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

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13 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
5 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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1answer
40 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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19 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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1answer
30 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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1answer
40 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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31 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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12 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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28 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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12 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 a distribution over the whole $\mathbb{Z}^+$, where $p(k) = \gamma_k$. We are interested in approximating $p(v)$ ...
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17 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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10 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
17 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
50 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
16 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
36 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
34 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
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 ...
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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
49 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
32 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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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
43 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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1answer
74 views

Question on Joint Posterior

Likelihood: $f(x^T, n^T|\theta^T) = \prod_{i=1}^{30} \binom{n^T_i}{x^T_i}{\theta^T}^{x^T_i}{(1-\theta^T)}^{n^T_i-x^T_i}$ Prior: $ log(\frac{\theta^T}{1-\theta^T})\sim N(\mu_T,\sigma_T^2) $ I am ...
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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. $X_{i,j}$ ...
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2answers
119 views

Mutual information of discrete RVs which converge in distribution to a continuous RV

$\mu_{X_n,Y_n}$ is a sequence of discrete joint-distributions on $\mathbf{R}^2$ that converge weakly to a continuous measure $\mu_{X,Y}$. That is, for any continuous function ...
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1answer
5k views

Expected Value and Variance of Monte Carlo Estimate of $\int_{0}^{1}e^{-x}dx$

Given the integral: $I=\int_{0}^{1}e^{-x}dx$, use standard Monte Carlo with 1000 random numbers and repeat the simulation 1000 times. (a) What is the expected value and variance of the simple Monte ...
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18 views

Continuity of Monte-Carlo simulations with uniformly distributed input parameters

Suppose a continuous and monotone function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ to be given. So, in the general case, if I slightly change parameters $a$ and $b$, the function ...
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0answers
16 views

Cdf of truncated distribution

Let $X$ be a random variable with density $f_x$ and distribution function $F_x$. Define the interval $I = (a,b)$. Given that we know these and the inverse distribution function $F^{-1}_x$, how can we ...
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Is the result of a Monte-Carlo simulation of a continuous function and with continuous input distributions again continuous?

Is the result of a Monte-Carlo simulation of a continuos function and with continuos input distributions again continuous? Suppose, we have a continuos function $f$ and a number of continuous random ...
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0answers
41 views

What is the benefit of stochastic models over deterministic models? [duplicate]

I have posted a similar question earlier and I guess this sounds naive to all of you, but nonetheless let me just ask: Consider I have a simple and deterministic model $M$, with a number of input ...
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1answer
69 views

Monte-Carlo simulation with sampling from uniform distribution

I used to work with Monte-Carlo simulations for a while. In my case, I generated random data for a variety of input parameters according to uniform distributions (with non-negative support), say for ...
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0answers
11 views

The exact usage of Sequential Monte Carlo for distributions over time?

I have wondered the usage of Sequential Monte Carlos and it is used as an alternative to Kalman filter for example. However I wonder if this can be also used for simulating a distribution over time? ...
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1answer
42 views

Average of Monte Carlo simulations of continuous functions again continuous?

I hope the following question is clear: Suppose, we have a continuous functions $f:\mathbb{N}^2 \rightarrow \mathbb{N}$. Now, suppose we run Monte Carlo simulations on the function, where the input ...
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1answer
18 views

Efficient methods for drawing random numbers and Monte Carlo for Tsallis q-Gaussians

I would like to draw random numbers from the q-Gaussian used in "Tsallis statistics." This is specifically the distribution $$ f(x) = {\sqrt{\beta} \over C_q} e_q(-\beta x^2) $$ where $$ e_q(x) = ...
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0answers
32 views

How to estimate a distribution from samples in a histogram

Given a r.v. $\tau$ , I've computed $\Bbb{P}(\tau >a)=e^{-Nx}(e^{Nxe^{-a}}-1) $ , where $N\in\Bbb{N}_{>1} $ and $ x\in \Bbb{R}_{>0} $ are just fixed parameters; say $N = 2 $ and $ x = 1$, ...
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1answer
28 views

Calculating success probability variance using Monte Carlo simulation

For a Monte Carlo simulation where each sample can produce a success value (1) or a failure value (0), what is the variance of the probability for success? Given n samples and r successes, the ...
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0answers
134 views

Monte Carlo Simulation

If we have a random variable $H$, such that $$ H=B_1G_2\min(1,G_1)+B_2\frac{\min(2,G_1 G_2+G_2)}{n-B_1-B_2}, $$ where $n$ is constant, $G_1$ and $G_2$ are independent lognormal with different ...
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1answer
41 views

Is there a such thing as a quasi-random shuffle?

I've recently experimented with Quasi-random numbers in monte-carlo applications. Is there a way to construct a quasi-random shuffle? By that I mean can I take a sequence $Q$ and shuffle it to produce ...
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0answers
53 views

Integrating sine with Monte Carlo / Metropolis algorithm

I'm learning Monte Carlo / Metropolis algorithm, so I made up a simple question and write some code to see if I really understand it. The question is simple: integrating sine over 0 to PI. The ...
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0answers
30 views

Biggest rectangle inside a random geometric shape

I'm looking for the most efficient algorithm to find the rectangle with the greatest area inside a random geometric shape. The rectangle can be also rotated of course. I am sure that there exists a ...
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3answers
682 views

Random directions on hemisphere oriented by an arbitrary vector

Hy, i'm writing a raytracer, and for that I need to generate n random vectors that are inside an hemisphere oriented by the surface normal. Ideally, I would also like being able to restrict the rays ...
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0answers
20 views

Opposite of Monte Carlo

In this lecture, at 1:08:35, the lecturer goes from $$\text{argmin}\frac{1}{N}\sum\limits_{i=1}^{N}\text{log}\frac{p(x_i|\theta_0)}{p(x_i|\theta)}$$ to ...
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0answers
20 views

Monte Carlo integration and variance

With the monte carlo integration of a function f(x), what do they mean with the variance? Is it the variance of the function we want to integrate? $I = ∫^{\inf}_{inf} f(x)p(x) dx$ (with p(x) some ...
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14 views

Numerical method to fit arbitrary 3D curve by distributing perturbing elements on a 2D grid

I am looking for help in choosing and possibly implementing an appropriate algorithm or method to solve the following problem: I have a surface that has a property $A(r)$ that I want minimized. I ...