All Questions
24 questions
1
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48
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Sample complexity of estimating a doubly stochastic matrix
Let $P\in\mathbb{R}^{n\times n}$ be a doubly-stochastic matrix. That is:
$$P(x,y)\geq 0,\quad \sum_xP(x,y)=1,\quad \sum_yP(x,y)=1.$$
I would like to know if lower and upper bounds on the sample ...
- 111
4
votes
1
answer
276
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About non-reversible Metropolis Hastings Markov chain
I am reading a paper about constructing a non-reversible Metropolis Hastings Markov chain from a reversible one as described at a high level in paragraph $3$ of page $1$.
But I don't understand how, ...
- 201
1
vote
1
answer
410
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Occupation times for two-state Markov processes
Consider a two-state Markov process in continuous time, with states labelled $A$ and $B$. The transition rates for going from state $A$ to $B$, and state $B$ to $A$ are $\alpha$ and $\beta$ ...
1
vote
1
answer
189
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If a Markov semigroup is eventually contractive, can we conclude that it admits a unique invariant measure?
Let $E$ be a separable $\mathbb R$-Banach space, $\rho$ be a complete separable metric on $E$, $\operatorname W_\rho$ denote the Wasserstein metric of order $1$ associated to $\rho$, $\mathcal M_1(E)$ ...
- 167
1
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0
answers
73
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Reduce the asymptotic variance for a class of Metropolis-Hasting estimates
I'm running the Metropolis-Hastings algorithm with state space $E$, target distribution $\mu=p\lambda$ and proposal kernel $Q$ to estimate $\mu(hf)$ for a fixed function $f:E\to[0,\infty)^3$ and a ...
- 167
1
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0
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56
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Minimizing the rate of geometric ergodicity of a Metropolis-Hastings kernel depending on a parameter
Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$.
I want to ...
- 167
1
vote
1
answer
170
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Stationary distribution of Markov Chain with departure
I have a Markov Chain of $N$ states. Such states represent the energy levels in a molecule.
The states' connectivity is as follows:
States $j\in\{0,\ldots,N\}$ transition to $k\in\{\max(j-M,0),...,\...
- 151
4
votes
5
answers
7k
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Proof of Bellman optimality equation for finite Markov Decision Processes
This question has already been posed in Cross Validated without receiving a correct formal answer, so I reformulate it here to gain attention of mathematicians. I am referring to chapter 3 of Sutton ...
- 171
1
vote
2
answers
302
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how to derive stationary distribution of maximal entropy random walk
I was reading the paper 0810.4113v2, burda, which analyzed the stationary distribution maximal entropy random walk on the irregular lattice. I am confused on some of the steps.
Description:
The ...
- 211
1
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1
answer
140
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Reference request: Cover times, Mixing Times and DGFF applied in statistics?
I am trying to find if in active research in statistics, there is interest in mixing times, cover times of graphs, and/or the discrete Gaussian free field?
I haven't found anything so far for the ...
- 11
5
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0
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485
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Hierarchical Random Walk (also known as Hierarchical Hidden Markov Model)
Let us consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/...
- 1,127
4
votes
2
answers
255
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The necessary sufficient condition for recurrence of a Markovian random walk
Suppose $\sigma_{1},\sigma_{2},...$are i.i.d random variables.$S_{0}=0$. Define $S_{n}=S_{0}+\sum_{i=1}^{n}\sigma_{i}$, then ${S_{n}}$ is a Markovian random walk.
I want to figure out the necessary ...
- 41
1
vote
0
answers
44
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Validating a probability density distribution forecast model for a Markov process
Let's say we have a Markov process $X_t$, and we come up with a forecast model that takes some information from outside world and says: "value $X_{t+1}$ has probability density distribution $P_t(x)$". ...
- 93
1
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0
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101
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How to fit a stochastic matrix to given data.?
Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
1
vote
2
answers
1k
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Gibbs sampler with linear constraints
My problem concerns the estimation of truncated multivariate normal distributions under constraints.
Let $X_1$ and $X_2$ two random variables following normal distributions $\mathcal{N_1}(m_1,\...
- 45
2
votes
1
answer
1k
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forward algorithm Hidden Markov Model
I am studying the the forward-backward algorithm used in Hidden Markov Models. I understand that you are trying to propagate through a sequence (and the available states) to find the most probable ...
- 123
0
votes
1
answer
408
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Generating independent random variable from two correlated random variables
Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...
- 1,109
3
votes
0
answers
494
views
Maximization of a total variation distance subject to another total variation distance in Markov chain
Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
- 1,109
3
votes
1
answer
188
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Markov Chains based on sampled transition probabilities [closed]
If I have a process that transitions between states with some set, unknown probability, I can sample to find the transition probability. This probability is a sample average, with a well understood ...
- 191
2
votes
1
answer
356
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The first eigenvalue of a branching process matrix
Let $M$ be the real square matrix of a typed branching process, such that $M_{ij}$ is the expected value of offspring of type $j$ emanating from type $i$.
We know that if the first eigenvalue if $M$ ...
4
votes
1
answer
213
views
Practical way to check for geometric convergence
Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution.
When I measure the convergence rate ...
- 101
2
votes
1
answer
421
views
Extending Wald's equation to two classes of i.d. random variables?
I try to adopt Wald's equation to a slightly more complex problem. In fact, after a full day, I found some solution now, but it has a confusing argument in the middle. Perhaps somebody can help me at ...
- 340
2
votes
1
answer
447
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MCMC with progressive demollification of delta distributions
Edit: I simplified the example to a canonical case for clarity.
Given an integral $\int_{\Omega}{g(\mathbf{x})}$ with a well-posed integrand $g(\mathbf{x})$ defined on some multidimensional space $\...
7
votes
2
answers
404
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Examples of Slowly Mixing Chains in Statistics
This should probably be community wiki, but I don't know how to set that myself.
I'm looking for examples or Markov chains that are used in statistics or statistical physics, and which are known to ...
- 71