# Questions tagged [monte-carlo]

Using (pseudo-)random numbers and the Law of Large Numbers to simulate the random behavior of a real system.

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### In sports modelling, are hot simulations better or cold simulations?

I'm thinking here largely of the context in which someone has an Elo rating model for a particular sport. To calculate things such as how often the team makes the Finals series, or wins the ...
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### Expected value of softmax transformation of Gaussian random vector

Let $\mathbf w_1,\mathbf w_2,\ldots,\mathbf w_n \in \mathbb R^p$ and $\mathbf v \in \mathbb R^n$ be fixed vectors, and $\mathbf x \sim \mathcal N_p(\boldsymbol{\mu}, \mathbf{\Sigma})$ be an $p$-...
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### Population Monte Carlo Algorithm

I am trying to wrap my head around the Population Monte Carlo Algorithm. I want to implement it for a mixture model, but I am uncertain on how to proceed. I am mostly looking for references or ...
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### Probability that one Gaussian RV exceeds all others

Imagine we have $k$ Gaussian RVs $$X_i \sim N(\mu_i, \sigma_i^2) \text{ for } i=1, \ldots, k$$ and we sample from each of them independently to produce a vector, $\vec{x} = (x_1, \ldots, x_k)$. For ...
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### Why do we use the log-derivative trick before Monte Carlo?

I still don't understand how we can approximate the gradient of an expected value... Indeed it's impossible to sample points and then to average the gradients of them as we have only samples... (How ...
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### Monte carlo method and Convergence in Distribution

Monte Carlo method, from what I could gather, allows one to obtain observations/draws from a possibly unknown statistical distribution. Let's say $T(X)\sim F$, where $T$ is a statistic, $X$ is a ...
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### Importance sampling: what is this bias?

I am experiencing what seems to be a bias in importance sampling, which, given that it's an unbiased procedure, should not be there. Consider linear regression $$y = X\beta+\epsilon$$ where there ...
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### Error bars for sum of Monte Carlo integrations

I want to calculate a quantity $I$ which is the sum of integrals $I_k$, $$I = \sum_k I_k\;,\quad I_k = \int_{\Omega} f_k(x)\,\textrm d x\;,$$ Every integral $I_k$ is evaluated independently by a ...
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### sampling from a posterior derived from hierarchical Bayesian using HMC

I have a complex pdf based on hierarchical Bayesian formalism where x depends on the priors w'and w'', and I consider hyper-prior for the latter's as w=Php(zeta,beta) where Php stands for the hyper ...
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### Confusion on three “types” of Markov Chain Monte Carlo for the same inference

This is a long question but I would be very grateful if someone can help or provide some reference! And I believe this is a common confusion among beginners of MCMC. Background Given a directed graph \$...