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The **Metropolis-Hastings (MH)** algorithm is one of the foundational MCMC methods. It generates a candidate point and determines whether to accept it based on an acceptance probability, which compares the likelihood of the current and proposed points.
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1. Start with an initial state $\mathbf{x}_0$.
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2. Propose a new candidate $\mathbf{x}^*$ based on a proposal distribution $q(\mathbf{x}^*|\mathbf{x}_t)$.
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2. Propose a new candidate $\mathbf{x}^{\*}$ based on a proposal distribution $q(\mathbf{x}^{\*}|\mathbf{x}_t)$.
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3. Calculate the acceptance probability:
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$$
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