it is a problem of
of direct probabilities! But it can be used
in general only only if probability means how much
we believe something − think the case of true values!):
- General case; linear combunations; linearizations.
- Caveat about the "propagation of CL intervals"
(or even of sound probabilistic interval!):
“Asimmetrie:
se le conosci le eviti”
- Montecarlo propagations.
- Central Limit Theorem: formulation, deep reason, applications...
and caveat.
Central limit at work
(figure).
- A simple (standard) Gaussian generator (although limited to
a few sigma's from the mean) based on he properties of
expected value, variance and central limit theory.
- n <- 100000; x <- rep(0,n); for (i in 1:n) x[i] <- sum(runif(12)) - 6
- To change mu and sigma: x * sigma + mu
References
- GdA, CERN A.T.:
lecture 4
(till p. 30).
- GdA, "Probabilità e incertezze di misura", Parte 3.
- GdA, "Asymmetric uncertainties: sources, treatment and possible dangers",
arXiv:physics/0403086
Exercise for the "vacation" week
- Install JAGS and rjags (e.g. starting
from here, but without
being confusing going through the examples there!);
- Just run this very basic script to perform
some simulations:
rjags_SimpleMC.R
(Pay attention about the second parameter of dnorm()!
− there is a reason...)
- Lezione 10 (23/2/15)
- Multidimensional pdf's. Some important theorems. Propagation
of uncertainties.
- More on Jags/rjags. Variations of the previous example to get familiar
with hyperparameters:
The Beta and Gamma distributions will be introduced at the due time.
For the moment just try
“beta distribution 20 10” or
“gamma distribution 10 0.2” on Wolfram Alpha
(also to learn, once more, that a probability distribution
might have paremeters defined differently in different envirenmonts!).
- Multivariate distributions. Marginal, conditional. Covariance and
correlation coefficience.
- More on linear combinations: expected value and variance.
- More on Central Limit Theorem and applications.
- Critical remarks about "large number laws"
and "the most misinterpreted theorem of probability theory"
(Bernoulli theorem, nothing more!).
- General rules to propagate uncertainties for discrete
and continuous variables.
- Bertrand 'paradox' and, expecially,
pdf of functions of variables:
VERY PRELIMINARYY draft (pdf)
(updated 9 march − Android app also updated.).
- Lezione 11 (27/2/15)
- More on transformation of variables. Monte Carlo. Beginning
probabilistic inference
- Covariance between linear combinations of variables:
→ from the covariance matrix of the input quantities
to the covariance matrix of the output quantities,
using the (rectangular) matrix of the coefficients.
- Linearization (left as home work).
An exercise (try to do it!):
- Information about a nominal A4 paper (input, 'X'):
a=29.53 cm, with σ(a)=0.03 cm;
b=21.45 cm, with σ(b)=0.04 cm;
assuming a Gaussian model a no correlation
[ρ(a,b) = 0]
- Quantities of interest (output, 'Y'):
- perimeter p=2×(a+b);
- area A = a×b;
- diagonal d = \sqrt(a^2+b^2);
- difference Δ=a-b.
- Find expected value of Y;
- Find covariance matrix of Y.
(In case of difficulties start only with p and
Δ, being the transformations linear.)
- Variants:
- How does the result change if we assume ρ(a,b) = 0.6?
- How does the result change if we assume that the value
of the diagonal is exactly 36.616 cm?
(Do it for both hypotheses about ρ(a,b).)
Note Answering to this last question is not that simple,
but at least try it! How would you proceed?
- A 'curious' tranformation: Y=F_{X}(X), very important
for simulations!
- Monte Carlo estimation of π
- points uniformely distributed on a plane, using e.g.
R runif();
- a curious method to "throw stones" (VERY first intro
to MCMC).
- Again on Bernoulli process and related phenomena:
random walk in different spaces; Brownian motion;
Maxwell distribution.
- Rayleigh distribution: special case with s=1, its relation
to the Gaussian distribution and a trick to make
`the best' Gaussian random number generator.
- Learning from data using probability theory:
Bayes Theorem and application to the six box
'toy experiment'
References, further readings... and more
[See previous years]
- Lezione 12 (2/3/15)
- Linearization. Multivariate distributions. Inference.
- Propagation of expected values, variances and covariances
of transformations linearized around the expected values
(and not around the modes!). Sensitivity coefficients.
Propagation of relative uncertainties (under the same hypotheses)
in the case on monomial expressions.
- Multinomial distribution: joint distribution, marginals,
correlation coefficients.
- Example: how probable was a particular
histogram?
- n=10000; N=10; p=0.5; x=rbinom(n,N,p); t=table(x); barplot(t)
- dmultinom(as.vector(t), n, dbinom(0:N, N, p))
- Alternative way to extract a "histogram at random"
(if we are not interested in the history of the extractions):
- n=10000; N=10; p=0.5; pm=dbinom(0:10, N, p); h=as.vector(rmultinom(1, n, pm))
- barplot(h, names=0:N)
- dmultinom(h, n, pm)
(All real events had very little probability!!)
- Bivariate Gaussian distribution: joint distribution, marginal,
conditional (expected value shifted; sigma squized).
- How to make use of the formula of the conditional
to generate values according to a generic
"normal bivariate" (i.e. a 2D Gaussian):
- n=10000; mx=10; sx=2; my=5; sx=1; rho=-0.6
- x=rnorm(n, mx, sx); y=rnorm(n, my+rho*sy/sx*(x-mx), sy*sqrt(1-rho^2))
- plot(x,y); cor(x,y)
- Conditional probability and Bayes' theorem. Examples.
Particle identification.
- Some problems:
- Three boxes with GG GS and SS rings. Choose at random a Golden
ring. Where it is better to make a second extraction
(in order to get again G)?
- Box with 1 ball, with P(B)=P(W)=1/2; add one white;
shake and extract: → white. What is the colour
of the ball inside?
References, further readings... and more
[See previous years]
- Lezione 13 (9/3/15)
- Inference, inference, inference (and prediction....)
- Cern AT, part 4, pp. 37-41
- Cern AT, part 5, pp. 4-11
- Paper on Bertrand 'paradox': expecially Appendix C on
inferential aspects
- Seminario a Roma 3 (feb 2012), pp. 29-35
- Extending the six-box problem: graphical model → Hugin.
- Several physics applications reviewd in terms of graphical
models: Berboulli trials (binomial models), Poisson processes
with background, fits (of various kinds, includinf systematics),
unfolding.
- Seminario a Roma 3, pp. 36-41, 45-46
- Playing with the Bayes' billard
- Seminario a Roma 3, pp. 47-51
- Seminar at Garching on "Probabilty, propensity and probability of propensity"
References, further readings... and more
[See previous years]
- More on p-values and significance:
^{(*)}In Physics it is even worst: none of the
claims of discoveries of NEW phenomena based on statistical significal
significance of the last 3-4 decades
was indeed a real discovery [The Higgs at LHC is not
something 'new'!]
- Lezione 14 (12/3/15)
- Parametric inference: binomial and Poisson model.
- Old problem: three boxes with two rings.
- Inference of p in a binomial model.
Extreme cases (x=0, x=n). Conjugate prior: beta distribution.
- Inference of λ of a Bernoulli process.
Special case of x=0. Conjugate prior: Gamma distribution.
- Gamma, Erlang and exponential distributions.
- Chi2 distribution as a special Gamma.
References, further readings... and more
[See previous years]
- Android app Probability distribution, a usefull vademecum.
- How to use R function integrate(), using a binomial model
with a conjugate beta:
- r.i = 1; s.i = 1; n= 100; nS = 60; nF=n-nS
- integrate(function(p) dbeta(p, r.i+nS, s.i + nF), 0, 1) # to check normalization
- Ep <- integrate(function(p) p*dbeta(p, r.i+nS, s.i + nF), 0, 1)[[1]]
- Ep2 <- integrate(function(p) p^2*dbeta(p, r.i+nS, s.i + nF), 0, 1)[[1]]
- sigma = sqrt(Ep2-Ep^2)
(Although in this case one can use the closed expressions of
expected value and variance.)
- Using a custom prior. For example measuring an efficiency
we could know that p(=ε) cannot exceed the geometric
efficiency of e.g. 98%, and our initial
beliefs are concentrated around 95% with a few percent
standard uncertainty. Let us imagine we get 100 successes over
100 trials:^{(*)}
- prior <- function(p) ifelse(p>0.98, 0, dbeta(p, 40, 3))
- norm <- integrate(prior, 0, 1)[[1]]
- n= 100; nS = 100; nF=n-nS
- u.posterior <- function(p) prior(p) * dbeta(p, 1+nS, 1+nF)
- norm <- integrate(u.posterior, 0, 1)[[1]]
- posterior <- function(p) prior(p) * dbeta(p, 1+nS, 1+nF) / norm
- integrate(posterior, 0, 1)
- Ep <- integrate(function(p) p*posterior(p), 0, 1)[[1]]
- sigma <- sqrt(integrate(function(p) p^2*posterior(p), 0, 1)[[1]] - Ep^2)
- integrate(posterior, 0.95, 1)
^{(*)}The parameters of the prior beta were found
by attempts using the forementioned vademecum
see screenshot
[the mode of the prior is 39/(39+2)=0.95].
- “I am not a Bayesian”
- Lezione 15 (16/3/15)
- Parametric inference: Gaussian model.
- Learning from data: role of models and priors
(do not believe the "Immaculate Observation").
- Inferring μ of a Gaussian given σ.
- Uniform prior; vague prior modelled by a Gaussian
(the Gaussian is `auto-conjugated').
- Inferring μ from a sample of independent observations:
sufficiency of the aritmetic average.
- The Gauss derivation of the Gaussian, ... and the
“Fermi's Bayes theorem”.
- Simultaneous inference on μ and σ.
- Predictive distribution.
- Case of observation at the edge or beyond the physical
region (“negative neutrino mass”).
- Comments on the Antropic Principle from a probabilistic
perpective:
- Calling M_{i} a cosmological model,
if P(Mankind|M_{i}) = 0, then
M_{i} is ruled out,
- All other speculations are methaphysics (in the bad sense of the term).
- An enlighting article on the subject:
What is the Ant, Sir? by Bernard Leikind (local pdf file).
References, further readings... and more
[See previous years]
- Lezione 16 (19/3/15)
- More on inference. Systematics. Bayesian networks.
- Simultaneous inference on μ and σ of a Gaussian model.
"Probabilità e incertezze di misura": Parte 4, from p.322
- Effects of systematic errors:
- Computational issues (introduction, up to importance sampling):
- OpenBUGS, a first overview.
- Lezione 17 (20/3/15)
- Applications of probability theory in data analysis.
- More on the way to handle systematics.
- ISO/BIPM recommendations on Type A and Type B uncertainties.
- Evaluation effects of systematics on results of measurements
using the propagation around the nominal values of
influence parameters (detailed example on offset and scale
uncertainties, included evaluation of correlations).
- Pay attention to the use of covariance matrix
if it comes from a linearization!!
- Systematic effects on the upper limits of intensities
of Poisson processes evaluated from counting experiments.
Special case having observed x=0 with uncertainty
on 'integrated luminosity' (the factor needed to go from
f(λ|x)
to f(r|x)).
- Predictive distributions in the case of binomial and Poisson model.
- Fits: introduction and very general scheme.
(In the probabilistic approach a fit is just parametric inference
on the parameters of the model.)
References, further readings... and more
[See previous years]
- Lezione 18 (23/3/15)
- Fits
- From the general scheme to the usual hypotheses
(errors only un vertical axes, independency, normality).
- Linear fit with unknown σ's.
- Linear case with known σ's and flat priors:
posterior f(m,c): normal bivariate.
- Recovering some 'usual methods' (at least as far
'best values' are concerned) as particular cases under
some special conditions (maximum likelihood; least squares).
- Estimation of the parameter under the hypotheses
of multivariate normal posterior:
minimization of χ^{2}; "Δχ^{2}=1"
rule;
inverse of covariance matrix from second derivatives.
- Detailed example of linear fit using the above approximations.
- Caveat about the use of frequentistic 'prescriptions'.
References, further readings... and more
[See previous years]
- GdA, “Fits, and especially linear fits, with errors on both axes, extra variance of the data points and other complications”
- GdA, “Le basi del metodo sperimentale”, Parte 4, Cap. 12,
vedi qui.
- Per metodi approssimativi per valutare le incertezze
dovute a errori sistematici di zero e di scala:
GdA, “Errori e incertezze di misura - Rassegna critica e proposte per l'insegnamento”, par. 23, vedi qui.
- Lezione 19 (27/3/15)
- Least square fits. MCMC applied to inferential/predictive problems
- Least square fits of linear models using linear algebra methods.
- Short intro to MCMC.
- Metropolis algorithm to sample an unnormalized pdf. Metropolis-Hasting
variation.
- Simulated annealing.
- Gibbs sampling.
- Example with OpenBUGS: inference of μ and σ in a Gaussian
model and prediction of a next observation:
- Example to make a linear fit, including extrapolation:
(Data generated with genera_dati.R,
which uses writeDatafileR, taken from the
WinBugs & R webpage of Alicia Carriquiry. [local backup file])
- Same fit in Jags/rjags, using same model, data and inits files:
- [ Variant with name of of `data' and `inits' lists
defined in the files: fit_jags0.R;
fit_data_J.txt;
fit_inits_J.txt ]
- To do data simulation and fit in a single file:
generate_and_fit.R
(Other examples here)
- Using least square formulae on the same points
- fit_least_squares.R
- Exercise: make the comparison
of Jags Vs least squares (also on the extrapolated point)
References, further readings... and more
[See previous years, in particular
Lezioni
20-21, 22.mo ciclo ]
- Lezione 20 (30/3/15)
- Last (but not least) issues
- Using MCMC chains for all previsions/inferences
depending on the model parameters.
- Playing with multivariate normals:
- implicit costruction of the joint distribution of
all model parameters and of observations in sevaral cases;
- 'multiconditional' distributions;
- inference from a sample, assiming σ known;
- effect of offset systematic;
- inferences and predictions from averages;
- role of constrains amobd quantities;
- propagation of evidence.
(details left to self study).
- Bayesian (multidimentional) unfolding;
- Model comparison and natural Ockam's razor;
- Which priors in frontier physics?
- Conclusions.
References, further readings... and more
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