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When considering the approximation $$\int_a^b f(x)\,\text dx\ \approx\ \frac{b-a}{N}\sum_{i=1}^N f(x_i)=\hat I$$ the lhs is an average of iid random variables, the $(b-a) f(X_i)$'s, with variance $$\text{var}\, \hat I = \dfrac{1}{N} \text{var}\, \{(b-a) f(X_i)\}=\dfrac{(b-a)^2}{N} \text{var}\, f(X_i)$$ This variance can thus be approximated by a convergent ...

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The most common situation I'm familiar with is fitting any kind of Bayesian (regression) model to data. It's usually very easy to write down the likelihood (=sampling distribution of the data for given parameter values), as well as some prior distributions for the model parameters. The posterior distribution is proportional to their product, but it is ...

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The reason it is such a useful method reduces to to Bayes' formula: $$p(x \vert \text{data}) = \frac{p(\text{data} \vert x) p(x)}{p(\text{data})}$$ Typically, we denote $p(\text{data} \vert x)$ the likelihood function, $p(x)$ the prior distribution and in your notation $C = p(\text{data})$ is the marginal likelihood of the data. As statisticians or machine ...

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First, let me summarize the principle. Let $\boldsymbol{y}$ be the data (sample) and $p(\boldsymbol{y}|\boldsymbol{\theta})$ be the distribution of $\boldsymbol{y}$, parameterized by $\boldsymbol{\theta}$. Let take prior $p(\boldsymbol{\theta})$ for $\boldsymbol{\theta}$. After observing the sample (with the joint distribution \$p(\boldsymbol{y}|\boldsymbol{\...

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This question is a bit similar to Why use Monte Carlo method instead of a simple grid? You are (partly) right, if the particles/randomness was nothing more than a simple way to do a grid search on a static grid generated by a random method rather than an evenly distributed grid, then the random method would indeed not be efficiënt. The inefficiency would be ...

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On a fundamental level, Monte Carlo is an approach to approximating integrals. It is not intrinsically a method for forecasting. It becomes a method for forecasting when your model defines a certain integral which corresponds to the forecast that you are trying to approximate. Because you have not explicitly defined a model, there is no way to tell if your ...

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