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Beautiful Probability

readthesequences.com

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Re: Beautiful Probability

#61

Post by pdonis »

lupire wrote: post #57
pdonis wrote: post #45

Earlier quoted context omitted.

> the number of roots even for the same quadratic equation may depend on "private" thoughts such as whether complex roots are of interest No. The number of roots that you care about might depend on your private thoughts; but the number of roots itself does not. It's a mathematical fact. It just might not be a mathematical fact that you actually care about. But what you care about is not part of math.

Why are you ignoring the quaternion roots? 3x3 matrix roots?

Normally "quadratic equation" means "over the complex numbers" (and the mention of "complex roots" in the post I responded to bears out that interpretation).

But yes, different mathematical models can give different answers for things like "number of roots of an equation". But that doesn't mean math depends on "private thoughts". It just means you need to specify which mathematical model you are talking about.

Re: Beautiful Probability

#62

Post by pdonis »

psychoslave wrote: post #59
pdonis wrote: post #49

Earlier quoted context omitted.

> This seems to assume that intentions "don't count" in some way, as if they were nonphysical Not nonphysical: just not part of the physical degrees of freedom that can affect things like how die rolls come up or how well a given treatment works on a patient. The experimenter's intentions (not about the stopping criterion, but about other things) can of course be upstream physical causes, so to speak, of things like…

> Not nonphysical: just not part of the physical degrees of freedom that can affect things like how die rolls come up or how well a given treatment works on a patient. For a dice that is not a concern (unless animism is taken into consideration), but when humans are on both side of the equation, how do you get rid of all the social and psychological effects that imply, including placebo and the desire to see the stud…

> when humans are on both side of the equation, how do you get rid of all the social and psychological effects that imply, including placebo and the desire to see the study bend in some direction, be it at some unconscious level?

You don't. But again, in the scenario under discussion, these are stipulated to be the same in both cases. (Or more precisely, the underlying unconscious factors involved have the same distribution in both cases.) So again, these kinds of "intentions" don't make the distributions different in the two cases.

Another comment is relevant here. The whole point of things like double blind studies in medicine is to make it the case that, whatever unconscious factors are involved along the lines you describe, they don't change the underlying distribution from which the sampled results are drawn. In the scenario as described in the article, it was assumed that all of these precautions were taken. That is part of the reason for the article's statement that the experimenters' intentions about the stopping criterion can't affect the results.

Of course if you know that in a particular case, those precautions were not taken, that changes how you view the results. But Bayesian analysis can cover this case too: you just expand your hypothesis space and your prior to include things like "the experimenters unconsciously influenced the results in different ways because of their different stopping criterion". The article excludes this possibility in the scenario it describes, but in the real world, yes, we know it is possible to have study designs that don't eliminate this failure mode, and our analysis should allow for that in cases where the study design was such that it might have happened.

Re: Beautiful Probability

#63

Post by psychoslave »

pdonis wrote: post #62
psychoslave wrote: post #59

Earlier quoted context omitted.

> Not nonphysical: just not part of the physical degrees of freedom that can affect things like how die rolls come up or how well a given treatment works on a patient. For a dice that is not a concern (unless animism is taken into consideration), but when humans are on both side of the equation, how do you get rid of all the social and psychological effects that imply, including placebo and the desire to see the stud…

> when humans are on both side of the equation, how do you get rid of all the social and psychological effects that imply, including placebo and the desire to see the study bend in some direction, be it at some unconscious level? You don't. But again, in the scenario under discussion, these are stipulated to be the same in both cases. (Or more precisely, the underlying unconscious factors involved have the same distr…

> But again, in the scenario under discussion, these are stipulated to be the same in both cases.

That's not what I read. The two studies imply that they are conducted under two very different mindsets, which will most likely also mean people will receive a different human treatment. At this point, the statistics you will get out of it is almost ornamental. Sometime the most significant information to extract from a description is not the one that is the most obvious that is pointed at as the thing you can quantify and draw comparisons.

Re: Beautiful Probability

#64

Post by pdonis »

psychoslave wrote: post #63
pdonis wrote: post #62

Earlier quoted context omitted.

> when humans are on both side of the equation, how do you get rid of all the social and psychological effects that imply, including placebo and the desire to see the study bend in some direction, be it at some unconscious level? You don't. But again, in the scenario under discussion, these are stipulated to be the same in both cases. (Or more precisely, the underlying unconscious factors involved have the same distr…

> But again, in the scenario under discussion, these are stipulated to be the same in both cases. That's not what I read. The two studies imply that they are conducted under two very different mindsets, which will most likely also mean people will receive a different human treatment. At this point, the statistics you will get out of it is almost ornamental. Sometime the most significant information to extract from a…

> The two studies imply that they are conducted under two very different mindsets, which will most likely also mean people will receive a different human treatment.

Even if the studies are conducted using the standard double blind protocol for medical and social science research, where the experimenters who have the different stopping criteria are not involved in any of the actual experimental activities? They don't do the random assignment of patients to treatment and control groups; they don't administer any of the actual treatments or placebos; they don't interact with the patients or the treatment personnel in any way during the experiment; and they don't inform anyone who is actually involved in the experimental activities of their intentions, regarding stopping criterion or anything else.

You're saying that even if all this is done--and, as I've said, this is standard procedure in medical and social science research--it is still impossible to prevent the experimenters' intentions--which aren't known to anyone else involved--from affecting the results? Remember that, as I've already said, the whole point of these double blind protocols is to prevent the experimenters' intentions from affecting the results. You're saying that's a fool's errand?

Re: Beautiful Probability

#65

Post by kgwgk »

derbOac wrote: post #56
AbrahamParangi wrote: post #18

I'm confused in that I don't see how this is troubling. Yes, the two experimenters rolled dice and got the same result, but it's as if one of them was rolling a 6 sided die and the other a 20 sided one. Each experiment is not a result per se but a sample from a distribution. How you infer the shape of that distribution based on the experiment is a function of the distribution of all courses your experiment could have…

I had the same reaction? We don't actually care at all about what happened in the two experiments per se, we care about the information provided by the experiments about future or other events. If somehow we learned that both experiments were totally unreplicable and a product purely of that time and location with no implications for anything else ever before or since we wouldn't care about them except maybe as a his…

> We don't actually care at all about what happened in the two experiments per se, we care about the information provided by the experiments about future or other events.

How can the experiments provide relevant information other than through what happened?

If what happened is exactly the same (first patient with such and such characteristics had this outcome, etc.) what information can be provided by the things that didn’t happen in either?

How could it matter that the things that didn’t happen in one experiment are different from the things that didn’t happen in the other when we are interested in the information provided by what did happen?

We don't actually care at all about the distribution of things that could have happened per se, we care about the information provided by the experiments about future or other events.

Re: Beautiful Probability

#66

Post by topologie »

xelxebar wrote: post #54
birdofhermes wrote: post #22

As other commenters have pointed out any given introductory chapter in a book on Bayesian statistics, including Jaynes’, is better exposition than this. I found _Probability Theory: The Logic of Science_ very easy to follow and very well-written. I had a similar experience when I finally found a copy of Barbour’s _The End of Time_ and discovered, much to my chagrin, that it wasn’t nearly as mystical or complicated as…

Jaynes is great, but The Logic of Science is a bit rough around the edges, with lots of errata. Jaynes died when the book was really just a very rough draft plus notes. Bretthorst had to go in and turn it into something publishable, not an enviable task by any means. Here's a list of errata and commentary, collected by a fan: https://ksvanhorn.com/bayes/jaynes/index.html .

Thank you for this!

I had spotted some errors here and there, but it's always good to have them in one place.

I think we are all in the same wagon when I say that even with those rough edges Jaynes' book is kind of a transformative experience for everyone who has already been "conditioned" to other Probability texts.

For example, for me Feller is a great intro to "start working with Probability," but Jaynes is where one starts actually "thinking in Probability."

The whole Maximum Entropy thing was mind blowing for me.

Re: Beautiful Probability

#67

Post by topologie »

birdofhermes wrote: post #22

As other commenters have pointed out any given introductory chapter in a book on Bayesian statistics, including Jaynes’, is better exposition than this. I found _Probability Theory: The Logic of Science_ very easy to follow and very well-written. I had a similar experience when I finally found a copy of Barbour’s _The End of Time_ and discovered, much to my chagrin, that it wasn’t nearly as mystical or complicated as…

100% with you on Jaynes and Barbour.

Jaynes' book is a game changer, but I particularly love that you mentioned Barbour and his work.

On Barbour's work: Apart from being an incredibly interesting book, I was amazed that he was a sort of "outsider" writing papers and books "on his own" (or at least outside of Academia) while making money through technical translations is just a really clever way to be able to explore any interesting avenues one might find. Einstein had the right idea too...

(Sadly, it's also something that wouldn't be as feasible nowadays, but who knows...)

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