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When the support of $q(x,\cdot)$ differs from the support of $p(\cdot)$ then the expectation of the ratio is not necessarily one. As an illustration, take \begin{align} p(x) &= \frac{1}{3}\Bbb I_{(1,4)}(x)\\ q(x,y) &= \frac{1}{3}\Bbb I_{(x-1,x+2)}(y) \end{align} Then the expectation of $$\Bbb I_{(1,4)}(x) \Bbb I_{(x-1,x+2)}(y)$$ under the density $p(...


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HMM can use EM, but typical applications also include Baum-Welch. GMM, k-means typically use it. MCMC is a simulation method, not EM. If EM is in some part of an algorithm, the convergence is sure for that part. EM finds you a local optimum depending on the initial values. If the initials change, you may end up with a different local optimum, and you don't ...


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The claim is equivalent to saying that $P$ generates a chain where, at $x$, with probability $1 - \rho ( x )$, you stay at $x$, and otherwise, you do move, and you do so according to $\tilde{P}$. Thus, in the Metropolis-Hastings case, you take \begin{align} \rho ( x ) &= 1 - r ( x), \\ \tilde{P} ( x, dy ) &= Q ( x, dy ) \cdot \frac{ \alpha ( x, ...


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If $K$ is a Markov kernel(with density $k$) with stationary distribution $P$ (with density $p$), then, if $(X_t)_t$ is a stationary Markov chain associated with $K$, \begin{align*}\mathbb E\left[\frac{p(X_{t+1})}{p(X_t)}\right] &=\int_{\mathfrak{X^2}} \frac{p(x_{t+1})}{p(x_t)} p(x_t)k(x_t,x_{t+1})\text{d}\lambda(x_t)\text{d}\lambda(x_{t+1}) \\ &= \...


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