Questions tagged [markov-chains]
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65
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Question about Markov Chains
The following question is taken from the book titled "Probability models for Computer Science" written by Sheldon M. Ross.
Question:
A particle moves along n + 1 vertices that are situated ...
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1answer
38 views
Q: Defining Markov-Chain Probability Matrix States | Specific Academic Paper
I am working on a uni project, for which i will model synthetic electricity consumption time series using Markov-Chains. I am reading through current academic literature on that topic and came across ...
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24 views
How long a graph random walk takes to hit every vertex?
I have a simply connected graph $G$. I start at a uniformly randomly chosen vertex, and from there, randomly walk through the graph by choosing a random edge to follow at each step.
On average, how ...
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4 views
Inferring reward function and transition model from optimal policy
Consider an MDP where the transition model and the reward function are unknown.
Consider an optimal policy $\pi^*$ generated from this MDP (say by some oracle who does know the transition model and ...
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24 views
Conditional dependence of Markov models
When studying Markov models, I see this equation but I don't understand where does it come from:
$$ P(x,y,z) = P(x) P(y|x) P(z|y) = \frac{P(x) P(x,y) P(z,y)}{P(x) P(y)} $$
Can anyone explain to me?
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15 views
Degree of regularity of a Markov chain
A Markov chain with transition matrix $P$ is termed regular if for some $n$, all entries of $P^n$ are positive.
Is there a known notion of degree of regularity quantified in terms of how soon all ...
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17 views
implementing a 802.15.4 math model based on a paper. How to verify it
I am implementing a mathematic model for predicting the congestion of 802.15.4 network based on the paper:
Performance analysis of IEEE 802.15.4 non-beacon mode with the unslotted CSMA/CA
Based on the ...
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1answer
204 views
Transition Function in MDP
I got a question about who and how sets the transation function values in markov decision processes?
I mean when some says that an agent, in real world grid, is going to step up by %80 and left/right ...
2
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1answer
76 views
Policy dependent on initial state distribution in finite horizon MDPs
Consider an MDP defined as the tuple $\langle S,A,R,P,\mu,\lambda\rangle$ where $S$ is the state space, $A$ the action space, $R:S\times A\times S\to\mathbb{R}$ the reward function, $P$ the transition ...
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1answer
24 views
Asymmetric Transition Probability Matrix with uniform stationary distribution
I am solving a discrete Markov chain problem. For this I need a Markov chain whose stationary distribution is uniform(or near to uniform distribution) and transition probability matrix is asymmetric.
...
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34 views
Is there any toolbox for Markov Random Field Structure Learning?
I need a toolbox or software that takes a dataset as input, detect independencies among its random variables and produces the relative Markov Random Field graphical structure from that. Can anyone ...
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17 views
Concrete practical examples for discrete time markov chain with transition rewards
Can anyone show examples for discrete-time markov chains with associated transition rewards?
I am searching for examples of industrial usage of markov chains where their expect and variance or ...
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1answer
90 views
Probability of terminating in a state in a probabilistic algorithm
Suppose i have a circular array of $n$ elements.
At time $t=0$ i am in position 0 of the array. The algorithm moves left or right with probability $p=1/2$ (since the array is circular when it moves ...
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0answers
28 views
Find consensus trajectory of how a genetic algorithm solves an optimization
I have implemented a genetic algorithm to find the evolutionary outcomes of a biological scenario. I simulate the evolution (i.e. optimization) of five traits in my model. I ran my code 100 times and ...
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0answers
60 views
Monopoly Matrix Coding
I am an IB student doing SL math. For my math IA i picked the topic of monopoly and the optimal way to play. However, in this IA there is a massive matrix that needs to be built to see how the long ...
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1answer
91 views
Understanding simulated annealing information theoretically
So I recently rediscovered simulated annealing though a path that others seem to be well aware of. I was aware of Metropolis-Hastings as a sampling algorithm that creates a Markov-Chain who's ...
2
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2answers
59 views
Modeling a set of probabilistic concurrent processes
I'm looking into discrete-time Markov chains (DTMCs) for use in analyzing a probabilistic consensus protocol. One basic thing I haven't been able to figure out is how to model a set of independent ...
3
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1answer
55 views
Algorithm for computing $Pr[s \vDash C \bigcup^{\geq n} B]$ for probabilistic verification
I'm having some difficulty trying to come up with an algorithm for computing $Pr[s \vDash C ~\bigcup^{\geq n} B]$ given a finite Markov chain where $S$ is the set of states, $s \in S$, $B,C \subseteq ...
2
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1answer
293 views
Calculating probability of reaching state in DTMC
Consider a highly-connected graph of states & transitions where each transition is marked with a weight (representing probability of occurring) and the graph satisfies the Discrete Time Markov ...
1
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1answer
68 views
Vorticity Matrix for Markov chain
I have a markov chain with $Q(u,v)$ as transition probability matrix and $\pi(u)$ as stationary distribution defined on state space $\Omega$. The dimension of matrix $Q$ is $nxn$ and vector $\pi$ is $...
0
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1answer
141 views
information theory, find entropy given Markov chain
There is an information source on the information source alphabet $A = \{a, b, c\}$ represented by the state transition diagram below:
a) The random variable representing the $i$-th output from this ...
1
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1answer
80 views
How to find optimal token set for compression?
By token I mean the smallest element of source data, that the compression algorithm works on. It may be a bit (like in DMC), a letter (like in Huffman or PPM), a word, or variable-length string (like ...
1
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1answer
32 views
Probabilisitc timed automaton
I am kind of new to timed automaton domain. I am trying to understand in which way they are different to Markov Decision Process. First I know there objectives is to solve the non-determinism of a MDP....
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1answer
75 views
Perturbing a Markov chain to be closer to a target stationary distribution
Suppose we are a given a Markov chain $A_0 \in \mathbb{R}^{n \times n}$ and a desired stationary probability vector $\pi_0 \in \mathbb{R}^n$. I would like to find a Markov chain that is as close as ...
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1answer
46 views
Sampling Multiple Times From a Markov Chain
In probability theory, the mixing time of a Markov chain is the time until the Markov chain is "close" to its steady state distribution. Lets fix some $\varepsilon\in (0,1]$ as our closeness parameter....
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41 views
Card Shuffling, Bounding Mixing time using Paths and Flows
I've been struggling with a problem that is very similar to a 2014 question posted here. The question in particular is 3(1) and 3(2).
To paraphrase, we are supposed to use paths and an encoding ...
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77 views
The relationship between MDP's end component and its induced MC recurrent class
Let's assume that I have an MDP $M$ which has a number of maximal end components. I also have a random policy (scheduler) $\pi$ that can convert the MDP $M$ into a Markov chain $m$. Can I argue that ...
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0answers
74 views
Game with Random Digits (Markov Chain / Coupling)
I've been self-studying Markov Chains and came across a problem online here: http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf
I'm not asking for anything too formal (I'...
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1answer
1k views
How many possible policies in a Markov Decision Process?
If a policy yields an action for a state, how come a 3-state MDP with 2 possible actions, i.e. $S = \{Hot, Mild, Cold\}$, $A = \{East, West\}$, has 8 possible policies? Isn't it 6 if there are 2 ...
1
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1answer
373 views
Why in simulated annealing, thermal equilibrium need to be meet for each temperature throughout the iterations?
I have read in this book, simulated annealing, on page 40-41, if temperature $t$ tend to $0$, then the stationary distribution will be distributed among optimal solutions. What i can not understand ...
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3answers
523 views
Why do we need Gibbs sampling (and MCMC)?
I just learned about Gibbs Sampling which is an MCMC method. Given a distribution $\pi$, we want to sample an item according to $\pi$.
Maybe my alternative suggestion would sound somewhat naive (even ...
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0answers
142 views
Second-order Markov text generation?
Looking at this video starting at 1:45, the author claims to be using a second-order approximation for a Markov text generation. He has one letter which he outputs followed by another letter which ...
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1answer
63 views
Maximize entrywise 1-norm of matrix product
I have $8$ sets $E_1, \ldots, E_8$, each one of size $256$, whose elements are $4\times 4$ matrices of (nonnegative) integers. I'd like to design an algorithm to find:
$$\underset{M_1 \in E_1, \...
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0answers
89 views
Solving the following recurrence relation derived from a Markov chain
I have the following system of recursive relations on $y_{i,j}$ that are derived from a Markov chain and that I am having difficulty in solving. For $i\ge 1$ and $j \ge 1$, we have
$$y_{i,j} \times (...
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1answer
1k views
Absorbing Markov Chains: An efficient algorithmic approach
Following this procedure I have successfully written a program to calculate the probability of ending in a given absorbing state given the initial state. The procedure is as follows:
Given the ...
3
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0answers
103 views
Form of conditional observation probabilities in a POMDP
Consider a partially observable Markov decision process (POMDP), see here for a complete definition.
My question is in relation to the conditional observation probabilities (denoted by $O(o|s',a)$ in ...
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0answers
189 views
Calculate expected values of "Craps Game" with the help of Prism Model Checker
I have modelled the craps game (https://en.wikipedia.org/wiki/Craps#Rules_of_play) as a dtmc with the prism model checker:
...
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195 views
probability matrix from digraph adjacency matrix
All I have in hand is a adjacency matrix of a digraph with equal weight on every edge, is there a very simple way to convert this to a state change probability matrix?
2
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1answer
335 views
Convergence of Markov model
I was learning Hidden Markov model, and encountered this theory about convergence of Markov model.
For example, consider a weather model, where on a first-day probability of weather being sunny was 0....
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0answers
62 views
Calssifying the Partitions of the problem Cycle Decomposition of Markov Chains
The book Cycle Representations of Markov Processes
solves the problem of Mapping Stochastic Matrices induced from a Markov Chain into Partitions using a $\lambda$-preserving ($\lambda$ is a Lebesgue ...
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0answers
190 views
How the conditional random fields algorithm work? [closed]
Im searching about conditional random fields for a long time and cant understand the concepts of this algorithm.
I understood markov chains, and understood that conditional random field is a more ...
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0answers
294 views
What's the relationship of max-flow-min-cut and Markov Random Fields?
I am trying to follow this paper [1]. There is a relationship between Markov Random Fields (MRF) to max-flow-min-cut. An MRF can be represented as an undirected graph, and you can find flow through it,...
3
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2answers
503 views
The transition function in a Markov decision process
A Markov decision process is typically described as a tuple $\langle A,U,T,R \rangle $ where
$A$ is the state space
$U$ is the action space
$T: A \times U \times A \mapsto [0,\infty) $ is the state ...
2
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1answer
102 views
Orderability of Belief States in a POMDP?
Consider a POMDP with integer states $1,2,\ldots,N$, where $N$ is finite. We thus have a complete order over the states.
It seems reasonable to think that belief states for this POMDP may be ...
1
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1answer
718 views
Markov Chain Mixing Time of the Complete Graph
I'm having a hard time understanding mixing time for Markov Chains on Complete Graphs (Kn).
We can define the probability matrix for Kn where Pi,j=probability of going from i to j (technically 1/...
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1answer
128 views
Clarification of the definition of a POMDP
From what I understand, a $MDP=(G, A, P, R)$ (markov decision process) is represented as:
A complete directed graph $G=(V, E)$
A set of actions $A_u$ for each vertex $u \in V$
A reward function $R$ ...
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0answers
185 views
Cheeger constant of a graph versus conductance of a Markov chain
Given some graph $G$ with vertices $V$ and edges $E$, its Cheeger constant $h(G)$ is well defined as
$$ h(G) = \min_{S\subset V,0<|S|\leq|V|}\frac{|\partial S|}{|S|}. $$
Given some doubly-...
7
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1answer
340 views
Can the solution to a POMDP be found using linear programming?
It is known that Markov decision processes (MDPs) can be solved using linear programming (see page 24 of Carlos Guestrin's PhD dissertation). The linear program is:
$$min_{V(x)} \sum_x \alpha(x)V(x)\\...
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197 views
Multicommodity flows with minimum congestion: NP-hard?
I have a question related to a paper of Chen, Lovasz and Pak [1].
The paper concerns the construction of the Markov chain with optimal mixing time on an arbitrary graph.
They prove the optimal bound (...
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1answer
83 views
Simulating continuous time semi-Markov state machine and changing transition probability on the fly
The problem that I'm trying to solve (well, I think that I almost did, but need a review from someone more experienced) is about changing probability of the transition for semi-Markov state machine ...
