Hmm. I'm not an expert, but some of this seems definitely not to be accurate. Some of the "Bullshit" turns out perhaps to be quite important.
Take the statement:
> Markov Chain is essentially a fancy name for a random walk on a graph
Is that really true? I definitely don't think so. To my understanding, a Markov process is a stochastic process that has the additional (aka "Markov") property that it is "memoryless". That is, to estimate the next state you only need to know the state now, not any of the history. It becomes a Markov chain if it is a discrete, rather than continuous process.
There are lots of random walks on graphs that satisfy this definition. Like say you have a graph and you just specify the end points and say "walk 5 nodes at random from this starting node, what is the expectation that you end up at a specific end node". This could be a Markov process. At any point to estimate the state you only need to know the state now.
But lots of random walks on graphs do not have the Markov property. For example, say I did the exact same thing as the previous example, so I have a random graph and a start and target node and I say "Walk n nodes at random from the starting node. What's the expectation that at some point you visit the target node". Now I have introduced a dependency on the history and my process is no longer memoryless. It is a discrete stochastic process and it is a random walk on a graph but is not a Markov chain.
An example of a Markov and non-Markov processes in real life is if I have a European option on a stock I only care about what the stock price is at the expiry date. But if I have a barrier option or my option has some knock-in/knock-out/autocallable features then it has a path dependence because I care about whether at any point in its trajectory the price hit the barrier level, not just the price at the end. So the price process for the barrier option is non-Markov.
low_tech_love 1 hours ago [-]
I won’t pretend to know the technical details (as the other replies do) but I want to make a point for the “pedagogical” effect here, which I agree with the author. The way I interpret the article, it’s not supposed to be a deep, theoretical treatise on the subject; more of an introductory, “intuitive” take on it. This works for those who need to either learn the concept to begin with, or refresh their memories if they don’t work with it every day. I think it’s a given that any intuitive take on a mathematical concept will always oversimplify things, with the underlying assumption that, if you actually need to know more, you’re going to have to dive deeper somewhere else. The most important thing I think is to help the reader build a rough conceptual understanding of the concept such that they can reason about it, instead of simply memorizing the terms.
NotAnOtter 2 hours ago [-]
Your overall point might be correct but your example does not prove your point.
A Makrov chain is just the path taken through the course of a markov process. The terms 'chain' and 'process' are sometimes conflated in this context, but this is the most common distinction I've seen. As such you can run a markov process for some number of steps N times, and then ask how many generated chains contain the property you are interested in. The process is memoryless but the chain is the result of the process and therefor contains memory.
I agree 'Random walk' is a superset of 'a markov process', but IMO when someone says Random walk - they normally make assumptions that qualify it as a markov chain. Therefor it's useful as a teaching to just call it a random walk.
seanhunter 2 hours ago [-]
Interesting. I'd not heard the term Markov Chain used to describe a path so I checked my copy of "All of Statistics" by Larry Wasserman and he (slightly) disagrees with both of us and says
"A Markov chain is a stochastic process for which the distribution of X_t depends only on X_{t-1}."
So he doesn't need it to be a discrete process but he also doesn't think it's a path. I guess the terminology is not 100% standardized. But anyhow thanks for making me think about this. Always interesting.
jmaker 10 minutes ago [-]
A Markov chain is commonly understood to be a time-discrete Markov process. Intuitively, it’s a “chain” because you can “single out” its states in time rather than intervals. That’s also Markov’s original definition. Instead of Wassermann one can look up the notion in Wikipedia. A path is a notion relevant to the state space of a process - it’s a realization of states at every single point in time for the time span of interest.
JohnKemeny 9 minutes ago [-]
But a random walk is precisely a stochastic process for which the _next state_ depends only on the _current state_. In terms of graphs (where _random walk_ comes from), the next node is decided by randomly selecting a neighbor of the current node.
bjornsing 2 hours ago [-]
Not all Markov processes have stationary distributions, and of those that do not all correspond to a non-normalized probability function.
It therefore has some merit to think about MCMC as a random walk on a graph rather than Markov processes, because the “graph” needs to have some properties in order for the Markov process to be useful for MCMC. For example every “node” in the “graph” needs to be reachable from every other “node” (ergodicity).
seanhunter 2 hours ago [-]
Could you explain further please? I agree with what you're saying but I don't' understand how it applies to what I said so there's definitely something I could learn here.
Edit: Thanks.
bjornsing 6 minutes ago [-]
Sure. As I see MCMC it’s basically a mathematical trick that lets you sample from probability distributions even if you only know the relative probability of different samples. It’s based on the observation that some Markov processes have a stationary distribution that is identical to a distribution you may want to sample from. But you need to carefully construct the Markov process for that observation to hold, and the properties you need to ensure are most easily understood as properties of a random walk graph. So the interesting subset of Markov processes are most easily understood as such random walks on a graph.
graycat 38 minutes ago [-]
Stochastic process with the Markov property: Past and future are conditionally independent given the present. The general version of conditionally independent is from probability theory based on measure theory and the Radon-Nikodym theorem (with von Neumann's novel proof in Rudin, Real and Complex Analysis), but an easier introduction is in Erhan Çınlar, Introduction to Stochastic Processes.
In a Poisson process the time until the next event has the Poisson distribution and, thus, from a simple calculus manipulation, is a Markov process.
E.g., time from now until a molecule of hydrogen peroxide H2O2 decomposes to water and oxygen is independent of when it was made from water and oxygen. That is the basis of half life, the same distribution until decomposition starting now no matter when the chemical or particle was created.
In WWII, searching at sea, e.g., for enemy submarines, was important, and then was Bernard O. Koopman, Search and Screening, 1946 with an argument that time to an encounter between two ships had a Poisson distribution, i.e., was a Markov process.
In grad school, there was a question about how long US submarines would last in war at sea. Well, take n Red ships and m Blue ships with, for each ship, position, speed, and detection radius and, for each Red-Blue pair, given a detection, probabilities of Red dies, Blue dies, both die, neither die (right, these four have to be non-negative and add to 1). Now have specified a Markov process that can evaluate with a relatively simple Monte-Carlo simulation.
Had written a random number generator in assembler using an Oak Ridge formula, typed quickly, and did the simulation. Had a review by a famous probability prof and passed when explained how the law of large numbers applied. So, some pure and applied math and computing worked, but some politics didn't!
seanhunter 31 minutes ago [-]
I once failed an interview with a hedge fund because they asked me a variant on that red ships/blue ships problem and at the time I knew nothing about probability. They also hadn’t given me lunch which didn’t help.
Interesting! I also make extensive notes of mathematical and computational concepts (and “buzzwords”) with a “no bullshit” title, and it works great. It’s great for quick refreshers once in a while.
emmelaich 5 hours ago [-]
Unfortunate that the equation rendering doesn't work in the body text.
pixelpoet 4 hours ago [-]
Works fine here, Firefox on Android
johnthesecure 4 hours ago [-]
I experienced this issue, but it works on another device, and after a refresh it worked on the original device. Good luck...
jokoon 3 hours ago [-]
Feels too long
I only want the python code
fedeb95 59 minutes ago [-]
it's better to understand the theory before putting up some faulty production code.
bilurrrbe 11 minutes ago [-]
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SamSeeder 2 hours ago [-]
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Rendered at 08:09:43 GMT+0000 (Coordinated Universal Time) with Vercel.
Take the statement:
> Markov Chain is essentially a fancy name for a random walk on a graph
Is that really true? I definitely don't think so. To my understanding, a Markov process is a stochastic process that has the additional (aka "Markov") property that it is "memoryless". That is, to estimate the next state you only need to know the state now, not any of the history. It becomes a Markov chain if it is a discrete, rather than continuous process.
There are lots of random walks on graphs that satisfy this definition. Like say you have a graph and you just specify the end points and say "walk 5 nodes at random from this starting node, what is the expectation that you end up at a specific end node". This could be a Markov process. At any point to estimate the state you only need to know the state now.
But lots of random walks on graphs do not have the Markov property. For example, say I did the exact same thing as the previous example, so I have a random graph and a start and target node and I say "Walk n nodes at random from the starting node. What's the expectation that at some point you visit the target node". Now I have introduced a dependency on the history and my process is no longer memoryless. It is a discrete stochastic process and it is a random walk on a graph but is not a Markov chain.
An example of a Markov and non-Markov processes in real life is if I have a European option on a stock I only care about what the stock price is at the expiry date. But if I have a barrier option or my option has some knock-in/knock-out/autocallable features then it has a path dependence because I care about whether at any point in its trajectory the price hit the barrier level, not just the price at the end. So the price process for the barrier option is non-Markov.
A Makrov chain is just the path taken through the course of a markov process. The terms 'chain' and 'process' are sometimes conflated in this context, but this is the most common distinction I've seen. As such you can run a markov process for some number of steps N times, and then ask how many generated chains contain the property you are interested in. The process is memoryless but the chain is the result of the process and therefor contains memory.
I agree 'Random walk' is a superset of 'a markov process', but IMO when someone says Random walk - they normally make assumptions that qualify it as a markov chain. Therefor it's useful as a teaching to just call it a random walk.
It therefore has some merit to think about MCMC as a random walk on a graph rather than Markov processes, because the “graph” needs to have some properties in order for the Markov process to be useful for MCMC. For example every “node” in the “graph” needs to be reachable from every other “node” (ergodicity).
Edit: Thanks.
In a Poisson process the time until the next event has the Poisson distribution and, thus, from a simple calculus manipulation, is a Markov process.
E.g., time from now until a molecule of hydrogen peroxide H2O2 decomposes to water and oxygen is independent of when it was made from water and oxygen. That is the basis of half life, the same distribution until decomposition starting now no matter when the chemical or particle was created.
In WWII, searching at sea, e.g., for enemy submarines, was important, and then was Bernard O. Koopman, Search and Screening, 1946 with an argument that time to an encounter between two ships had a Poisson distribution, i.e., was a Markov process.
In grad school, there was a question about how long US submarines would last in war at sea. Well, take n Red ships and m Blue ships with, for each ship, position, speed, and detection radius and, for each Red-Blue pair, given a detection, probabilities of Red dies, Blue dies, both die, neither die (right, these four have to be non-negative and add to 1). Now have specified a Markov process that can evaluate with a relatively simple Monte-Carlo simulation.
Had written a random number generator in assembler using an Oak Ridge formula, typed quickly, and did the simulation. Had a review by a famous probability prof and passed when explained how the law of large numbers applied. So, some pure and applied math and computing worked, but some politics didn't!
2022 (233 points, 53 comments) https://news.ycombinator.com/item?id=33332437
2016 (125 points, 20 comments) https://news.ycombinator.com/item?id=12537043
2015 (114 points, 17 comments) https://news.ycombinator.com/item?id=9331808
I only want the python code