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Sandra Day O'Connor College of Law
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Dec 20, 2016

In addition to taking CS 294-115 this semester , I also took STAT 210A, Theoretical Statistics. Here’s a description of the class from the Berkeley Statistics Graduate Student Association (SGSA):

All of the core courses require a large commitment of time, particularly with the homework. Taking more than two of them per semester (eight units) is a Herculean task.

210A contains essential material for the understanding of statistics. 210B contains more specialized material. The 210 courses are less mathematically rigorous than the 205 courses. 210A is easier than 205A; 210B is very difficult conceptually, though in practice it’s easy to perform adequately. Homework is time-consuming.

210A: The frequentist approach to statistics with comparison to Bayesian and decision theory alternatives, estimation, model assessment, testing, confidence regions, some asymptotic theory.

And on our course website we get the following description:

Stat 210A is Berkeley’s introductory Ph.D.-level course on theoretical statistics. It is a fast-paced and demanding course intended to prepare students for research careers in statistics.

Both of these descriptions are highly accurate. I saw these before the semester, so I was already prepared to spend many hours on this course. Now that it’s over, I think I spent about 20 hours a week on it. I thought this was a lot of time, but judging from the SGSA, maybe even the statistics PhD students have to dedicate lots of time to this course?

Our instructor was Professor William Fithian, who is currently in his second year as a statistics faculty member. This course is the first of two theoretical statistics courses offered each year aimed at statistics PhD students. However, I am not even sure if statistics PhD students were in the majority for our class. At the start of the semester, Professor Fithian asked us which fields we were studying, and a lot of people raised their hands when he asked “EECS”. Other departments represented in 210A included mathematics and engineering fields. Professor Fithian also asked if there was anyone studying the humanities or social sciences, but it looks like there were no such students. I wouldn’t expect any; I don’t know why 210A would be remotely useful for them. Maybe I can see a case for a statistically-minded political science student, but it seems like 215A and 215B would be far superior choices.

The number of students wasn’t too large, at least compared to the crowd-fest that’s drowning the EECS department. Right now I see 41 students listed on the bCourses website, and this is generally accurate since the students who drop classes usually drop bCourses as well.

It’s important to realize that even though some descriptions of this course say “introduction” (e.g., see Professor Fithian’s comments above), any student who majored in statistics for undergrad, or who studied various related concepts (computer science and mathematics, for instance) will have encountered the material in 210A before at some level. In my opinion, a succinct description of this course’s purpose is probably: “provide a broad review of statistical concepts but with lots of mathematical rigor , while filling in any minor gaps.”

Blog | Virta Health

Next Virta Health Changing Status Quo in Chronic Disease Care with Strong Cardiovascular Outcomes from Type 2 Diabetes Reversal Study
Rich Wood, PhD Amy McKenzie, PhD Jeff Volek, PhD, RD Stephen Phinney, MD, PhD on May 2, 2018
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Understanding the Total Risk ‘Forest’ Rather than Focusing on the ‘LDL Tree’

For the last 5 decades, most medical and nutrition scientists have focused on low density lipoprotein (LDL) cholesterol as a primary cause of coronary heart disease. Characterized as ‘bad cholesterol’, literally thousands of studies have been done using drugs or diet to reduce LDL cholesterol and thereby hoping to reduce heart attacks and mortality. While cholesterol lowering therapy has become the standard of care for some individuals with well-defined heart disease risk, this focus on cholesterol in general—and LDL cholesterol in particular—remains very controversial.

Part of this controversy stems from our tendency as scientists to reduce a problem to its simplest component(s). Unfortunately the standard measure of blood LDL cholesterol is easy but inaccurate (Volek Phinney 2011), and even when the various components of the blood LDL are accurately measured (Reaven 1993) they represent only a fraction of the lipid and other biomarkers of heart disease risk. In other words, in the interest of ‘keeping it simple’, we have wrongly ignored the rest of the ‘forest’ of other risk factors while focusing on the LDL cholesterol ‘tree’.

A turning point in understanding the limitations of LDL and heart disease came with the publication of the Lyon Diet Heart Study (de Lorgeril 1994, 1999). This randomized trial pitted a standard low fat diet against a Mediterranean diet for people with a prior heart attack. The study was halted after 2.7 years because there was a dramatic reduction in repeat heart attack incidence and overall mortality in the Mediterranean diet group. But to everyone’s astonishment, there was no difference in LDL cholesterol changes between the Mediterranean and low fat diet groups. At least for this diet study, the standard calculated LDL value did not seem to matter that much. Nor did most of the other standard biomarkers of that era that they measured, indicating that some very important drivers of coronary disease risk were going unmeasured. In the intervening 20 years, however, the range of factors linked to heart attack risk has expanded greatly – i.e., the forest has gotten a lot bigger.

In our recently published 1-year results from the IUH/Virta diabetes reversal study, we reported a small but significant rise in the average blood LDL cholesterol level in our patients on a well-formulated ketogenic diet (Hallberg, 2018). At the same time, however, we noted major reductions in a number of coronary disease risk factors including weight, blood pressure, and of course HbA1c. But now we have published a much more extensive review of these risk factor changes for this group of about 200 patients after a year of Virta Treatment (Bhanpuri 2018). As a snap-shot of this ‘tree view’ versus a ‘forest view’, here are two contrasting figures that demonstrate just how complex this picture is, but also the pattern of how these other risk factors change independent of the variable changes in LDL.

23 November 09

Filed under: Filtrations and Processes , Stochastic Calculus Notes — George Lowther @ 12:20 AM Tags: math.PR , Stochastic Calculus , Fashion Unisex Womens Mens Summer Antiskid Flax Flat Mules Slip on Slippers Lightweight Open Toe Cozy Linen Flip Flops Home House Indoor Bedroom Bath Spa Slide Scuff Sandals Slippers Shoes Footwear Crossed Band Green nd9hYrk

The previous post introduced the notion of a stopping time . A stochastic process can be sampled at such random times and, if the process is jointly measurable, will be a measurable random variable. It is usual to study adapted processes, where is measurable with respect to the sigma-algebra at that time. Then, it is natural to extend the notion of adapted processes to random times and ask the following. What is the sigma-algebra of observable events at the random time , and is measurable with respect to this? The idea is that if a set is observable at time then for any time , its restriction to the set should be in . As always, we work with respect to a filtered probability space . The sigma-algebra at the stopping time is then,

The restriction to sets in is to take account of the possibility that the stopping time can be infinite, and it ensures that . From this definition, a random variable us -measurable if and only if is -measurable for all times .

Similarly, we can ask what is the set of events observable strictly before the stopping time. For any time , then this sigma-algebra should include restricted to the event . This suggests the following definition,

The notation denotes the sigma-algebra generated by a collection of sets, and in this definition the collection of elements of are included in the sigma-algebra so that we are consistent with the convention used in these notes.

With these definitions, the question of whether or not a process is -measurable at a stopping time can be answered. There is one minor issue here though; stopping times can be infinite whereas stochastic processes in these notes are defined on the time index set . We could just restrict to the set , but it is handy to allow the processes to take values at infinity. So, for the moment we consider a processes where the time index runs over , and say that is a predictable, optional or progressive process if it satisfies the respective property restricted to times in and is -measurable.

Lemma 1 Let be a stochastic process and be a stopping time.

Proof: If is progressive then, as proven in the previous post, the stopped process is also progressive and, hence, is adapted. It follows that is -measurable which, from the definition above, implies that is -measurable.

Furthermore, is -measurable and is zero when restricted to the set for all , so is also -measurable.

Now, consider a predictable process . Write for the predictable sigma-algebra on . That is, the subsets of which are predictable when restricted to and such that is -measurable. Then, is -measurable. By the functional monotone class theorem , it is enough to prove the result for processes of the form for some pi-system of sets generating .

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