Questions tagged [markov-chains]
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Are these representations of the Bellman Equation equivalent?
I've found two slightly different Bellman Equations. Are they totally equivalent?
I see the one on the bottom has an s' in the reward. Does this or anything else about the groupings change anything ...
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In Viterbi's algorithm what is the difference between the observation space and the sequence of observations?
Wikipedia has an explanation for the viterbi algorithm, in particular it describes the following summary of the inputs:
What is the difference between O and Y? Also I am trying to understand what the ...
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How to draw a Markov chain model for this simulation
A node of a data communication network which can transmit data at 4 different rates r, 2r, 3r & 4r uses the mechanism described below to control congestion: At any given time the node is ...
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Counting number of copies of a given tree T in a graph G. Looking for a randomised algorithm which is an FPRAS
I'm looking for a randomised algorithm (specifically an epsilon-delta approximation) which takes as input a graph G, a subgraph T (which is a tree), and outputs an approximation to the total number of ...
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Observable Markov Model: Expected number of observations
I have a question that asks me "What is the expected number of observations in a state?" with the note:
$$\sum^{\infty}_{d=1}d a^{d=1} = \frac{1}{(a-1)^2}\text{ when } |a| < 1$$
Prior to ...
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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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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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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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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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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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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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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 ...
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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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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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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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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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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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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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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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Understanding simulated annealing information theoretically
So I recently rediscovered simulated annealing through 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 whose ...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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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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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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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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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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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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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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 ...
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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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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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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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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:
$$\underset{M_1 \in E_1, \...
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Solving the following recurrence relation derived from a Markov chain
I have the following system of recursive relations on $y_{i,j}$ that are derived from a Markov chain and that I am having difficulty in solving. For $i\ge 1$ and $j \ge 1$, we have
$$y_{i,j} \times (...
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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 ...
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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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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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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?
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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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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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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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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,...
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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 ...
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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 ...
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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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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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