lezioni per i Dottorati di Ricerca in Fisica (31

(G. D'Agostini)

Ultima lezione: ven 15, ore 10:00, Aula Atlas.

Il corso sarà di **40 ore**, con inizio
**11 gennaio 2016, ore 16:00**

- Programma indicativo.
- Per ulteriori indicazioni vedi i corsi degli anni precedenti
(in particolare quelli evidenziati in
**grassetto**)- 21.mo ciclo,
**22.mo ciclo**23.mo ciclo, 24.mo ciclo, 25.mo ciclo, 26.mo ciclo, 27.mo ciclo,**29.mo ciclo**, **30.mo ciclo**(list of contents in English)

- 21.mo ciclo,
- Nota: alcuni argomenti e applicazioni potranno dipendere dagli interessi dei dottorandi.

- 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.

Nr. | Giorno | Orario | Aula |

1 | Lun 11/01 | 16:00-18:00 | Rasetti |

2 | Gio 14/01 | 14:00-16:00 | Rasetti |

3 | Lun 18/01 | 16:00-18:00 | Rasetti |

4 | Mar 19/01 | 16:00-18:00 | Rasetti |

5 | Mer 20/01 | 16:00-18:00 | Rasetti |

6 | Lun 25/01 | 16:00-18:00 | Rasetti |

7 | Mer 27/01 | 16:00-18:00 | Rasetti |

8 | Ven 29/01 | 16:00-18:00 | Rasetti |

9 | Lun 1/02 | 16:00-18:00 | Rasetti |

10 | Mer 3/02 | 16:00-18:00 | Rasetti |

11 | Ven 5/02 | 16:00-18:00 | Rasetti |

12 | Lun 7/03 | 16:00-18:00 | Rasetti |

13 | Gio 10/03 | 10:00-12:00 (in punto) | Sala Riunioni ATLAS (Stanza 232) |

14 | Lun 14/03 | 16:00-18:00 | Rasetti |

15 | Gio 17/03 | 10:00-12:00 (in punto) | Sala Riunioni ATLAS(Stanza 232) |

16 | 31 mar | ||

17 | 4 apr | ||

18 | 15 apr |

**Lezione 1 (11/1/16)****Introduction to the course:**- Entry self-test.

Only statistical results will be shown. - 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. 5-8).

- C'è statistica e statistica
- “
*An unsophisticated forecaster uses statistics as a drunken man uses lamp-posts − for support rather than for illumination*” (Andrew Lang, cited in Famous Forecasting Quotes) - La statistica secondo Trilussa

- “Claims of discoveries based on sigmas”
- HASCO
Summer School, sl. 1-20.

(Vedi anche Incontri di Frascati 2012, incluso video e supporto R)

- HASCO
Summer School, sl. 1-20.
- A recent,
**case (LHC, Dec. 2015)**

- Entry self-test.
**Lezione 2 (14/1/15)****R language short tutorial. More on fake discoveries based on “statistics”**- Continuation of the entry test: → on reading errors,
and uncertainties on digital readings.
- An Android app to check your personal(!) skill in reading
analog instruments (in steady conditions):
ErroriLettura.apk

No dogmatism, please! - Other apps from Google app store:
- Vernier Caliper Simulator (try to read the tenths without using the vernier)
- Length Fraction Calculator, another useful app to check the ability to read the tenths (and less).

- An Android app to check your personal(!) skill in reading
analog instruments (in steady conditions):
ErroriLettura.apk
- P-values and ... supercazzole

→ See exchange of mails in the pdf file posted on the web page (not linked). - First intro to the R language
- Some examples from R
- Producing Simple Graphs with R
- RStudio

- About falsification and its "statistical variations": p-values.
- HASCO Summer School, sl. 21-42.

- Some issues of the lecture in 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?

- 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)

- rnorm(1, sample(1:2)[1], 0.5)
- esempi_IdF2012.R
- gaussiane_IdF2012.R
- chi2_IdF2012.R

- A
- R at Coursera (running courses, or starting soon, or still available):
- Data Science Specialization [10 short courses
^{(*)}]: - Mathematical Biostatistics Boot Camp 1 (perhaps still available)
- Statistics One
- https://www.coursera.org/specializations/genomic-data-science

*might*help in getting a job. - Data Science Specialization [10 short courses
- What can you do with R? Quite a lot.
^{(*)}Here is the the alphabetic list of the MANY packages available

^{(*)}But most R packages are written in C! So those who wrote R or contribute to it didn't do because they were/are unable to program in C, but because doing normal work in C is heavy and very inefficient.

- Continuation of the entry test: → on reading errors,
and uncertainties on digital readings.
**Lezione 3 (18/1/16)****More on p-values. Measurements, uncertainty, probability.**- P-values
- What p-values do not mean
- Doing mistakes (and cheating!) with p-values:
- “A lot of what is published is incorrect” (Lancet editorial)
- p-hacking, or cheating on a p-value (and how R can help...)
- Significant (at xkcd - well done!)
- P > 0.05? I can make any p-value statistically significant with adaptive FDR procedures
- The Ease of Cheating With Statistics
- False-positive psychology: undisclosed flexibility in data collection and analysis allows presenting anything as significant.

- “Instead of heeding impressive-sounding statistics, we should ask what scientists actually believe”
- Psychology Journal Bans Significance Testing

*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)- About errors in reading analog instruments (and related topics):
- R code to show the effect of rounding
- residui_da_lettura.R:
- residui_da_lettura.eps,
- residui_da_lettura.png (converted with 'convert', although direct png from R is preferable)

- ErroriLettura.apk (reproposed)
- GdA,
*Errori e incertezze di misura - Rassegna critica e proposte per l'insegnamento*(vedi qui), pp. 7-23 e 82-84.- Android app to check your ability to interpolate between marks: ErroriLettura.apk (screenshot).

- 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.

- R code to show the effect of rounding
- About
*Frequentistic*“Confidence Intervals” and “Confidence Levels” that theyimply**do not***levels of confidence*(in the human sense!):- G. Cowan, "Statistical data analysis", Chapter 9,
on "Statistical errors, confidence intervals and limits"
("
**/dott-prob_22/CI/**" -- not linked: go to the Ciclo 22 and guess how to reach the web page): for(we shall come back to it later).__self study__ - “Definizione dell'intervallo di confidenza” (da dispense in italiano, autori omessi...).
__Do check__on your preferred books and lecture notes.- About the meaning of upper/lower limits (e.g. the famous
pre-LHC M
_{H}> 114 GeV at 90% C.L. - Why the (tens of) thousand confidence limits of the last 3-4 decades to not agree with expectations from coverage?
- Neyman about the
*confidence intervals*(“Carry out your experiment... ”)

(Fisher used to call the confidence intervals*“that technological and commercial apparatus”*) - The ultimate confidence intervals calculator (Android version: Ultimate_CI_Calculator.apk).

- G. Cowan, "Statistical data analysis", Chapter 9,
on "Statistical errors, confidence intervals and limits"
("

**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 (available here; local copy)

Additional matter on the subject;:- P-values are not error probabilities (Hubbard and Bayarri)
- Why We Don’t Really Know What “Statistical Significance” Means: A Major Educational Failure (Hubbard and Scott Armstrong)
- Why P Values Are Not a Useful Measure of Evidence in Statistical Significance Testing (Bayarri and Lindsay).
- Further Wikipedia references

- Farneticante (per diversi aspetti)
lettera aperta al direttore del CERN, apparsa
il 15 gennaio su Il Fatto Quotidiano:
a parte i diversi punti discutibili, sui quali non entro,
**Esercizio**: trovare i**due**errori nella frase “Per semplificare, dire che il bosone di Higgs è stato osservato con intervallo di confidenza 6 sigma, vuol dire che c’è una probabilità di 4 miliardesimi che il bosone di Higgs non esista.”**Soluzione**.

- P-values
**Lezione 4 (19/1/16)****Continuing on measurament, uncertainty, probability**- Arbitrary probabilistic inversions implicit in the construction
of frequentistic 'confidence intervals', that
__do not__provide a "confidence". [Very*insightful*quote from Mathematical Biostatistics Boot Camp 1 (Lecture 8 video, starting at about the 20th minute):*“ If you happen to be taking a statistics test, there's a trick to get around this mental games associated with fictitious repetitions of experiments and so on. The trick is just to say we're 95% confident the interval contains mu, and statisticians have said well that's enough hedging to count as a legitimate instance of the strict definition. So, if you're taking a statistics test don't say there's a 95% chance that the interval that I just calculated contains mu, cuz your teacher might yell at you about that. But if you say you're 95% confident, they'll begrudgingly give you credit.”*-- see here for the complete, official transcript of the video.]- The dog and the hunter.
- Implicit assunsions and case in which they do not hold.
- Objective methods are often arbitrary and unjustified (and often the solveproblems difference from those practitioners have).
- Pure empirical information is not Science!

Concerning the*misterious*sqrt(2), try to play with radice_di_due.R. - Remarks of standard (old?) analysis methods in first years
laboratory:
- rule of the “half scale spacing”;
- “theory” of maximum bounds (and their propagation);
- “always draw error bars in plots!”;
- lines of minimum and maximum slope;
- on the propagation of the “statistical”
errors:
- contradiction with “standard definition of probability” (that is apply probabilistic formulae to object about which probability statements are esplicitly forbitten);
- issues of evaluating the sigmas (it is not necessary to make many measurements!);
- oversimplifications due to not taking into account correlations;
- and, anyway, linearization is sometimes a bad assumption.

- Learning from data, learning about models and their parameters, forecasting future observations: the inferential predictive process.
- Deep source of uncertainty: causal links. The essential problem of the experimental method.
- From true values to observations... and back.
- A simple experiment (with most issues of real
experiments): the
**six box problem**. - Comparison of probability of black/white from box
known composition wrt box of unkown composition:
**Ellsberg paradox**(name not mentioned during the lecture). - What is probability?
- Probability and bets.

**References**- GdA,
*Errori e incertezze di misura - Rassegna critica e proposte per l'insegnamento* - GdA,
*Introducing Bayesian Reasoning in Measurements with a Toy Experiment* - GdA, 2005 CERN Academic Training (second lecture);
- GdA,
*Bayesian reasoning*(or CERN Yellow Report): Chapter 1. - xkcd comics on betting:
- Did the sun just explode?;
- (Opera's) Neutrinos

- And another one to remind that
*correlation does not necessarly imply causation*:

- Arbitrary probabilistic inversions implicit in the construction
of frequentistic 'confidence intervals', that
**Lezione 5 (20/1/16)**- Playing with R. “Statistica.”
Fundamental aspects of probability.
- Some training with R
- Remarks on descriptive statistics, and reminders on the different meanings of "statistica" (particularly in Italian, see C'è statistica e statistica)
- On the “n-1” in the definition of the
standard deviation calculated from a sample
- interference between descriptive and inferential statistics;
- irrelevant correction is n is large;
- when n is such small to make a great difference (e.g. 2 or 3) is more important our prior idea on what σ could possibly be that what we learn from data.

- xkcd comics with a boxplot showing an outlier
- Uncertainty and probability. Degree of belief
or
*“a quantitative measure of the strength of our conjecture or anticipation”* - About the importance of state of information in evaluating
probabilities (probability is
__always__conditional probability!):- two variants of the 3 box problem (one of which corresponds to the famous Monty Hall, whose 'explanation' is made exagerately long in the wiki);
- the two envelopes problems:
- equiprobability is often misused!
- do not confuse wishes and beliefs!

- Meaning and role of subjective probability (that is far from being arbitrary). Anticipation of possible objections.
- Basic rules of probability, and meaning of the relation between joint probability, conditional probability and probability of the conditionand (certainly not a 'definition' of conditional probability!).

**References**- When frequentistic gurus talk about the “probability” of a true value to be in an interval: see here.
- GdA, 2005 CERN Academic Training: last part of second lecture; third lecture till p. 25.

**Seminars**Friday 22 January on the 750 GeV diphoton excess at LHC:

[→ play special attention to probabilistic statements about the meaning of the excess]- Reports from CMS and Atlas;

[note in the abstract “This excess is not statistically conclusive...”:**when is an excess statistically conclusive?**] - Some theoretical speculations

**Lezione 6 (25/1/16)****More on basic rulesof probability. Uncertain numbers. Intro to Monte Carlo**- Again on models transferring the past to the future
(motivated by a conversation with a colleague about “smoothing
background”):
- Cern Academic Training, Lecture 2, pp. 13-20

(see in particulare the Hume's quote); - 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).

- Cern Academic Training, Lecture 2, pp. 13-20
- Probability and odds.
- Basic rules of probability from
*coherence*. - Expected gain.
- Remarks on combinatorics.
- Set, events and rules of probability.
- On the interpretation of
*P(E) = Σ*: probability of probability._{i}P(E|H_{i})×P(H_{i}) - Conditional event and fourth basic rule of probability derived from coherence (and more ramarks on its meaning).
- Events dependent/independent
*in probability*. - Back to the six box problem and comparison of the box
of unknow composition Vs that of precesely known composition.
- More on Ellsberg's paradox, and on Ellsberg.
- Probability of the sequences of colors (WW, WB, BW, BB)
from the two boxes (after reintroduction):
- the probabilities are all equal for the box of
**known**composition; - Instead, for the box on
**unknown**composition they aren't:- during the extractions we are learning something;
- → the subsequent colors are non independent!
*P(W*_{1},W_{2}) = P(W_{1})×P(W_{2}|W_{1})

*n*realizations of the*same*event”: all events are differents! (at most analogue)

- the probabilities are all equal for the box of

**Uncertain numbers**and... “random numbers”- Remarks about randomness (à la von Mises).
- Probability functions of discrete uncertain numbers:
*f(x)*and*F(x)* - Introduction to Monte Carlo “random numbers” generators:
- reweighing of events;
- 'hitting' the steps of
*F(x)*with (pseudo-)randomly generated numbers; - extension to the continuum;
**hit/miss**technique for continuous variables for- random number generation;
- evaluation of integrals.

- Monte Carlo sampling in R:
- Some examples of
**Jags**used (via**rjags**) just as a MC generator

**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.

- Again on models transferring the past to the future
(motivated by a conversation with a colleague about “smoothing
background”):
**Lezione 7 (27/1/16)****Randomness and Monte Carlo. Dependence/independence. From the Bernoulli trials to the Poisson precess.**- Playing with a circle:
- Bertrand "paradox", with a practical test:
- “Draw a chord at random”;
- list the instructions (pseudocode) to extract chords “at random” starting from a uniform random generator.

- Estimating π by sampling.
- A curious game (trowing stones), as a very first introducion to MCMC.

- Bertrand "paradox", with a practical test:
- Logical and 'stochastic' (probabilistic) dependence/indipendence.
- Bernoulli trials: geometric and binomial (and Pascal).
- Poisson process: relation between Poisson distribution and esponential distribution.

**references etc.**see previous course.

- Playing with a circle:
**Lezione 8 (29/1/16)****More on uncertain numbers and 'propagation on uncertainies'**- Probability function of the sum of the numbers resulting from two dice,
with comments on R

(distr_due_dadi.R, including command to save the plot as eps file) - More on exponential distribution and its relation with exponential and geometric. Decay life times.
- General scheme of the ditributions deriving from the bernoulli process: schema_distribuzioni.pdf
- How much to believe that X in within E[X]+-σ? General considerations (and caveat!); Markov and Cebicev disequalities.
- Properties of the 'operators' E[] and Var[] under linear transformations.
- Bernoulli theorem and its misinterpretations and mistifications.
- Generalities on probability distributions of continuous variables.
- Uniform and triangular distribution (both symmetric and asymmetric, stressing their conceptual and pratical importance!)
- Remarks on the propagation of uncertainties, expecially in the case of asymmetric distributions. 'Eye opener' example: somma_triangolari_asimmetriche.pdf (see arXiv:physics/0403086 for details.)
- Normal distribution.
- Propagation on uncertainties: genaral consideration and general multivariate formulae for discete and continuos variables.
- Examples of application to the sum of uniform continuous variables and of asymmetric triangular distributions. (See remarks and reference three items above.)
- Propagation by sampling (`Monte Carlo').
- Sum of two triangular-asymmetrically distributed variables
(see above) performd in R (using triang.R):
- somma_triangolari_asimmetriche_sampling.R
- somma_triangolari_asimmetriche_sampling.png (example of resulting histograms)

- Sum of two triangular-asymmetrically distributed variables
(see above) performd in R (using triang.R):

For**references etc.**see previous course and dispense in italiano.

- Probability function of the sum of the numbers resulting from two dice,
with comments on R
**Lezione 9 (1/2/16)****More on propagations (linear, independent). Gaussian 'tricks'. Central limit theorem and applications.**- Details on Exponential and decays.
- Gaussian tricks to evaluate expected values and variance
of pdf "assumed approximately normal" (to be used
*cum grano salis*). - On the general formula to 'propagate' the pdf
one input quantity to the pdf of one output quantity:
- proposed exercises on simple transformations, line linear, x^2, sqrt(x), x^4. Graphical interpretation of the 'distorsion' of the densities;
- a curious transformation, Y=F(X), to show that a quantity defined as the cumulative of another one is unformely distributed between 0 and 1.

- Linear transformations of independent variables:
general rules and Central Limit Theorem (CLT).

Remark: it is assumed to be applied to cumulative distributions (defines on the real axis also for discrete distributions). - Some applications:
- Expected value and variance of the binomial, of the Pascal and of the Erlang (a Gamma with k integer).
- Reproductive property of the Binomial and of the Poisson distribution, and its relevance for the use of the CLT.
- Distribution of the arithmetic average;
- A simple rough Gaussian random generator;
- Measuremt errors.

- Words of caution: we are dealing with probabilities not with certainties! (If you don't like to live in a continuous state of uncertainty take in consideration to mobe to the mathematics department...)
- Remark on the use of probability theory to perform propagations of quantities: we have to assume that probability statement can be applied to the (uncertain) values of physics quantities. (Intellectual schizophrenia not tolerated!)
- From Bernoulli process to random walk... and more:
- From binomial distribution to expected random walk;
- 'Gambler ruin' problem (just sketched).
- Pallinometro (Quinconce de Galton, or Bean machine, also available as Android app): comments on the 'didactic paradox' (of the ol, mechanical boards, like the Roman 'pallinometro'.)
- Brownian motion in 1D.
- Measurement errors (
**errors**, not uncertainties!) as a random walk the signal space. - A (limited) random walk in the velocity space: euristic derivation of the Maxwell distribution on velocities.
- A Maxwell distribution in 2D: Rayleigh distribution.

- Use of the Rayleigh distribution, with s=1, to make a 100% effcient renerator of (pairs of) Gaussian numbers from pairs of uniformly distributed numbers.

For

**references etc.**see previous course and dispense in italiano.

**Intereresting links**on Bayes, Laplace (“the man who did everything”), Turing and more (as a preparation to the inferential part of the course):- "The
Theory That Would Not Die: How Bayes' Rule Cracked the
Enigma Code
^{(*)}, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy": - Sharon Bertsch McGrayne: "The Theory That Would Not Die" | Talk at Google

^{(*)}For a nice, very well done app simulating the Enigma machine: Enigma Simulator.

**Software recommended**(not as important as Jags/rjags for applications in Physics):

**Lezione 10 (3/2/16)****Inference**- On previous lectures:
- Clarification about the distribution of
*r*(distance from the origin) in the case of points in a plain whose coordinates are i.i.d. ~ N(0,1): r_distr_unif_Vs_norm.R. - Exercise to get the pdf of
*t*, time to wait_{n}*n*events in a Poisson process of constant intensity*r*. (Hint: use the general tranformation rule with the Dirac delta.)

- Clarification about the distribution of
- The “Laplace's” Bayes rule.
- Application to the six box toy model:
- analysis of the experiment with R;
- analysis with Hugin Expert:

For

**references etc.**see- Second part of the slides of talks at Incontri di Fisica and HASCO (see above);
- GdA, Teaching statistics in the physics curriculum: Unifying and clarifying role of subjective probability

**How to install VGAM**and to use**rrayleigh()**

[Unfortunatly it does not work with install.packages("VGAM"),

at least in the version I have. The following has worked under Linux]- Download VGAM_1.0-0.tar.gz (or later version) from the developer site or from the CRAN.
- Enter in R as root, then

install.packages('VGAM_1.0-0.tar.gz', repos = NULL, type="source")

[check that VGAM_1.0-0.tar.gz is in the working directory, or use the proper path] - q()
- Then, as an example, as normal user:
- library(VGAM)
- n= 10000; r <- rrayleigh(n, 1); phi=runif(n, 0, 2*pi)
- x <- r*cos(phi); y <- r*sin(phi)
- plot(x,y,pch=19, cex=0.2, col='cyan')

Trovato il mitico articolo del Corriera sul protone che aveva la venerabile età di 10

^{25}anni:

- On previous lectures:
**Lezione 11 (5/2/16)****Inferring hypoteses and model parameters**- More on the six box problems:
- Simulating extractions. Comparisong between the probabilities of the next extraction calculated using probability theory with those extimated by past frequences.
- Analysis of the extractions without reintroduction.
- “Untangled boxes.”

- Inferring white ball proportions.
- The Bayes' billard
- A-B-C of parametric inference (with flat priors)
- Inferring
*p*of Bernoulli processes from: the sequence of successes/failures; the number of successes.- Laplace' succession rule;
- probabilistic meaning of the expected value
of
*f(p)*; - on the evaluation of the probability from past relative frequency

- Inferring λ of a Poisson distribution from the number of counts.
- Inferring μ of a Gaussian distribution
from a single observation:
*f(x*and the (not really) 'famous' sqrt(2) factor in the standard deviation._{f}|x_{p})

- Inferring

**References**- GdA,
*Bayesian Reasoning* - GdA,
*1999 CERN Yellow Report* - GdA, Dispense

- More on the six box problems:
**Lezione 12 (7/3/16)****More on parametric inference. Conjugate priors.**- Inferring "Bernoulli'p" from different kinds of experiments. Meaning of E(p). Special cases. Predictive distribution. Beta distribution as prior conjugate.
- Inferring λ of the Poisson distribution. Special cases.

*Bayesian Reasoning*

**Lezione 13 (10/3/16)****More on the inference of $\lambda$. Multivariate distributions**- Gamma distribution as conjugate prior of the Poisson distribution.
- Relation of the Gamma to other distributions.
- Multivariate distributions: overview and definitions.
- Bivariate normal distribution.
- Multivariate normal distribution.

**References**- GdA,
*Bayesian Reasoning* - GdA,
*Probabilità e incertezze di misura*(dispense)

**The p-value 'revolution'**^{(*)}**The American Statistical Association's statement**^{(*)}As explained in the Nature paper, “This is the first time that the 177-year-old ASA has made

explicit recommendations on such a foundational matter in statistics.”

**Lezione 14 (14/3/16)****Propagation. Inferring Gaussian μ. Systematics.**- Covariance matrix of linear combinations.
- Exercise on A4 paper: Exercise_A4_1.pdf.
- Repeate exercise using matrix formalism.

- Warnings concerning propagation of uncertainties.
- On the combination of "confidence intervals" (or of "probable intervals").
- Linearization.
- Other exercise on A4 paper: Exercise_A4_2.pdf.
- Special case of monomial espressions: propagation of relative uncertainties (with special case of independent variables).
- Inferring μ of a Gaussian from a single observation (assuming σ):
- case of flat prior;
- Gaussian prior conjugate: combining result;
- predictive distribution

- The Gauss derivation of the Gaussian.
- Introduction to the treatmenet of uncertaintyes due to systematic.
- Special case of uncertainty on `the zero' of an instrument
('offset uncertainty'):
- global uncertainty;
- correlation induced on the result of measurements obtained with the same instrument.

**References**- GdA,
*Bayesian Reasoning* - GdA,
*1999 CERN Yellow Report* - GdA, Dispense
- For some relevant pages (in Latin) of the Gauss' derivation of the Gaussian see here.
- GdA, On the use of the covariance matrix to fit correlated data
- GdA, W. de Boer and G. Grindhammer,
Determination
of α
_{s}and the Z_{0}Mass From Measurements of the Total Hadronic Cross-section in e+e− Annihilation - GdA, Asymmetric Uncertainties: Sources, Treatment and Potential Dangers
- GdA and M. Raso, Uncertainties due to imperfect knowledge of systematic effects: general considerations and approximate formulae

- Covariance matrix of linear combinations.
**Lezione 15 (17/3/16)****Gaussian inference from a sample. Intro to MCMC and to Gibbs sampler****References etc.**- Dispense, parte 4.
- C. Andrieu at al.
*An introduction to MCMC for Machine Learning*, pdf. - As far as OpenBUGS is concerned see
info here, plus
the following examples
- Model: gauss_Nm_sigma_pred.txt
- Dati: gauss_Nm.dat.txt
- Inits: gauss_Nm.in.txt
- Resulting chain ('coda'): bugs_out.txt and bugs_ind.txt
- Example of how to analyze the chain in R: gauss_Nm_sigma_pred.R
- Example of how to generate data with R: genera_dati.R

**Lezione 16 (31/3/16)****More on systematics. Rejection sampling and importance sampling. Unfolding****References etc.**- GdA, Bayesian Reasoning
- GdA,Errori e incertezze di misura - Rassegna critica e proposte per l'insegnamento
- GdA, On the use of the covariance matrix to fit correlated data
- C. Andrieu at al.
*An introduction to MCMC for Machine Learning*: pdf. - GdA, A Multidimensional unfolding method based on Bayes' theorem
- GdA, Improved iterative Bayesian unfolding

**R code**- importance-sampling.R
- JAGS and
rjags: Try to install them
- Getting Started with JAGS, rjags, and Bayesian Modelling (link not checked, but it looks ok)

**Lezione 17 (4/4/16)****Introduction to MCMC**

(vedi referenze della volta scorsa)

**Lezione 18 (15/4/16)****Fits of linear models. More on Metropolis. Simulated annealing. Examples with Jags/rjags**

- More on fits: arXiv:physics/0511182
- Examples of rjags:
- simpleMC_1.R
- simpleMC_2.R
- simpleMC_3.R
- inf_lambda_pred.R (model: inf_lambda_pred.bug)
- inf_lambda_pred_N.R (model: inf_lambda_pred_N.bug)
- inf_r.R (model: inf_r.bug)
- inf_r_bkg_measured.R (model: inf_r_bkg_measured.bug)

- Other scripts (or files of R commands):

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