Questions tagged [metropolis-hastings]
A special type of Markov Chain Monte Carlo (MCMC) algorithm used to simulate from complex probability distributions. It is validated by Markov chain theory and offers a wide range of possible implementations.
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Derivation of acceptance probability from Linero, Yang (2018)
I am wondering how this paper Bayesian Regression Tree Ensembles that Adapt to
Smoothness and Sparsity by Linero & Yang (2018) derived the acceptance probability for $\sigma$.
The authors give $\...
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Metropolis Hastings Proposal with Gradient and Hessian Information
I need to sample a high-dimensional parameter vector from a distribution where the gradient, the Hessian and the inverse of the Hessian of the log-likelihood are very cheap to compute. Are there any ...
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What is the use of MCMC techniques in this context?
Currently I'm learning about MCMC techniques. .
Exercise
A researcher wants to sample $\theta$ from a discrete distribution with density function
$$
f(\theta)= \begin{cases}p=1 / 6 & \theta=0 |\\ ...
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Using both discrete and continuous moves in Metropolis-Hastings
I want to sample a continuous distribution $f$ using the Metropolis-Hastings algorithm. Can I define my transition kernel as being sometimes discrete and sometimes continuous as long as I use the ...
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Simulate SDE without error
Let
$d,k\in\mathbb N$;
$\sigma\in C^1(\mathbb R^d,\mathbb R^{d\times k})$ be Lipschitz and $\Sigma:=\sigma\sigma^\ast$;
$(W_t)_{t\ge0}$ be a $k$-dimensional Brownian motion;
$\lambda$ denote the ...
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Is Metropolis-Hastings ever more efficient than rejection sampling in 2 dimensions?
I know that Metropolis-Hastings is an MCMC (Markov Chain Monte Carlo) method that is very useful in higher dimensions. The advantages it has over something like simple rejection sampling are that ...
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Non-reversible MCMC based on diffusion dynamics
Given that the current sample location is $x\in[0,1)^d$, I would like to take the next sample $y$ as $$y=x+b(x)\Delta t+\sigma(x)\sqrt{\Delta t}\xi\;;\;\;\;\xi\sim\mathcal N_{0,\:I_d}\tag1$$ for some ...
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Can I use the output of an MCMC algorithm as the input of an independent Metropolis-Hastings algorithm?
Can I and if so, how can I, use the output of a MCMC method as the input for an independent Metropolis-Hastings algorithm? Maybe this question reduces to: How can I get (independent? or at least "...
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How does Rao-Blackwellization of the Metropolis-Hastings algorithm work?
I've read the paper A vanilla Rao-Blackwellization of Metropolis-Hastings algorithms, but I don't get what their actually suggested estimator is.
To give some detail, we are considering the following ...
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Metropolis Hastings Algorithm and Breaking Reversibility in MCMC
If the goal is to sample from a distribution $\pi$ it is common to build a Markov Chain with stationary distribution $\pi$. Solving this problem using Markov Chain Monte Carlo is essentially ...
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Metropolis Hastings Proposal Distribution [closed]
I am trying to sample from a bivariate normal distribution given above. If the proposal distribution is q(theta|theta prime) and theta prime = theta + U, where U is uniformly distributed over [a,b], ...
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Estimate $\int f\:{\rm d}\lambda$ and $\int g\:{\rm d}\mu$, where whenever $(X,Y)\sim\mu$, then $X\sim\lambda$, by a single Markov chain
Say I want to estimate $$I=\int f\:{\rm d}\lambda+\int g\:{\rm d}\mu.$$ Now, I'm using Metropolis-Hastings to sample from $\lambda$ and from $\mu$, separately. Assume $\mu$ is defined on a product ...
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Sampling from the posterior with a constraint on the posterior mean
Background
Under certain assumptions we know that being given the posterior mean and a family of conditional distributions, we can uniquely determine the joint distribution. I quote one of the ...
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How to draw from a uniform distribution over a large state space via MCMC
Motivating question
I have a high-dimensional state space $\Omega \subseteq \mathbb R^n$ with an admissible subset $S\subseteq \Omega$, which is connected. I would like to draw a uniform random sample ...
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How should we stratify the space for Metropolis-Hastings?
Say I'm running Metropolis-Hastings with target density $p$. What I would like to do is divide the space $E$, on which $p$ is defined, into a disjoint union $E=\bigcup_iE_i$ and run a separate ...
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Running Metropolis-Hastings algorithm with changing proposal kernel; each time the kernel is changing starting the algorithm afresh. Does it work?
I have a Markov kernel $Q$ from which I would like to generate proposals for the Metropolis-Hastings algorithm. The problem is: When the proposal is accepted, the "internal state" of $Q$ ...
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Can we run the Metropolis-Hastings algorithm with a proposal also generated by the Metropolis-Hastings algorithm?
In the Metropolis-Hastings algorithm, depending on the current state $x$, I have a distribution $\rho_x$ and I want to use a sample from $\rho_x$ as the proposal in the next iteration. (I guess ...
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What motivates a good proposal distribution for a target distribution? (Metropolis - Hasting sampling)
In Metropolis - Hasting sampling, the proposal distribution does not necessarily have to have a form similar to that of the target distribution from which attempts are made to sample from.
For one, I ...
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Metropolis - Hastings sampling: histogram shapes looks sane but bin values are off
The target distribution is of the form:
$ p(x) = x^{-6}.e^{\frac{-2.475}{x}}$ with a support in the interval $[0.0, 2.0]$.
This gives a plot like
Now, to choose a proposal kernel, I think a lognormal ...
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Computational aspect of the Metropolis-Hastings algorithm
One of the examples online is about how to write the Metropolis-Hastings algorithm from scratch. This tutorial uses a linear regression model as an example. Estimate three parameters: an intercept, a ...
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What is "critical slowing down" using MCMC for a Gibbs-Boltzmann distribution?
When sampling from a probability density function of the form $$p(x)=e^{-\beta E(x)},$$ where $E$ is considered to be the energy of a system and $\beta=1/T$ is the inverse of a temperature parameter $...
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Test whether a Metropolis-Hastings chain is "sufficiently near" equilibrium using the autocorrelation function
Let $(E,\mathcal E)$ be a measurable space, $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(X_n)_{n\in\mathbb N_0}$ be an $(E,\mathcal E)$-valued process on $(\Omega,\mathcal A,\...
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Why is it easy for the Gibbs sampler to take long time to converge to target distribution?
This is related to Gelman's Bayesian Data Analysis 3rd Edition pg 300 first paragraph of Section 12.4. The book says the following.
"An inherent inefficiency in the Gibbs sampler and Metropolis
...
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Metropolis Hastings Algorithms: How to measure the performance of algorithms? (Multidimensional)
I am working on a project and I am trying to measure the performance and compare two MCMC algorithms. The one is Random-Walk MH and the second one is PCN.
I thought of maybe comparing the mean ...
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Expression / R code for theconditional posterior PDF for ordinal regression model (for Metropolis-Hastings step within Gibbs sampler)
I am trying to derive an expression for the PDF (conditional posterior density) of the cutpoint parameters of an ordinal regression model so that I can use a Metropolis-Hastings step to sample the ...
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Ordinal regression - 'induced Dirichlet' conditional posterior distribution
I am trying to implement the 'induced Dirichlet' prior model proposed by Michael Betancourt (from section 2.2 of his ordinal regression case study here: https://betanalpha.github.io/assets/...
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How to use Gibbs sampler to simulate normal-normal hierarchical models?
This is related to Gelman's BDA 3rd Edition Chapter 11, Sec 3. The book says the following.
"The Gibbs sampler is the simplest of the Markov chain simulation algorithms, and it is our first ...
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How can Metropolis-Hastings use the function it is trying to approximate?
The MH algoithm is used to obtain samples from a probability distribution $f$ that is difficult to sample from directly. The process as described in this answer is:
Pick a initial random state $x_0$.
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Implementing Metropolis-Hasting for multiple variables
Working through a lecture exercise on MCMC methods. I have a dataset containing the outcome of N chess games $-$ in the format Winner, Loser $-$ between M players.
When 2 players $p_1$ and $p_2$ play ...
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Is there a variant of the Metropolis-Hastings algorithm with proposal and/or acceptance function depending on the history?
Is there a version of the Metropolis-Hastings algorithm where either
the proposal kernel; or
the acceptance function
might depend on the whole history (or at least a part of it) of the chain so far?
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Beginner's example: fitting algorithm to obtain the estimated posterior distributions of the input model parameters using black box models
Some years ago when I was a student I used the DiffeRential Evolution Adaptive Metropolis (DREAM) algorithm (Vrugt 2016) in MATLAB with the goal of fitting experimental data (e.g., the concentration ...
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Necessity of Metropolis Hastings algorithm for given posterior distribution
Let's say that we have calculated the posterior distribution of a parameter of interest given the data of a binomial experiment $N=70,x=34$ which the probability of event occurrence $\theta$ follows ...
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MCMC acceptance formula clarification
Metropolis - Hastings : Data Science Concepts youtube shows the acceptance probability $A(a \rightarrow b)$ is $Max(1, \frac {f(b)}{f(a)})$.
Is it correct or it should have been $Min(1, \frac {f(b)}{f(...
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MCMC MH - Intuition of dividing by proposal distribution at acceptance ratio calculation
Understanding Metropolis-Hastings algorithm youtube shows the step dividing the $g(\theta)$, which is propotional to the posterior distribution $P(\theta)$, by the conditional proposal distribution $q(...
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Metropolis Hastings Algorithm for Student-$t$ distribution
I am self-studying Metropolis-Hastings Algorithm and I am reading the book from C.Robert & G.Casella "Introduction Monte Carlo Methods with R.
In page 196 there is the problem 6.10 that asks:
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Can we use the Metropolis-Hastings with a discretized proposal?
Consider the Metropolis-Hastings algorithm with proposal density $q(x,y)$ and target density $p(y)$ with respect to some reference measure $\lambda$. If we don't use the proposals $Y\sim q(x,\;\cdot\;)...
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Is there a variant of the Metropolis-Hastings algorithm where the acceptance probabiltiy can depend on all states generated so far?
I wasn't able to find anything on google, but is there a variant of the Metroplis-Hastings algorithm where the acceptance probability (not the proposal kernel) in the $i$th iteration might depend on ...
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Time complexity of Metropolis-Hastings and potential speed-up?
The MH algorithm essentially involves generating a sample destination state from a proposal distribution, computing the acceptance probability as a function of that sample, and checking whether a ...
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Is there a Quasi-Monte Carlo variant of the Metropolis-Hastings algorithm?
If we run the Metropolis-Hastings algorithm for a target distribution $\mu$ with proposals from a quasi-Monte Carlo sequence $(y_n)_{n\in\mathbb N}$ (such as a Sobol sequence) and the generated chain ...
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Generate samples by the Metropolis-Hastings algorithm which have a minimum distance between one another
Can we generate samples by the Metropolis-Hastings algorithm with target density $p(x)$ and proposal kernel density $q(x,y)$ such that the samples satisfy a certain constraint?
Take the simple example ...
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Calculating acceptance probability in Metropolis-Hastings algorithm
In the Metropolis-Hastings algorithm, acceptance probability is given as
$$
\alpha = \min \left( 1,\frac{f(\theta^{'}|y)q(\theta|\theta^{'})}{f(\theta|y)q(\theta^{'}|\theta)} \right)
$$
which ...
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The guidelines for choosing different MCMC algorithms [closed]
MCMC has several types of algorithms: Metropolis-Hastings, Gibbs, Adaptive MH, Hamiltonian Monte Carlo. What are their respective pro/cons, and how to choose them in the Bayesian analysis?
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Adaptive metropolis within Gibbs
I once saw some study claims to use the algorithm referred to as "Adaptive Metropolis within Gibbs". Are there any formal introduction on this algorithm? What are the difference between this ...
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Why do we sample from the uniform distribution in Metropolis-Hastings for acceptance?
For each iteration of the MH, sample $x'=q(x|x')$, then the acceptance probability is computed:$$A=\min(1,a)$$
where
$$
\alpha=\frac{p(x')q(x|x')}{p(x)q(x'|x)}
$$
Now, I've seen that the algorithm ...
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Slice sampling in Particle Gibbs with Ancestral Sampling
Bear with me as I am not from statistical background. My question is about the implementation of PGAS algorithm as given in Lindsten et. al 2014 concerning sampling in state-space models. The two ...
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Metropolis-Hastings or other MCMC method with an unknown asymmetric proposal distribution?
When working with the Metropolis-Hastings algorithm, we can work with an asymmetric proposal density $g(x^\prime | x)$ provided we know the distribution in order to calculate the ratio $\frac{g(x|x^\...
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parallelizing log-sum-exp
I have some approximate likelihoods: $L_1, \ldots, L_n$. Each is quite expensive to calculate. They're approximate because they use random numbers. Each of them is being calculated on the same data ...
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Computing the Hastings ratio for multinomial distribution as a proposal distribution in Metropolis-Hastings accept-reject step
I have a question concerning calculating the Hastings ratio in a specific case (multinomial proposal distribution).
I consider a discrete vector $M$ with integer values that sum up to some number $N$. ...
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Parametrizing and Sampling Multivariate Garch Parameters Metropolis-Hastings MCMC
My question is how to sample multivariate GARCH parameters from a proposal distribution (multivariate normal) for a Metropolis-Hastings algorithm. Considering the different dimensions of the parameter ...
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Metropolis - Hastings algorithm on a set of countable sequences
I want to simulate $\sigma$ from a measure $\pi(\sigma)$ through the Metropolis-Hastings algorithm, where $\sigma$ is a sequence of 0's and 1's on $S = \{0, 1\}^n$, the set of all sequences of 0's ...
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