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统计代写|贝叶斯分析代考Bayesian Analysis代写|Non-symmetric drivers and the general Metropolis algorithm

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统计代写|贝叶斯分析代考Bayesian Analysis代写|Non-symmetric drivers and the general Metropolis algorithm

In some cases, applying the Metropolis algorithm as described above may lead to poor mixing, even after experimentation to decide on the most suitable value of the tuning constant.

For example, if the random variable of interest is strictly positive with a pdf $f(x)$ which is positively skewed and highly concentrated just above 0 (for example, if $f(x) \rightarrow \infty$ as $x \downarrow 0$ ), proposing a value symmetrically distributed around the last value may lead to many candidate values which are negative and therefore automatically rejected.

In such cases, the support of $X$ may not be properly represented, and it may be preferable to choose a different type of driver distribution, one which adapts ‘cleverly’ to the current state of the Markov chain.

This can be achieved using the general Metropolis algorithm which allows for non-symmetric driver distributions. As before, let $g(t \mid x)$ denote a driver density, where $t$ denotes the proposed value and $x$ is the last value in the chain. Then at iteration $j$, after generating a proposed value from the driver distribution,
$$x_j^{\prime} \sim g\left(t \mid x=x_{j-1}\right),$$
the acceptance probability is
$$p=\frac{f\left(x_j^{\prime}\right)}{f\left(x_{j-1}\right)} \times \frac{g\left(x_{j-1} \mid x_j^{\prime}\right)}{g\left(x_j^{\prime} \mid x_{j-1}\right)} .$$

Note 1: Previously, when $g(t \mid x)$ was assumed to be symmetric,
$$\frac{g\left(x_{j-1} \mid x_j^{\prime}\right)}{g\left(x_j^{\prime} \mid x_{j-1}\right)}=1 \text {. }$$
Note 2: To calculate $p$, the best strategy is to let
$$p=\exp (q)$$
after first computing
\begin{aligned} q & =\log f\left(x_j^{\prime}\right)-\log f\left(x_{j-1}\right) \ & +\log g\left(x_{j-1} \mid x_j^{\prime}\right)-\log g\left(x_j^{\prime} \mid x_{j-1}\right) . \end{aligned}

统计代写|贝叶斯分析代考Bayesian Analysis代写|The Metropolis-Hastings algorithm

We have introduced Markov chain Monte Carlo methods with a detailed discussion of the Metropolis algorithm. As already noted, this algorithm is limited and rarely used on its own because it can only be used to sample from univariate distributions. Typically, other methods will be better suited to the task of sampling from a univariate distribution.

We now turn to the Metropolis-Hastings (MH) algorithm, a generalisation of the Metropolis algorithm that can be used to sample from a very wide range of multivariate distributions. This algorithm is very useful and has been applied in many difficult statistical modelling settings.

First let us again review the Metropolis algorithm for sampling from a univariate density, $f(x)$. This involves choosing an arbitrary starting value of $x$, a suitable driver density $g(t \mid x)$ and then repeatedly proposing a value $x^{\prime} \sim g(t \mid x)$, each time accepting this value with probability
$$p=\frac{f\left(x^{\prime}\right)}{f(x)} \times \frac{g\left(x \mid x^{\prime}\right)}{g\left(x^{\prime} \mid x\right)}$$
(or $p=\frac{f\left(x^{\prime}\right)}{f(x)}$ in the case of a symmetric driver).
Each proposal and then either acceptance or rejection constitutes one iteration of the algorithm and may be referred to as a Metropolis step.
Performing $K$ iterations, each consisting of a single Metropolis step, results in a Markov chain of values which may be denoted $x^{(0)}, x^{(1)}, \ldots, x^{(K)}$.
Assuming that stochastic equilibrium has been attained within $B$ iterations ( $B$ standing for burn-in) the last $J=K-B$ values may be renumbered so as to yield the required sample, $x^{(1)}, \ldots, x^{(J)} \dot{\sim}$ iid $f(x)$.

The Metropolis-Hastings (MH) algorithm is a generalisation of this procedure to the case where $x$ is a vector of length $M$ (say).

贝叶斯分析代写

统计代写|贝叶斯分析代考Bayesian Analysis代写|Non-symmetric drivers and the general Metropolis algorithm

$$x_j^{\prime} \sim g\left(t \mid x=x_{j-1}\right),$$

$$p=\frac{f\left(x_j^{\prime}\right)}{f\left(x_{j-1}\right)} \times \frac{g\left(x_{j-1} \mid x_j^{\prime}\right)}{g\left(x_j^{\prime} \mid x_{j-1}\right)} .$$

$$\frac{g\left(x_{j-1} \mid x_j^{\prime}\right)}{g\left(x_j^{\prime} \mid x_{j-1}\right)}=1 \text {. }$$

$$p=\exp (q)$$

\begin{aligned} q & =\log f\left(x_j^{\prime}\right)-\log f\left(x_{j-1}\right) \ & +\log g\left(x_{j-1} \mid x_j^{\prime}\right)-\log g\left(x_j^{\prime} \mid x_{j-1}\right) . \end{aligned}

统计代写|贝叶斯分析代考Bayesian Analysis代写|The Metropolis-Hastings algorithm

$$p=\frac{f\left(x^{\prime}\right)}{f(x)} \times \frac{g\left(x \mid x^{\prime}\right)}{g\left(x^{\prime} \mid x\right)}$$
(或$p=\frac{f\left(x^{\prime}\right)}{f(x)}$在对称驱动程序的情况下)。

Metropolis-Hastings (MH)算法是该过程的一般化，其中$x$是长度为$M$(例如)的向量。

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