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The quote from the article in Encyclopedia of Biostatistics is awash in undefined terminology sometimes about peripheral issues. Clean, logical, all terms well

The quote from the article in Encyclopedia of Biostatistics is awash in undefined terminology sometimes about peripheral issues. Clean, logical, all terms well defined and explained, with plenty of advanced content, is in (with some markup using TeX) Erhan \c Cinlar, {\it Introduction to Stochastic Processes,\/} ISBN 0-13-498089-1, Prentice-Hall, Englewood Cliffs, NJ, 1975.\ \ The author was long at Princeton. He is a high quality guy. As I was working my way through grad school in a company working on US national security, a question came up about the survivability of the US SSBN fleet under a special scenario of global nuclear war but limited to sea. Results were wanted in two weeks. So, I drew from Çinlar's book, postulated a Markov process subordinated to a Poisson process, typed some code into a text editor, called a random number generator I'd written in assembler based on the recurrence X(n+1) = X(n) 5^15 + 1 mod 2^47 and was done on time. A famous probabilist was assigned to review my work. His first remark was that there was no way for my software to "fathom" the enormous "state space". I responded, at each time t, the number of SSBNs left is a random variable, finite, with an expectation. So, I generate 500 sample paths, take their average, use the strong law of large numbers, and get an estimate of their expected value within a "gnat's ass" nearly all the time. "The Monte Carlo puts the effort where the action is." The probabilist's remark was "That is a good way to think of it." Need to do some work with Markov chains, simulation, etc.? Right, just read some Çinlar, not much in prerequisites (he omitted measure theory), get clear explanations, no undefined terminology, from first principles to some relatively advanced material, and be successful with your project.

wuschel 4y ago
Thank you for the recommendation. The book looks serious to me - it reminds me my math lectures during my chemistry studies in university. How much time would you think does it take to work through it to get some value/insight? Is this even the right material for such endeavour?
- graycat 4y ago
  For background, need only some routine calculus and the most elementary probability theory. The chapter on the Poisson process, early in the book, is nice to amazing, e.g., his axiomatic derivation. You should be able to get the main ideas in 1-2 evenings of fun, pleasant reading. Science is awash in Poisson processes; it is nice to understand more about them. That they have no memory seems to me like maybe a thin thread could pull to say some things about what is going on in the internals of some of those cases of Poisson processes. I.e., you are looking at some system with some unknown internals, some working mechanism, but the evidence is that it has no memory. The system has some internal state but no memory? How can that be? So, we have some possibly severe limitations on how the internals work. Now I'm concentrating mostly on my startup and deliberately setting aside pursuing such questions -- maybe later. But you, maybe in chemistry, quantum lifetimes, time until a nucleus splits, ..., glitches in complicated computer systems, .... Similarly for the other chapters. If you take the book fully seriously, e.g., carefully follow all the proofs and work most of the exercises, it's a good one semester course.
  - wuschel 4y ago
    > Science is awash in Poisson processes; it is nice to understand more about them (...) Agreed :) Although I never dared to explore them in the depth you are describing in your text. > If you take the book fully seriously, e.g., carefully follow all the proofs and work most of the exercises, it's a good one semester course. Thank you for an approximate time quantification. As a side question - judging from you comments you seem to the right person to ask: What kind of approach would you recommend to get a level in statistics strong enough in theoretical and practical aspects for industrial application? Would you be so kind to recommend any courses or books? > Now I'm concentrating mostly on my startup Out of curiosity: What are of industry are you operating in? Cheers!
heintje_ghulam 4y ago
Thank you for the book recommendation. I have gained a lot from reading your math study recommendations over the past few years. I wish I had more time/motivation to fully follow-through with them.
- graycat 4y ago
  As in the book, a Poisson process is an arrival process -- omitting the math, some of which is amazing, the vanilla example is Geiger counter clicks. One of the special things, actually an easy calculus exercise, is: Suppose the clicks come on average once a second. Suppose have not gotten a click for time t >= 0. E.g., maybe t is 2 seconds. Now what is the average time to the next click, less than the 1 second? Nope, the average now (a statement in conditional probability) is STILL the 1 second. For a Markov chain, that is a discrete time, discrete state space Markov process. Halt: In a Markov process, at any time t, the past and future are conditionally independent given the present. E.g., for predicting the future, if know the present, then also knowing the past does not help (or hurt, really). So, Markov is an enormous simplification. In particular, a Poisson process is a Markov process. For the discrete time, discrete state space Markov process, i.e., a Markov chain, subordinated to a Poisson process, the time to the next change (jump) is a Poisson process. A discrete time, discrete state space Markov process does not have to have anything to do with a Poisson process and, instead, can have, say, one jump an hour on the hour. But with subordinated to a Poisson process, the expected time to the next jump (arrival), the parameter of the process, can depend on the state. E.g., in the war at sea problem, as the ships sink, the rate at which they sink can slow down. There was a WWII calculation by Koopmans that gave the encounter rate of multiple ships at sea. The formula used the area of the sea, the number of ships, etc. So, that is where I got the encounter rates. Then the main input was the number of Red ships, the number of Blue ships, the speed of each ship, and for any Red-Blue encounter the probability the Red dies, the Blue dies, both die, or neither die. That was enough data to generate sample paths (war simulations) of the assumed scenario. Silly, sure, but okay for one grad student in a rush in just two weeks! The above may be enough for you to read and understand the details in the book quickly. But for more the book also covers limiting distributions (as time evolves, the probability distribution of the states approaches a limit), Kolmogorov equations, branching processes, etc. An example of branching processes is some of the simple cases of nuclear fission. Now you have saved quite a lot of time and gotten ready for a fairly easy time in the book!

wuschel 4y ago

Thank you for the recommendation. The book looks serious to me - it reminds me my math lectures during my chemistry studies in university. How much time would you think does it take to work through it to get some value/insight? Is this even the right material for such endeavour?

graycat 4y ago
For background, need only some routine calculus and the most elementary probability theory. The chapter on the Poisson process, early in the book, is nice to amazing, e.g., his axiomatic derivation. You should be able to get the main ideas in 1-2 evenings of fun, pleasant reading. Science is awash in Poisson processes; it is nice to understand more about them. That they have no memory seems to me like maybe a thin thread could pull to say some things about what is going on in the internals of some of those cases of Poisson processes. I.e., you are looking at some system with some unknown internals, some working mechanism, but the evidence is that it has no memory. The system has some internal state but no memory? How can that be? So, we have some possibly severe limitations on how the internals work. Now I'm concentrating mostly on my startup and deliberately setting aside pursuing such questions -- maybe later. But you, maybe in chemistry, quantum lifetimes, time until a nucleus splits, ..., glitches in complicated computer systems, .... Similarly for the other chapters. If you take the book fully seriously, e.g., carefully follow all the proofs and work most of the exercises, it's a good one semester course.
- wuschel 4y ago
  > Science is awash in Poisson processes; it is nice to understand more about them (...) Agreed :) Although I never dared to explore them in the depth you are describing in your text. > If you take the book fully seriously, e.g., carefully follow all the proofs and work most of the exercises, it's a good one semester course. Thank you for an approximate time quantification. As a side question - judging from you comments you seem to the right person to ask: What kind of approach would you recommend to get a level in statistics strong enough in theoretical and practical aspects for industrial application? Would you be so kind to recommend any courses or books? > Now I'm concentrating mostly on my startup Out of curiosity: What are of industry are you operating in? Cheers!

heintje_ghulam 4y ago

Thank you for the book recommendation. I have gained a lot from reading your math study recommendations over the past few years. I wish I had more time/motivation to fully follow-through with them.

graycat 4y ago
As in the book, a Poisson process is an arrival process -- omitting the math, some of which is amazing, the vanilla example is Geiger counter clicks. One of the special things, actually an easy calculus exercise, is: Suppose the clicks come on average once a second. Suppose have not gotten a click for time t >= 0. E.g., maybe t is 2 seconds. Now what is the average time to the next click, less than the 1 second? Nope, the average now (a statement in conditional probability) is STILL the 1 second. For a Markov chain, that is a discrete time, discrete state space Markov process. Halt: In a Markov process, at any time t, the past and future are conditionally independent given the present. E.g., for predicting the future, if know the present, then also knowing the past does not help (or hurt, really). So, Markov is an enormous simplification. In particular, a Poisson process is a Markov process. For the discrete time, discrete state space Markov process, i.e., a Markov chain, subordinated to a Poisson process, the time to the next change (jump) is a Poisson process. A discrete time, discrete state space Markov process does not have to have anything to do with a Poisson process and, instead, can have, say, one jump an hour on the hour. But with subordinated to a Poisson process, the expected time to the next jump (arrival), the parameter of the process, can depend on the state. E.g., in the war at sea problem, as the ships sink, the rate at which they sink can slow down. There was a WWII calculation by Koopmans that gave the encounter rate of multiple ships at sea. The formula used the area of the sea, the number of ships, etc. So, that is where I got the encounter rates. Then the main input was the number of Red ships, the number of Blue ships, the speed of each ship, and for any Red-Blue encounter the probability the Red dies, the Blue dies, both die, or neither die. That was enough data to generate sample paths (war simulations) of the assumed scenario. Silly, sure, but okay for one grad student in a rush in just two weeks! The above may be enough for you to read and understand the details in the book quickly. But for more the book also covers limiting distributions (as time evolves, the probability distribution of the states approaches a limit), Kolmogorov equations, branching processes, etc. An example of branching processes is some of the simple cases of nuclear fission. Now you have saved quite a lot of time and gotten ready for a fairly easy time in the book!