Questions on Monte Carlo methods, methods that require the repeated generation of (pseudo-, quasi-)random numbers for computing their results.

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11 views

Estimator of expectation value for standard normal distribution

In the case of a standard normal distribution, I just read that a good estimator for E[f(x)] is $\frac{1}{M}\sum_{i=1}^M f(X_i)$ (where each $X_i$ is standard normally distributed and independent). ...
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1answer
34 views

Log normal simulation.

I want to calculate numerically the expectation of a lognormal random variable $Y=e^X$, where $X$ is normally distributed with mean $m$ and variance $V$. The expectation is known as ...
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2answers
21 views

Show that this MC is ergodic?

Suppose I have a Markov Chain with States, $S = {1,2,3,4}$ and a PTM given by $P =$ $\begin{pmatrix} .25 & .25 & .25 & .25 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 ...
1
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1answer
29 views

Limiting Distribution of a Gibbs Distribution

I know that the Gibbs distribution at a particular state, x, is given by $\frac{e^{-\beta E_i}}{\sum_j e^{-\beta E_j}}$ with $\beta = \frac{1}{T}$, but I do not understand what a limiting ...
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4answers
95 views

Monte Carlo double integral over a non-rectangular region (Matlab)

I want to evaluated the following integral using Monte Carlo method: $$\int_{0}^{1}\int_{0}^{y}x^2y\ dxdy $$ What I tried using Matlab: ...
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0answers
23 views

Gaussian processes and bias

I would like to simulate two Gaussian processes along a time grid. Ideally, I would like to see the average of the samples, for each grid point, to be close to the mean. Using the antithetic method, I ...
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0answers
19 views

Correlation Matrix Question

Why is this not a possible correlation matrix for any three random variables X, Y, and Z? $\begin{pmatrix} 1 & -1 & -1 \\ -1 & 1 & -1\\ -1 & -1 & 1\end{pmatrix}$
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1answer
30 views

Correlation Matrices proofs

(*) says that the diagonals of $R$ are $1$ and the non-diagonals are the correlation, $p$. I planned on simplifying both equations until they're equivalent, but I'm not sure how I could go about ...
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1answer
25 views

Generating Normals with specific means and variances

Suppose I wish to generate normals $X, Y, Z$ with the correlation matrix R but with means $0, 1, 2$, and variances $4, 16, 25$, respectively. How would you do this? The only way I know of doing ...
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1answer
42 views

Related problem to covering a circle with random arcs

I have a problem setup wherein we have (the following are all integers) a sequence of length $G$, and $N$ reads of length $L$. I'm interested in the problem where we consider the sequence to be ...
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0answers
23 views

MCMC and Metropolis-Hastings problem(s)

What does it mean for a particular state to be a "ground" state or a "stable" state? I should make clear that this is final exam review material and not homework. Also, how does one compute a ...
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1answer
65 views

A Question Regarding Markov Chains and Ergodicity

Suppose the Markov chain with Probability Transition Matrix, $P$ = ($p{_x}{_y}$) is ergodic and $p{_m}(x, y) > 0$ for all states $x$ and $y$. If $n ≥ m$, show that $p_n(x, y) > 0$ for all ...
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0answers
8 views

Monte-Carlo estimation of Mutual Information over AWGN channel

I'm trying to solve a problem I was tasked with. Basically I have to generate a 100k 16QAM inputs and transmit them over a AWGN channel. With this I have to use the Monte-Carlo estimation to figure ...
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1answer
31 views

Debugging a Metropolis Hastings Algorithm Simulation

I have some questions about the Metropolis Hastings algorithm: Wikipedia says: ...choose an arbitrary probability density g(x|y) which suggests a candidate for the next sample value x, given ...
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0answers
15 views

Monte Carlo divergence

I'm running MonteCarlo simulations to compute the most likely voltage values in a grid. The process is to generate a random load from a load values distribution, and simulate the load with the power ...
2
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1answer
29 views

discretized Brownian motion

These are the definitions I'm working with: A (standard) Brownian motion in $\mathbb{R}$ is a stochastic process $W(t)$ $(t \geq 0)$ such that the following properties hold: $W(0) = 0$ almost ...
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0answers
16 views

Computing the partition-function of an exponential family member

I am working on an Monte Carlo Expectation Propagation problem. In that context I got the following property: $ I = \sum\limits_i w^{(i)} \log p_\eta(x^{(i)}) $ where $\{w^{(i)}\}_i$ are weights, ...
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0answers
11 views

Quasi-Monte Carlo with Conditional Distributions

I want to estimate $E(f(X))$ using quasi-Monte Carlo where $X = (X_1,\ldots,X_n)$ is a random vector and $$ X_i\sim f(\cdot; \theta), \quad \text{independent}, $$ where $\theta \in \mathbb{R}$ is some ...
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1answer
27 views

Generating correlated standard normals

Suppose I want to generate three standard normals $X, Y, Z$ with correlation matrix given by $R$= $ \begin{pmatrix} 1.0 & 0.2 & 0.2 \\ 0.2 & 1.0 & 0.2 \\ 0.2 & 0.2 & ...
3
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2answers
82 views

Error Estimates. L1 or L2 norm?

I simulate random walk on a divide difference grid to solve heat equation 1D. I want to prove numerically that this method has $Ν^{-1/2}$ error accuracy. My problem is that I don't know which norm ...
1
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0answers
19 views

Generate Correlated Normals

I want to generate normals $X,Y,Z$ with the correlation matrix $R$ but with means $0, 1, 2$ and variances $4, 16, 25$ respectively. How can I do this? Is it possible to apply Cholesky?
2
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1answer
25 views

Geometric Brownian motion with exponential of sum of iid's

Glasserman's "Monte Carlo Methods in Financial Engineering" on p. 265 states that the geometric Brownian motion can be modelled with : $$S(t_n)=S(0) \exp(\sum_{i=1}^n X_i)$$ where $X_i$ are iid. I ...
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0answers
6 views

Accuracy Rebonato Swaption Approximation Formula among Different Strikes

Can somebody explain me if the Rebonato swaption volatility approximation formula is accurate for only ATM strikes, and if yes why? Can it also be used for ITM and OTM strikes? My foundings: Let $0 ...
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2answers
55 views

References on probability theory, stochastic processes, Monte Carlo and convex optimisation, with similar writing style to Terence Tao

I learned a lot from prof Tao's notes and books because unlike many authors, he seems to prefer writing more words, explanations and intuitions rather than just mathematical formulae. His approach is ...
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0answers
21 views

Reducing sequential correlations in Metropolis Algorithm

In our last lab, we use MCMC method to simulate a walker walking in the phase space. Using the Metropolis method, a walker at its currect position will sample another point inside a cube (centered at ...
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0answers
14 views

Help with Hidden Markov model and SMC methods

So its quite a long background i don't really know where to start but here goes. The background is as follows: Background Observation model As the target is moving, it measures the signal (RSSI) ...
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0answers
35 views

Theoretical interpretation of simulating from a distribution

Suppose there is a random variable $X$ with marginal density $p_X$. However only the conditional densities $\{p_{X\mid\Theta}(\cdot\mid\theta):\theta \in \mathbf{T}\}$ are known directly, where ...
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0answers
35 views

Can someone help me balance this game (probability question) [closed]

A team of 9 vs a team of 1. Each round each of "the 9" roll a die to "attack" and "the 1" rolls 9 dice to "defend", the nine dice are preassigned to attackers before the roll, "the 1" cannot choose ...
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1answer
43 views

How would you simulate Brownian motion with a die?

You can simulate Brownian motion on $[0, 1]$ for instance by splitting it into $K$ intervals and at each time $k \Delta t$ add $N(0, \Delta t)$ to your running total, where $\Delta t = 1/K$. If you ...
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0answers
13 views

Main differences between Monte carlo and Law of propagation of variance

I need to know the main differences/limitations between/of Monte-Carlo simulation technique and law of propagation of variance. Can someone briefly describe it or is there any good reference or link ...
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0answers
30 views

Ways to sample a complicated PDF on an hemisphere

I want to generate samples on the upper real unit hemisphere with the following PDF (it's not really a PDF because I can't guarantee that it integrates $1$) $$\frac{\sum_{i=0}^{n}c_i(\text{ ...
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0answers
78 views

Monte Carlo with non uniform weighting

So, I just want to check if what is in my mind is in fact true. Assume, that we have are given a distribution $p_{z}(k)$ over the whole $\mathbb{Z}^+$. We are interested in approximating $p_v(v)$ over ...
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0answers
20 views

Variance of Importance Weights in Importance Sampling

What is the variance of importance weights in Importance Sampling? Is it good to reduce the variance of importance weights and why? And what is the optimum proposal in importance sampling? I assume ...
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0answers
13 views

How to determine the transition probability in Sequential Importance Sampling (SIS) for Particle Filter

Given a state-space model \begin{align} x_k &= f_k(x_{k-1}, v_{k-1}),\\ z_k &= h_k(x_k, w_k), \end{align} where $x_k \in {\mathbb R}^{n}$ and $y_k \in {\mathbb R}^{m}$ are the system state ...
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1answer
14 views

How to generate “independent” quasi random numbers

I am studying Monte-Carlo simulations using quasi random numbers and encounter the following problem: I am given a set of 1D quasi-random numbers $(X_i)$ over $[0,1)$, and would like to generate ...
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0answers
19 views

Exercises on the following topics on Markov Chains

We are being taught the following topics in Markov Chains: 1) Markov Chain Monte Carlo: Hard Core model, Counting random q-colourings of a graph 2) Total variation distance for a Simple Symmetric ...
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2answers
75 views

Monte Carlo Simulation- Simulating Sum of a DICE. Matlab CODE.

Hello everyone, I try to solve the following problem: Use Monte Carlo simulation to approximate the sum of the 100 consecutive rolls of a fair die. My work in math lab is: ...
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0answers
48 views

$\theta=\int_{0}^{2}3e^{-3x}dx$ Compute theta using Monte-Carlo Method

$\theta=\int_{0}^{2}3e^{-3x}dx$ Using Monte-Carlo method estimate the confidence intervals for the integral above. Use a distribution different from the uniform distribution to minimize the ...
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1answer
37 views

Quasi Monte Carlo or Gaussian Hermite quadrature for a statistical model with random effect

In my likelihood function, I need to integrate a random effect out as follows $$\int g(x,c)\exp(-c^2/2)dc .$$ Since the likelihood function is really complicated, I need the approximation to be fast ...
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0answers
37 views

Adequacy of Monte-Carlo simulations

Suppose we have a number of independent random variables of the form $X_1 \sim U[a_1,b_1], X_2 \sim U[a_2,b_2], X_3 \sim U[a_3,b_3]$. Now, suppose we generate a random variable $Y$ as follows: $$Y = ...
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1answer
20 views

Measurement, lognormal distribution, Monte-Carlo

I do have problems to understand the lognormal distribution. So, I do have one measurement M, measured with a sensor having a std S. As the sensor is not too accurate I want to build up a list of ...
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1answer
21 views

Why does this MCMC algorithm to estimate parameters of a linear equation not converge to the posterior distribution?

As a kind of proof of principle I'm trying to estimate the parameters of a linear equation (before moving on to ODEs) using Markov Chain Monte Carlo sampling. The post that I am following can be found ...
2
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2answers
64 views

Montecarlo estimate of a integrand from 0 to $\infty$

I have a question about monte carlo estimation of integrals. Suppose I am told to estimate using monte carlo, the integral: $$f(y) = \int_{0}^{y}\frac{4}{1+x^{2}}dx$$ I want to estimate $f(\infty)$. I ...
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1answer
52 views

Monte Carlo estimation of a constant?

I am currently learning monte carlo and I dont quite understand it. In the question I am given, I am asked to estimate $\pi$. So I am to write a Matlab code that computes a numerical estimate of ...
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0answers
26 views

Integrating a function containing a conditional Gaussian mixture

I wonder if the following integral has an analytical solution. $$ \int_{-\infty}^{\infty}\frac{w_1 N_1(x)}{\sum_{i=1}^{n} w_i N_i(x)} N_0(x) dx $$ where $w_1, \ldots, w_n$ are positive constants, ...
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1answer
35 views

Sampling uniform equilibrium distribution with Markov Chain Monte Carlo

I'm wanting to sample the discrete uniform distribution over $n = 10$ integers using MCMC. My question concerns the transition probability matrix, $P$. As I understand it, any symmetric, irreducible ...
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0answers
13 views

Finding marginal posterior distributions (Gibbs Sampling)?

When using Gibbs sampling I need to find the conditional distributions of the parameters. In all textbooks and examples they seem to unanimously suggest that "it's obvious". Take for example page 56 ...
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2answers
46 views

why we use uniform distribution on accept reject method?

the accept-reject method have the following algorithm: Given known random number generators $U \sim Unif(0,1)$ and $X \sim g$, we can generate $Y \sim f$ by the following algorithm. Let $c$ be a ...
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0answers
24 views

Monte Carlo integration to solve coefficients of an orthogonal series - reusing the set of random points

I'm trying to approximate a function by summing a series of orthogonal functions. $f(x) \approx \sum_i a_i \phi_i(x)$ Since the set of functions $\phi_i(x)$ are orthogonal with respect to each ...
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0answers
35 views

How to derive the conditional given the following joint probability

I encountered this question while reading about MCMC methods to solve image reconstruction problems. Consider a black and white image where $-1$ corresponds to white and $+1$ to black. $X_{i,j}$ ...