Messages
Check near the bottom of this course webpage to find both the firetimers exam set, 4-xii-2025, and notes on solutions for these exam questions.
Nei, kandidatene f?r ikke anledning til ? ha med seg eget h?ndkle p? eksamen (kun dette ene arket med h?ndskrevnde notater), s? det kan jo l?nne seg ? l?re denne ene formelen utenat:?
https://www.facebook.com/groups/1589206911336271/posts/4283599865230282/
Our very last teaching-discussing session is on Monday December 1st: there will be Q-and-A opportunities.?
Exam date: December 4th, 9-13. Time: December 4 at 9:00 AM (4 hours), Silurveien 2 Sal 3B. Bring calculator and 1 page with handwritten notes.?
Curriculum, as agreed upon & announced a few times:?
All exercises and all stories worked through in the course, from Hjort-Stoltenberg "Statistical Inference: 600 Exercises, 100 Stories" (preliminary version, 2025). The chapters are "only" 7 and 14, and we've been though perhaps 12 stories. You may backtrack through all messages given, from August to late November, to check these.?
Asgeir and Linus have kokt i hop this list of themes, and I tell them it looks good -- we may also throw in mixture priors, Dirichlet processes, basic maximum likelihood theory.?
* Bayesian master recipe
* decision theory (loss functions, risk, bayes estimators, Stein, minimaxity, admissibility)...
Mon Nov 24: we do Exam firetimers 2017, 3 and 4, and otherwise talk & discuss & ask & answer.?
We've had various "ProcMod" themes in the FocuStat project, From Processes to Models, and here's a talk by Celine Cunen:?
https://www.mn.uio.no/math/english/research/projects/focustat/talks/eysm2019_cc.pdf
Cromwell's Rule: "I beseech you, in the bowels of Christ, think it possible that you may be mistaken", he said, in 1650. So Lindley says: we should always have a little epsilon-part of the prior for the unforeseen -- otherwise the Bayesian will never ever detect it, if something's wrong. If the prior says "lots of swans are different, here's a four-parameter model, and all swans are white", then the Bayesian will NEVER detect that some swans are black.?
Mon Nov 17: we did (a) the L_c(u) = \exp(cu) - 1 - cu loss function, with Bayes and minimax; (b) a bit on different loss functions, and do the details for Lemma 0: g(a) = \E\,|X-a| is minimised for a_0 = the median; (c) mixture priors (where Nils didn't have time to put in a simple example; will do Wed Nov 19).
The class is invited to point to themes, and then we discuss them, with Q + A or more.?
For Wed Nov 19 and Mon Nov 24: we use some time, perhaps not too much on Exam 2004 (yes, it's old), 1 and 2:?
https://www.mn.uio.no/math/english/research/projects/focustat/lecture-notes-and-exercises-for-various-courses-gi/exam_stk4020_2004.pdf
" Del II er en muntlig eksamen, og avholdes mandag 48. desember (ogs? kalt 17. januar); n?rmere opplysninger om dette blir gitt senere."
We're soon rounding off!, having essentially been through all cofre material in the course, also having worked through a quite long list of exercises and stories. We meet a few more times, with themes asked for by you, the students -- we did minimaxity the other day, for instance.
For the coming few weeks: Find Olaf Bull's "Til de dristige", and reflect on its relevance for minimax strategies. Do exercises 7.6, the linex loss function, with concrete steps for the classic case of Y a \N(\theta, 1), and a \N(0,\sigma^2) prior for \theta. Then 7.7.?
We did Celine Cunen's Submarine (exam set 2018): show that the Bayes estimator there is the same as the sample mean of the two data points, and that it is unbiased. Which assumption, in the lemma which says that Bayes estimators are never unbiased, is not met??
Thanks to our four wondrous PhD candidates for their intriguing presentations. I am also satisfed & grateful for your efforts with the OBLIG. I've placed com205c on the page, with mcmc for plants, Oblig #3. You may play with the code a bit to check further issues.
From Mon 10 onwards we're partly in repetisjonsmodus, looking back at what we've learned, etc., so please come with questions, themes you would like more comments on, etc.
We also go through:?
Exam 2015, #1, two models; Exam 2018, #1, poisson-gamma and more; 7.20; 7.29 (i).?
I'm uploading com217b, for Bayesian kriging, Gaussian processes (part of Ch14 terrain): how to interpoplate Z(x), with uncertainty, when it is only observed in a few positions.
Mon Nov 3: PhD twenty-minute presentations no. 3 and 4!
Nex:
Exam 2013, #2, Rayleigh.
Oblig 2021, #3, Galton 1889, 111 husbands and wives (and their temper).
Exam 2013, #3, do the data come from f_0 or f_1?
Berit used "copulas" in her presentation, but I prefer, of course, copulae (sorry, can't get this in neat table format here):
Case Singular Plural Meaning
Nominative copula copulae the link / the links
Genitive copulae copulārum of the link / of the links
Dative copulae copulīs to/for the link / to/for the links
Accusative copulam copulās the link (object) / the links (object)
Ablative copulā copulīs by/with/from the link / by/with/from the links
Vocative copula copulae O link! / O links!
Check out com216a, for exercise 14.11 and Story #18, alpha waves, exhibiting Dirichlet process simulations for two groups of data. Make your own supplement, reading off, from F and G given data, the posterior for \delta = \med(F) - \med(G), the difference between the two medians.
Wednesday 29 Oct: two 20-minute PhD presentations, then Nils does more with Ch 14: the above, plus a bit on Gamma processes.
We're soon finished with the curriculum!, and will use the remaining weeks for refreshing & reviewing & repetition & consolidating knowledge, with more earlier exam things, etc.
We shall use perhaps 6 or 7 teaching hours for the appropriate of Ch14, Bayesian nonparametrics, starting Week 11. We do 14.11 and its continuation, alpha waves for two groups of prison inmates, Story #18: simulate from Dirichlet processes, prior & posterior, to answer "all questions". Then we do exercises 1, 4, 7, 10, 15, with one more story.
After this we round off, and spend some time consolidating our fabulous knowledge (and preparing for the four-hour exam December 4).
20 minute presentations by the PhD persons:
Wed 29 Oct: Berit ?, copulae; Juan Paulo, bank.
.
Mon 3 Nov: Vetle V, cosmos; Henrik N, forensic science.
We did this per blackboard -- check com213a, uploaded now. So
# mean median mode
## after first five: 84.8 75 59
## after ten numbers: 68.1 65 59
ja s? sannelig, posterior mode unchanged, from 5 known numbers to 10 known numbers. The 90 percent credibility interval narrows down from [52, 151] to [55,90].
Since LW called from Japan (!) just before Monday's class, where Bayes was hinted at and FIC discussed, I mentioned things that are discussed in this FocuStat blogpost, by Celine Cunen & Nils Lid Hjort:
https://www.mn.uio.no/math/english/research/projects/focustat/the-focustat-blog!/whales.html
** OBLIG, prepare your pdf's, read Shakepeare and Goethe and Chekov, I look forward to seeing your reports, by Mon Oct 27. Check early enough that the Canvas system works.
** Are there any PhD Candidates on board? Yes, 20 minutes presentations will come:
.. Wed 29 Oct: Berit ?, copulae; Juan Paulo, bank.
.. Mon 3 Nov: Vetle V, cosmos; Henrik N, forensic science.
** Let Y be pois(\theta), with loss (\delta - \theta)^2/\theta. We did this today, with minimaxity and more. But for the parameter restriction \theta \ge 0.50, find & plot the risk function for the truncation-from-0-to-1 estimator, and comment.
** Simulate something, along the lines of (i) \theta_i from some distribution, wide or narrow, (ii) Y_i is \N(\theta_i,1), say for n = 100. Plot (y_i, \hat y_{i,\stein}), for today's exercise 9.23.
** The rest of the Abel 1902 envelopes Story #42.
** 7.14, 7.11.
Soon, next week: we'...
Tusen takk til Berit ? who spotted a ... confusing aspect of OBIG Exercise 1(d). So I have now repaired & invented another question (the pdf now available is the corrected one). Thanks for your patience & forgiveness.
(d)
Via simulations, provide similar inference summaries
for the ratio $\kappa=M/H$, where $M=\half(\theta_1+\theta_2)$
and $H=2/(1/\theta_1 + 1/\theta_2)$ are the usual mean
and the harmonic mean: comment on what you find.
S?ren klype, I had forgot to specify (a, b) in Exercise 1(c). The pdf is now repaired, with the line " For the prior, set (a, b) = (1.11, 1.11). " Takk, Henrik, da blir det Pr("godkjent" p? deg) \ge 0.88.
Dessuten f?yde jeg til ordene "hjemme-hos" helt til slutt, da jeg skj?nte at noen ikke hadde skj?nt at dette er hjemme hos Stigler, ikke p? hans kontor:
(Chicago, October 2025; amateur hjemme-hos photo by Nils)
** 1. The OBLIG is out, Tue Oct 14 -- happy struggle & work hard og det g?r deg godt og du vil leve lenge i landet (Ef. 6:3). Reports to be delivered, each student with 1 single pdf, to the Canvas system, by Monday Oct 27 at 13:58.
** 2. There's a new Special Edition chapter9-only at the website. We do exercises 12, 13, 14, 15, perhaps 17, and start with these Mon 13 Oct, in addition to Exam Project 2013, 3.
** 3 (Monday Oct 13): I've uploaded com206a, for the leukemia dataset of Exam Project 2013 #3.
** 4. Revised programme for the coming few teaching days: In Ch9, 12, 13, 14, 15. Also, do this: Y is Poisson (theta), with the parameter to be estimated with loss (hat theta - theta)^2/theta. Show that Y has constant risk 1, and that it is minimax. Then do Story #42, Bayesian parts, Abel envelopes.
Check out:
https://www.facebook.com/nilslidhjort/posts/pfbid0HuNEXsjFy4XWj5Bn7ptzL7EbouiweLqDcEtrkoNNb2APNS3V5Yv9eG72FQsMfiCWl
Extra: first, check that you find about Fisher's zeta transformation, inside Ch 5: with R_n the correlation coefficient, we have sqrt(n) (hat zeta - zeta) close to N(0, 1): we can "transform to normality" and back. Suppose you observe R_n equal to 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60 for 8 experiments, each with n = 30. Do the transformation to normality; estimate zeta_1, ..., zeta_8 there, via the multiparameter Stein recipes of Ex 7.36 that Emil has gone through with you; and then transform back to correlation scale. Also attempt a full Bayesian analysis.
Exam 2021, 1, 2, 3.
Exam project 2013, 3.
When back from Chicago I'll post *Chapter 9* on the website, from Hjort-Stoltenberg, from which there will be four of five exercises included in the curriculum,
So, check out the R code I upload, when needed (and I do not have a very clear algorithm for what I upload and what I keep in my laptop, so to speak). You might e.g. look at, run, compile, modify, play with *com364a* du jour: it does various things related to Celine Cunen's Exam 2018: (i) maximum likelihood, with precision; (ii) using these to get to empirical Bayes \hat\lambda_{i,EB}; (iii) MCMC for (a, b) given data, starting from a rather noninformative prior; (iv) using the MCMC output to have full Bayes analysis for Celine's seven \lambda_i.
Similar mathematics, similar problems, similar code, might be needed for handling many other multiparameter setups.
Apropos studielivets mange dimensjoner:
" Det var nok til Stor Lokal Ber?mmelse, hin gang, i den era der enhver flink student satte sin lille ?re i ? v?re toppforberedt, ikke n?dvendigvis til hver forelesning, men til ukeoppgavene p? gruppene. Men Ungdommen N?tildax (osv.). " ...
(a) The Oblig will be from t_1 = Wed Oct 15 to t_2 = Mon Oct 27; more details to come.
(b) The four PhDs are required to give us 20 minute presentations, inside the two-hour teaching windows, and we come back to practicalities; these will be given after the Oblig. Thanks for initial thoughts, Berit, Vetle, Juan Pablo; please write me a mail, Henrik.
(c) Nils is in Chicago for a conference first days of October; wondrous substitutes will be Dennis Christensen (Oct 1) and Emil Stoltenberg (Oct 6).
(d) Programme for exercises & more:
Do the full Bayesian version of Story #26, Blood on the Tracks; do an mcmc for posterior of (p, q) for the one-locus theory. If your prior says 0.50, 0.50 for Theories One and Two, how do you conclude after having seen the data? The Theory One probabilities are p(2-p-2q), q(2-q-2p), 2pq, (1-p-q)^2; find the posterior for NAB.
Exam 2018, #2, seven expo, Celine eks...
The OBLIG will be from Wed Oct 15 to Mon Oct 27. It will be be properly announced and linked to at the website.
The PhD candidates: write to me!, for a potential 20 minute presentation during the teaching hours.
Mon Sep 29: Nils!, then off to Chicago for a week.
Wed Oct 1: Dennis Christensen!, Exam 2024, #1 (updating), Oblig 2021, #4 (rats), perhaps BvM wth 7.34, 7.35.
Mon Oct 6: Emil Stoltenberg!, 7.36, Exam project 2015, #2 (geometrics), and more.
Mon Sep 22 and Wed Sep 24, and next week: as much as we can from:
Exam 2015, #4 uniform with a prior
Exam project 2010, #1, Myshkin kabal
Exam project 2015, #3, British mining changepoint
Exam 2013, #1, Mr Savage the Bayesian
Exam 2013, #2, f1 or f2
Oblig 2024, #2, Artemis
Oblig 2021, #4, rats and mcmc
Exam project 201...
Mon Sep 22 & Wed Sep 24:
Exam 2015, #4 uniform with a prior
Exam project 2010. #1 Myshkin kabal
More Jeffreys and vip-Jeffreys: 7.26, 7.29, 7.30.
Mon Sep 15 & Wed Sep 17: Story #17 cigarettes (Bayesian part); normal-normal and gamma-normal things 7.19, 7.21; then Jeffreys 7.25, 7.26, and soon enough vip-Jeffreys exercises 7.29.
You should also do this, a simplified version of the cigarettes problem: generate n = 100 datapoints from y = \beta x + \eps, for some true \beta_0, with the x_i and \eps_i standard normal. The prior for \beta is flat on [0,\infty) (i.e. we do not tolerate \beta negative). Code up an MCMC doing this, and check how \beta given data looks, for true \beta_0 clearly positive; for \beta_0 close to zero; and for \beta_0 a bit negative.
We'll also do more of earlier exam, exam projects, oblig exercises soon, and the next to work through here are:
Exam 2017...
Wed Sep 10: Exam 2021, #2, #4, Exam 2017, #4.
Nils is at a Human Rights in Adacemics conference on that day, but you should absolutely meet up: Kushagri Tandon will take you through these three earlier exam exercises (and I'll make sure that *something* from these will turn up, one way or another, for the December 2025 conference).
Next couple of weeks: Story #17 (cigarettes & cancer); rest of 7.19; hack you way through 7.21 (normal-normal is nice and clear with known sigma, more messy but doable with unknown sigma); 7.23, then 7.24 Jeffreys (with examples) and after that again 7.29 Nils-Emil vip-Jeffreys. We'll also fill up with more from earlier exam, earlier Oblig, earlier exam projects.
https://dnva.no/detskjer/2025/05/symposium-menneskerettigheter-og-akademisk-frihet
In connection with Story #85, an MCMC take on Magic Squares, I've uploaded com5d to the site. Use it, play with it, constuct magic squares of sizes 4 x 4, 5 x 5, 6 x 6, perhaps higher. This is achieved via the MCMC for f(x) = \exp[-\lambda Q(x)] / constant, with positive \lambda. You may also play with *negative \lambda*, to encourage outcomes as far as possible from magic-ness: sums of rows, columns, diagonals, far away from the magic number (34 for 4 x 4, 111 for 6 x 6, etc.). The exercise is then to find the Bayesian MAP, maximising Q(x) as score: the worst ever squares.
I think it's pretty hard to do this by pure math -- so run chains and see what happens. My current guesses, for 4 x 4 and 6 x 6 (after tne minutes of playnig) are 114 and 540. If you can beat these numbers, tell me, and I'll give you tyve kroner (les Rudolf Nilsen).