(Immagine generata a solo scopo illustrativo)
The Metropolis-Hastings algorithm is a prominent MCMC metodo for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult. At each iteration, it generates a candidate for the next sample based on the current sample. This candidate is then accepted or rejected with a certain probability, ensuring the resulting chain converges to the desired distribution.
The algorithm works by constructing a Markov chain whose stationary distribution is the target distribution [latex]P(x)[/latex]. Let the current state be [latex]x_t[/latex]. First, a candidate state [latex]x'[/latex] is generated from a proposal distribution [latex]Q(x’|x_t)[/latex], which can be any distribution that is easy to sample from (e.g., a Gaussian centered at [latex]x_t[/latex]). Second, an acceptance probability [latex]\alpha(x’, x_t)[/latex] is calculated: [latex]\alpha(x’, x_t) = \min\left(1, \frac{P(x’)Q(x_t|x’)}{P(x_t)Q(x’|x_t)}\right)[/latex]. A random number [latex]u[/latex] is drawn from a uniform distribution on [latex][0, 1][/latex]. If [latex]u \le \alpha[/latex], the candidate is accepted, and the next state is set to [latex]x_{t+1} = x'[/latex]. Otherwise, the candidate is rejected, and the chain remains at the current state, [latex]x_{t+1} = x_t[/latex].
The brilliance of this acceptance ratio is that it only requires knowing [latex]P(x)[/latex] up to a constant of proportionality, since any normalization constant cancels out in the fraction [latex]frac{P(x’)}{P(x_t)}[/latex]. This is crucial in Bayesian statistics, where the posterior distribution is often known only up to the intractable marginal likelihood. The original Metropolis algorithm (1953) is a special case where the proposal distribution [latex]Q[/latex] is symmetric, i.e., [latex]Q(x’|x_t) = Q(x_t|x’)[/latex], simplifying the acceptance probability to [latex]minleft(1, frac{P(x’)}{P(x_t)}right)[/latex]. Hastings’ 1970 generalization allowed for asymmetric proposal distributions, significantly broadening the algorithm’s applicability and efficiency.
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Algoritmo Metropolis-Hastings
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