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Markov Chain Monte Carlo
  • Concept of Distribution
  • Monte Carlo Simulation
  • Markov Chain
  • Metropolis Algorithm
    Gibbs Sampler
  • Dynamics of Gibbs Models
  • Gibbs Distribution
  • Ising and Potts Models
    Bayesian Models
  • Joint Distribution
  • Frequentist Approach
  • Bayesian Approach
  • Dirichlet Distribution
  • Mixture Model
  • MCMC scheme
    Data Mining and MCMC
  • Relative Report Rate
  • Graphical Model
  • MCMC scheme
    Explore MCMC with R
  • Normal Mixture
  • Searching for Maxima
  • Bernoulli Trials
  • Random Beta Walk
  • Metropolis Algorithm
  • Potts Model

    • Joint Distribution

    When a pair $ (X,Y)$ of random variables is considered, a joint density function $ f(x,y)$ is used to compute probabilities constructed from the random variables $ X$ and $ Y$ simultaneously. Given the joint density function $ f(x,y)$ , the distribution for each of $ X$ and $ Y$ is called the marginal distribution. The marginal density functions of $ X$ and $ Y$ , denoted by $ f_X(x)$ and $ f_Y(y)$ , are given respectively by

    $\displaystyle f_X(x) = \int_{-\infty}^\infty f(x,y) dy$    and $\displaystyle \quad f_Y(y) = \int_{-\infty}^\infty f(x,y) dx. $

    Conditional probability distributions. Suppose that two random variables $ X$ and $ Y$ has a joint density function $ f(x,y)$ . If $ f_X(x) > 0$ , then we can define the conditional density function $ f_{Y\vert X}(y\vert x)$ given $ X = x$ by

    $\displaystyle f_{Y\vert X}(y\vert x) = \frac{f(x,y)}{f_X(x)}. $

    Similarly we can define the conditional density function $ f_{X\vert Y}(x\vert y)$ given $ Y = y$ by

    $\displaystyle f_{X\vert Y}(x\vert y) = \frac{f(x,y)}{f_Y(y)} $

    if $ f_Y(y) > 0$ . Then, clearly we have the following relation

    $\displaystyle f(x,y) = f_{Y\vert X}(y\vert x) f_X(x) = f_{X\vert Y}(x\vert y) f_Y(y). $

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