I've been sick a lot recently, in part thanks to having small children. In any case, one thing I've been doing is revisiting Chess. I honestly am pretty clumsy at Chess but it's one of those things I always felt I should probably do. When I was younger most of my friends played stronger games than me, and it was hard to enjoy when you were getting beaten all the time. Now, thanks to Moore's law and clever programming, even the very very very top players are useless against a 4 core laptop computer running Stockfish.
So we can all agree now that it's no fun getting blasted out of the water every time, but also we can use computers to make things better and more interesting for humans, since that's what they're for right?
There are lots of proposals for randomized or alternative starting position Chess games. For example Chess 960 (Fischer random chess) is a variant with 960 possible starting positions. The idea is to avoid making Chess a game where a big advantage comes from memorizing opening moves in some opening database. I'm more or less for this in my play. I enjoy playing Chess well enough, but I have absolutely NO interest in poring over variation after variation in a big book of opening theory. I think some people like this stuff, so for them, they can of course continue to play regular chess.
On the other hand, for people like me, consider the following method of starting the game:
- Set up the board in standard starting position.
- Using a computer, play N random legal pairs of moves (turns). possibly with or without capture.
- Using a chess program on the computer, find the "score" for this position.
- Accept the position if the chess program decides that the score is within of (where positive is good for white, negative is good for black, this is standard output from chess engines), otherwise go to step 1.
- Assign the color randomly to the human (or to one of the humans if you're not playing against a computer).
- Start the game by allowing white to move.
Note, this variation also can be used to handicap games by accepting a starting position if it is within of some handicap value , and then assigning the weaker player to the color who has the advantage. It's also possible to play random moves and then allow the computer to move moves until the score evens out properly if you can get support from the chess engine. Finally, it's also possible to search a large database of games for one in which after moves the position evaluates to within of the appropriate handicap value, rather than generating random moves.
I suspect to would be the appropriate number of moves to use.
Now, who will implement this in SCID or the like?
Frequentist statistics often relies on p values as summaries of whether a particular dataset implies an important property about a population (often that the average is different from 0).
In a comment thread on Gelman's blog (complete with a little controversy) I discussed some of the realistic problems with that, which I'll repeat and elaborate here:
When we do some study in which we collect data and then calculate a value to see if it has some particular property, we calculate the following:
Where is a functional form for a cumulative distribution function, and are sample statistics of the data .
A typical case might be where is the sample average of the data and is the sample standard deviation, is the number of data points, and is the standard t distribution CDF with degrees of freedom.
The basic idea is this: you have a finite population of things, you can sample those things, and measure them to get values . You do that for some particular sample, and then want to know whether future samples will have similar outcomes. In order for the value to be a meaningful way to think about those future samples you need:
- Representativeness of the sample. If your sample covers a small range of the population's total variability, then obviously future samples will not necessarily look like your current sample.
- Stability of the measurements in time. If the population's values are changing on the timescale between now and the next time you have a sample, then the p value is meaningless for the future sample.
- Knowledge of a good functional form for . When we can rely on things like central limit theorems, and certain summary statistics therefore have sampling distributions that are somewhat independent of the underlying population distribution, we will get a more robust and reliable summary from our p values. This is one reason why the t-test is so popular.
- Belief that there is only one, or at least a small number of possible analyses that could have been done, and that the choice of sample statistics and functional form are not influenced by information about the data: represents in essence a population of possible p values from analyses indexed by , when there are a wide variety of possible values for , the fact that one particular p value was reported with "statistical significance" only indicates to the reader that it was possible to find a given that gave the required small .
The "Garden of Forking Paths" that Gelman has been discussing is really about the size of the set independent of the number of values that the researcher actually looked at. It's also about the fact that having seen your data, it is plausibly easier to choose a given analysis which produces small values even without looking at a large number of values when there is a large plausible set of potential .
Gelman has commented on all of these, but there's been a fair amount of hoo-ha about his "Forking Paths" argument. I think the symbolification of it here makes things a little clearer, if there are a huge number of values which could plausibly have been accepted by the reader, and the particular value chosen (the analysis) was not pre-registered, then there is no way to know whether is a meaningful summary about future samples representative of the whole population of things.
What problems are solved by a Bayesian viewpoint?
Representativeness of the sample is still important, but if we have knowledge of the data collection process, and background knowledge about the general population, we can build in that knowledge to our choice of data model and prior. We can, at least partially, account for our uncertainty in representativeness.
Stability in time: A Bayesian analysis can give us reasonable estimates of model parameters for a model of the population at the given point in time, and can use probability to do this, even though there is no possibility to go back in time and make repeated measurements at the same time point. Frequentist sampling theory often confuses things by implicitly assuming time-independent values, though I should mention it is possible to explicitly include time in frequentist analyses.
Knowledge of a good functional form: Bayesian analysis does not rely on the concept of repeated sampling for its conception of a distribution. A Bayesian data distribution does not need to reproduce the actual unobserved histogram of values "out there" in the world in order to be accurate. What it does need to do is encode true facts about the world which make it sensitive to the questions of interest. see my example problem on orange juice for instance.
Possible Alternative Analysis: In general, Bayesian analyses are rarely summarized by p values, so the idea that the values themselves are random variables and we have a lot to choose from is less relevant. Furthermore, Bayesian analysis is always explicitly conditional on the model, and the model is generally something with some scientific content. One of the huge advantages of Bayesian models is that they leave the description of the data to the modeler in a very general way. So a Bayesian model essentially says: "if you believe my model for how data arises, then the parameter values that are reasonable are ". Most Frequentist results can be summarized by "if you believe the data arise by some kind of simple boring process, then you would be surprised to see my data". That's not at all the same thing!
Boobies. There I had to say it. This is a post about boobies, and math, and consulting with experts before making too many claims.
In this click bait article that I found somehow searching on Google News for unrelated topics, I see that some "Medical Anthropologists" are claiming that Bras seem to cause breast cancer (not a new claim, their book came out in 1995, but their push against the scientific establishment is reignited I guess). At least part of this conclusion seems to be based on the observation from their PDF
Dressed To Kill described our 1991-93 Bra and Breast Cancer Study, examining the bra wearing habits and attitudes of about 4,700 American women, nearly half of whom had had breast cancer. The study results showed that wearing a bra over 12 hours daily dramatically increases breast cancer incidence. Bra-free women were shown to have about the same incidence of breast cancer as men, while those who wear a bra 18-24 hours daily have over 100 times greater incidence of breast cancer than do bra-free women. This link was 3-4 times greater than that between cigarettes and lung cancer!
They further claim "bras are the leading cause of breast cancer."
That's pretty shocking data! I mean really? Now, according to http://seer.cancer.gov/statfacts/html/breast.html there are about 2 Million women in the US living with breast cancer, and 12% overall will be diagnosed throughout their lives. There are around 150M women in the US overall. So
However, in our sample That's 50 times the background rate (ok 37.5 if you do the math precisely).
Doesn't it maybe seem plausible that in winnowing through the 1% of women living with breast cancer and are still alive, or even the 5 or 6 percent who have been diagnosed in the past but are still alive (figure half of women who are alive today who will at some point be diagnosed have already been diagnosed at this point) that maybe, just maybe they could have introduced a bias in whether or not their sample wears bras?
So "looking for cancer patients causes us to find bra wearing women" is actually maybe the more likely story here? Perhaps "cancer patients who were non bra wearers were overwhelmingly more likely to have died from their breast cancer, and so we couldn't find any of them?" That's somehow not as reassuring to the non-bra-wearers in the audience I think.
Symbolically: pretend BC and Bra are independent. We conclude or not wearing a bra reduces your chance of surviving by a factor of 10 or so if P(Bra) ~ 0.9? Put on those bras ladies! The exact opposite of their conclusion!
I personally suspect something else spurious in their research. But nothing in their PDF convinces me that they know what they are doing.
Note that wikipedia has some discussion of their book.
A friend of mine posted a link to news articles about a recent Surgeon General warning about sunscreen
Quoting from that article:
Skin cancer is on the rise, according to the American Cancer Society, with more cases diagnosed annually than breast, prostate, lung and colon cancer cases combined.
On Tuesday, the United States surgeon general issued a call to action to prevent the disease, calling it a major public health problem that requires immediate action. Nearly 5 million people are treated for skin cancer each year.
But let's dissect this a little bit. First of all, most skin cancer is unlikely to really hurt you. It's melanoma that is the real concern.
cancer.gov gives melanoma rates according to the following graph:
As for melanoma itself, clearly the diagnosed cases are on the rise, but look at the deaths per year. Totally flat. This is consistent with improved diagnosis procedures without any change in actual incidence in the population. Furthermore looking deeper on the melanoma site we see that 5 year survival rates have increased from 82% in 1975 to 93% in 2006, this is also consistent with earlier diagnosis (so that the 5 year rate measures from an earlier point relative to the initiation of the melanoma).
How about melanoma by state? Climate should be involved right? More sun exposure should mean more melanoma?
As you can see, states in the southwest have lower rates, and states in the northwest and northeast have higher rates. The cluster of southeastern states with high rates are interesting too.
Vitamin D is suspected to be involved in several health affects related to cancer, so overall exposure to sun may be beneficial. However, I think that the data is also clear that intense exposure to sun from tanning beds, intentional tanning, and repeated sunburn is bad for your skin.
Sun exposure, like other exposures such as alcohol, may have nonlinear risk effects. At moderate exposures you are better off than at zero exposure (in Oregon rain or Maine winters) or heavy exposure (leading to repeated sunburn and high melanoma risk).
So, is the advice to slather on sunscreen good? I don't think the conclusion is so clear cut, but I don't have any in-depth study data to go on either. All I can tell you is that I'll continue to avoid getting sunburn by covering up and using zinc based sunblock when I'm outside for long periods, but I'll continue to get regular sun exposure without sunblock in mornings, evenings, and mid day during non-summer months.