Probability, Mathematical Statistics, Stochastic Processes(randomservices.org)
randomservices.org
Probability, Mathematical Statistics, Stochastic Processes
http://www.randomservices.org/random/
13 comments
> Fun fact to get you started, Nakamoto suggested in the original Bitcoin paper that blocks would be added to the Bitcoin blockchain according to a homogenous Poisson process (spoiler alert: it's definitely not).
Care to elaborate why it’s not a Poisson process?
Care to elaborate why it’s not a Poisson process?
It's definitely not a homogenous Poisson process, mainly because of the random changes to mining difficulty and propagation delays. There's a good paper here looking at block arrival times and fitting some different models - https://arxiv.org/pdf/1801.07447.pdf
Is Figure 8 an unconditional empirical CDF of inter-arrival times? Apart from the heavy right tail (which covers ~0.01% of the data), it looks pretty exponential to me. If I'm understanding what I'm seeing, it sounds like like the homogeneous Poisson assumption was pretty solid. Especially considering its purpose. Maybe it would have been more accurate to say "there's a mixture of two Poissons: the bulk and the network disruption". But I think that possibility would occur to most people reading the paper at the time.
Also, Figure 7 seems to show very little change in mean block inter-arrival time.
In fairness the authors say, "Performing the Lilliefors test on the LR data rejects the null hypothesis that block mining intervals are exponentially distributed, at a significance level of α= 0.05." But this isn't physics. We want to know how useful the approximation is, and whether there is a similarly tractable one with better predictive power.
Also, Figure 7 seems to show very little change in mean block inter-arrival time.
In fairness the authors say, "Performing the Lilliefors test on the LR data rejects the null hypothesis that block mining intervals are exponentially distributed, at a significance level of α= 0.05." But this isn't physics. We want to know how useful the approximation is, and whether there is a similarly tractable one with better predictive power.
> Is Figure 8 an unconditional empirical CDF of inter-arrival times?
My understanding is that it's the inter-arrival times after some cleaning and resampling. If I've understood correctly, when they resampled the data, they did so uniformly between the neighbours of the points they omitted, which would actually make the data appear more like an exponential distribution.
> Especially considering its purpose. Maybe it would have been more accurate to say "there's a mixture of two Poissons: the bulk and the network disruption".
Could be. Could also follow a power law or a phase type distribution.
> But this isn't physics. We want to know how useful the approximation is, and whether there is a similarly tractable one with better predictive power.
It's worse, it's math :-) I take your point though, it all comes down to what you're trying to do. If inter-arrival times did follow an exponential distribution with parameter $\lambda$, then we'd have finite variance and I'd be pretty confident that I could build a performant predictive model. The presence of a heavy right tail makes me think otherwise.
My understanding is that it's the inter-arrival times after some cleaning and resampling. If I've understood correctly, when they resampled the data, they did so uniformly between the neighbours of the points they omitted, which would actually make the data appear more like an exponential distribution.
> Especially considering its purpose. Maybe it would have been more accurate to say "there's a mixture of two Poissons: the bulk and the network disruption".
Could be. Could also follow a power law or a phase type distribution.
> But this isn't physics. We want to know how useful the approximation is, and whether there is a similarly tractable one with better predictive power.
It's worse, it's math :-) I take your point though, it all comes down to what you're trying to do. If inter-arrival times did follow an exponential distribution with parameter $\lambda$, then we'd have finite variance and I'd be pretty confident that I could build a performant predictive model. The presence of a heavy right tail makes me think otherwise.
Its been a while, but I just read an article arguing the case that Len Sassaman was a Satoshi. It was a neat article, so I watched one of Len's Defcon talks about remailers from waaay back in the day.
In his talk, Len mentioned that most remailer security analysis assumes homogeneous Poisson email arrivals. He pointed out how bad an assumption that is for email.
I still think it was a solid assumption in the Bitcoin white paper.
https://leung-btc.medium.com/len-sassaman-and-satoshi-e483c8...
In his talk, Len mentioned that most remailer security analysis assumes homogeneous Poisson email arrivals. He pointed out how bad an assumption that is for email.
I still think it was a solid assumption in the Bitcoin white paper.
https://leung-btc.medium.com/len-sassaman-and-satoshi-e483c8...
It wouldn't be if block additions caused more additions, which is perhaps plausible?
Calculus 2,3 and Real Analysis can also be understood with undergrad calculus.
:)
:)
I remember reading. That equation and it never made sense interms of where it actually fit in bitcoins code.
Thanks for elaborating
Thanks for elaborating
What an excellent site, it's full of probability, stochastic and other such stuff that I need to brush up on as well topics that I know little about.
As a resource, I reckon it'd be pretty hard to beat as its range of subject matter is about as comprehensive as most would ever want.
That said, each topic is covered extensively and in depth which may be a problem for someone who is approaching a new aspect of the subject for the first time or who just wants a general overview of the subject. (Some of the topics I thought I was reasonably familiar with set me back a little with the sheer death of detail.)
As a resource, I reckon it'd be pretty hard to beat as its range of subject matter is about as comprehensive as most would ever want.
That said, each topic is covered extensively and in depth which may be a problem for someone who is approaching a new aspect of the subject for the first time or who just wants a general overview of the subject. (Some of the topics I thought I was reasonably familiar with set me back a little with the sheer death of detail.)
On a side note, randomservices.org is a great resource. Used it as a reference often through my statistics undergrad.
agreed. it is a very very good reference for a lot of stats and probability at even the grad level.
I found this recently and was super impressed. A really great (and well-organized) reference!
Fun fact to get you started, Nakamoto suggested in the original Bitcoin paper that blocks would be added to the Bitcoin blockchain according to a homogenous Poisson process (spoiler alert: it's definitely not).