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Probability, Mathematical Statistics, Stochastic Processes

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Re: Probability, Mathematical Statistics, Stochastic Processes

#4

Post by cesarosum »

To the programmer, developer or casual visitor looking at this and wondering whether it's worth the time and effort to dig into this, it is. Most of what's covered here can be understood with undergrad calculus, and will give you a solid basis for understanding and modelling random phenomena you may encounter in your studies, work or hobby.

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).

Re: Probability, Mathematical Statistics, Stochastic Processes

#5

Post by drexlspivey »

cesarosum wrote: post #4

To the programmer, developer or casual visitor looking at this and wondering whether it's worth the time and effort to dig into this, it is. Most of what's covered here can be understood with undergrad calculus, and will give you a solid basis for understanding and modelling random phenomena you may encounter in your studies, work or hobby. Fun fact to get you started, Nakamoto suggested in the original Bitcoin paper…

> 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?

Re: Probability, Mathematical Statistics, Stochastic Processes

#6

Post by Schiphol »

drexlspivey wrote: post #5
cesarosum wrote: post #4

To the programmer, developer or casual visitor looking at this and wondering whether it's worth the time and effort to dig into this, it is. Most of what's covered here can be understood with undergrad calculus, and will give you a solid basis for understanding and modelling random phenomena you may encounter in your studies, work or hobby. Fun fact to get you started, Nakamoto suggested in the original Bitcoin paper…

> 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?

It wouldn't be if block additions caused more additions, which is perhaps plausible?

Re: Probability, Mathematical Statistics, Stochastic Processes

#7

Post by cesarosum »

drexlspivey wrote: post #5
cesarosum wrote: post #4

To the programmer, developer or casual visitor looking at this and wondering whether it's worth the time and effort to dig into this, it is. Most of what's covered here can be understood with undergrad calculus, and will give you a solid basis for understanding and modelling random phenomena you may encounter in your studies, work or hobby. Fun fact to get you started, Nakamoto suggested in the original Bitcoin paper…

> 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?

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

Re: Probability, Mathematical Statistics, Stochastic Processes

#8

Post by hilbert42 »

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.)

Re: Probability, Mathematical Statistics, Stochastic Processes

#9

Post by quasirandom »

cesarosum wrote: post #7
drexlspivey wrote: post #5

Earlier quoted context omitted.

> 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?

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.

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