The Ghost of Statistics Past
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The Ghost of Statistics Past
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Re: The Ghost of Statistics Past
#2Computing with frequentist statistics just means making a bunch of simplifying assumptions, setting some things constant to make computation tractable. The author correctly hints at that in the middle of the article, but then glosses past it.
Frequentist vs Bayeisan interpretation is like different interpretations of quantum mechanics. It has no impact on the calculations.
Novice self-labelled "Bayesians" overlook the reality that, as wikipedia explains:
> where appropriate, Bayesian inference (meaning in this case an application of Bayes' theorem) is used by those employing a frequentist interpretation of probabilities.
Re: The Ghost of Statistics Past
#3Computing with frequentist statistics just means making a bunch of simplifying assumptions, setting some things constant to make computation tractable. The author correctly hints at that in the middle of the article, but then glosses past it. Frequentist vs Bayeisan interpretation is like different interpretations of quantum mechanics. It has no impact on the calculations. Novice self-labelled "Bayesians" overlook th…
I would care to interject.
First of all, you are right on several points.
* Most of mathematics is the same in both schools of though, and the interpretations is not different.
* Some basic ideas (i.e. the nature of probability) are quite different, and this is where most of the argument (Frequentist vs. Bayesian) comes from.
However, this second point has a major impact on calculations. So I disagree here:
* The notion of a _prior_ (probability of the hypothesis, P(H)) is essentially nonsensical in the Frequentist view. Any frequentist would just call it 'bias'. However, for a Bayesian, this is the degree of belief that you put in you system before you do any measurements. Practically, it is either non-informative (you don't give more belief to any hypothesis a-priori), or comes from earlier data. The prior gives you a natural way to incorporate multiple experiments. I think that is a large difference in calculation (or at least the structure of calculation).
More importantly, Bayesian statistics inspires (and is enabled by) Markov Chain Monte Carlo inference. It is the main mathematical machinery used for today's Bayesian data analysis, and is impossible in the frequentist framework. This approach allows you to scale to very complicated (i.e. feature-rich, multiparameter) data, explore very complex (i.e. non-convex, hard to optimize) probability surfaces. All of this stuff is very hard (if at all possible) in the frequentist framework.
So there are difference. But people don't really argue about them. There is no grand flame war, or anything of that sort.
Oh, and contrary to what the article suggests, no statistician likes p-values.
Re: The Ghost of Statistics Past
#4Perhaps the post is alluding to some different use of statistics than I'm used to - but isn't it normal to view a certain set of outcomes as a sample from some larger population, and not as eg: a sequence. In the case of a given coin, and a controlled sequence of flips, we can ask different questions, like what is the probability of heads following heads vs, heads following tails? (Eg: when a certain experimenter flips the coin, do the side that is up at the start of the experiment affect the outcome? Will someone that mechanically flips a coin 10s or 100s of times in a row end up with such a similar mechanical motion that the coin tends to spin approximately the same number of times from "flip" until it lands?).
I think the author mixes up the mental models involved, when mixing "throwing away information", "sequence" and "a (typical) fair coin" and "a (typical) unfair coin".
There's this coin, this sequence, and those typical fair/unfair coins.
In as much as I've been able to grasp anything about proper statistics, it's the idea that without some idea of the population and sample type (eg: can we expect a Poisson distribution?) -- most modern statistics makes no sense. And one can look at things differently. Just like Newtonian physics is correct at the same time as quantum physics is correct (most of the time). But sometimes we need to change our theoretical model to be more precise (quantum) in order to predict and model behaviour mathematically.
Re: The Ghost of Statistics Past
#5Perhaps the post is alluding to some different use of statistics than I'm used to - but isn't it normal to view a certain set of outcomes as a sample from some larger population, and not as eg: a sequence . In the case of a given coin, and a controlled sequence of flips, we can ask different questions, like what is the probability of heads following heads vs, heads following tails? (Eg: when a certain experimenter fl…
A sequence corresponds to multiple draws from that population, and a statistic (number of successes / number of draws) is a summarization of those draws in a single number. The author isn't really arguing that a binomial distribution is not the right way to model this problem, but that the way frequentist statistics weighs the evidence is faulty.
Edit: although, actually, you're right that the author does at one point say "Why should all sequences containing exactly 12 heads be treated the same?" but then goes on to model everything with the binomial distribution anyway.
Re: The Ghost of Statistics Past
#6The author seems to imply that frequentists are not just wrong but borderline malicious – they use all sorts of ad hoc procedures, have no theoretical justification for anything they're doing, and so on and so on. But in fact, there is an extensive body of theory about when you can "throw away information" (cf. minimally sufficient statistics), the difference between P(D|H) and P(H|D) (cf. likelihood theory) and so on. His undergraduate textbook might not talk about all that, but a graduate textbook sure would -- it's certainly not lore full of implicit assumptions that everyone's forgotten.
A much better explanation of what is happening to statistics is this: flipping P(D|H) to P(H|D) used to be really hard, so we came up with all sorts of tricks to either approximate it or to get by without having to bother about P(H|D) at all, things like p-values. Now there's computers and really good sampling algorithms so we don't have to use the approximations anymore. But some people still prefer the approximations because they're used to them. There. No incompetence or malice involved.
Other than that, if you ignore the tone of the article, it's an insightful read if you're new to Bayesian statistics and want to understand what's the point.
Re: The Ghost of Statistics Past
#7The coin flipping example used in the article is pretty weak. We are convinced that the coin is unfair when we see "HHHHHHHH" but not "HHTHTHTHH" because the former event is more likely given an unfair coin. We choose the hypothesis that makes the data most likely. I doubt the author's of the undergraduate textbook were unaware of maximum likelihood estimation. I suspect that the post's author simply did not read far enough into his textbook and got hung up on a simplified model given in an early chapter.
Any hypothesis we test about the coin's "fairness" is implicitly a test of how close p is to 1/2. The only difference between a frequentist analysis and a bayesian analysis in this case would be that the latter might impose a prior on p (although in reality we would likely impose a uniform prior, rendering the two analyses identical).
Re: The Ghost of Statistics Past
#8Perhaps the post is alluding to some different use of statistics than I'm used to - but isn't it normal to view a certain set of outcomes as a sample from some larger population, and not as eg: a sequence . In the case of a given coin, and a controlled sequence of flips, we can ask different questions, like what is the probability of heads following heads vs, heads following tails? (Eg: when a certain experimenter fl…
A sequence corresponds to multiple draws from that population, and a statistic (number of successes / number of draws) is a summarization of those draws in a single number. The author isn't really arguing that a binomial distribution is not the right way to model this problem, but that the way frequentist statistics weighs the evidence is faulty. Edit: although, actually, you're right that the author does at one poin…
Since each draw is independent from each other (I don't think anyone disputes that), the order is irrelevant. You want to think about it as a set, not as a sequence.
Re: The Ghost of Statistics Past
#9The author seems to imply that frequentists are not just wrong but borderline malicious – they use all sorts of ad hoc procedures, have no theoretical justification for anything they're doing, and so on and so on. But in fact, there is an extensive body of theory about when you can "throw away information" (cf. minimally sufficient statistics), the difference between P(D|H) and P(H|D) (cf. likelihood theory) and so o…
The author is also wrong when he states, "We used the entire sequence, not just the number of heads", but the the calculation only uses the number of heads and tails, and not the sequence.
The calcuation used:
> sum[p^12*(1 - p)^8 | p Does not have anything marking position.
You would get the same results with the sequence
HHHHHHHHTTTTTTTTTTTT as TTHHTHTTHHTHTTTTHTTH
To say "In other words, we’ve shown it’s fine to forget the particular sequence and only count the number of heads after all. What is not fine is doing so without justification." is a confused conclusion from his statements, he hasn't actually justified it, he's just used a calculation that doesn't use position (but he claimed it did) then when it got to the same answer ignoring position then said "so position doesn't matter!".
But you would have a hard time believing sequence 1 above was random.
Re: The Ghost of Statistics Past
#10One reason people might not follow this method for inference is that it depends heavily on the experimenter's personal beliefs about the world. The "improvements" here only lead to more accurate inference if the author's assumptions about the world are true. In business applications you often just want to get to a conclusion and make a decision, so this method makes sense. In scientific publication you want to verify results with a larger community with minimal assumptions. When you calculate a p-value and print it in publication, you might not be giving much information, but at least that information is objective and invariant to readers' personal beliefs. Making inference based on p-values you can at least say "In the long run I will incorrectly reject the null hypothesis 5% of the time with this method", while there is no such similar statement for a method that depends on the experimenter's personal beliefs.
In addition, the probability being calculated here is a little misleading in that it doesn't fit with the traditional definition of "probability of X". Despite the same s notation, P(H|D) is not the same type of probability as P(D|H). The coin is either biased or not, so there isn't actually any random process there and the statement "the probability this coin is fair is 50%" makes little sense under the traditional definition.