我正在尝试在Python中实现Metropolis算法（Metropolis-Hastings算法的简单版本）。
这是我的实现：
def Metropolis_Gaussian(p, z0, sigma, n_samples=100, burn_in=0, m=1):
"""
Metropolis Algorithm using a Gaussian proposal distribution.
p: distribution that we want to sample from (can be unnormalized)
z0: Initial sample
sigma: standard deviation of the proposal normal distribution.
n_samples: number of final samples that we want to obtain.
burn_in: number of initial samples to discard.
m: this number is used to take every mth sample at the end
"""
# List of samples, check feasibility of first sample and set z to first sample
sample_list = [z0]
_ = p(z0)
z = z0
# set a counter of samples for burn-in
n_sampled = 0
while len(sample_list[::m]) < n_samples:
# Sample a candidate from Normal(mu, sigma), draw a uniform sample, find acceptance probability
cand = np.random.normal(loc=z, scale=sigma)
u = np.random.rand()
try:
prob = min(1, p(cand) / p(z))
except (OverflowError, ValueError) as error:
continue
n_sampled += 1
if prob > u:
z = cand # accept and make candidate the new sample
# do not add burn-in samples
if n_sampled > burn_in:
sample_list.append(z)
# Finally want to take every Mth sample in order to achieve independence
return np.array(sample_list)[::m]
当我尝试将我的算法应用于指数函数时，它只需要很少的时间。然而，当我在t分布上尝试它时需要很长时间，考虑到它没有进行那么多计算。这是您复制我的代码的方法：
t_samples = Metropolis_Gaussian(pdf_t, 3, 1, 1000, 1000, m=100)
plt.hist(t_samples, density=True, bins=15, label='histogram of samples')
x = np.linspace(min(t_samples), max(t_samples), 100)
plt.plot(x, pdf_t(x), label='t pdf')
plt.xlim(min(t_samples), max(t_samples))
plt.title("Sampling t distribution via Metropolis")
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')
plt.legend()
这段代码需要很长时间才能运行，我不知道为什么。在我的Metropolis_Gaussian代码中，我试图提高效率
函数pdf_t定义如下
from scipy.stats import t
def pdf_t(x, df=10):
return t.pdf(x, df=df)
我回答了一个similar question previously。我在那里提到的许多事情（不计算每次迭代的当前可能性，预先计算随机创新等）都可以在这里使用。
对您的实施的其他改进是不使用列表来存储您的样本。相反，您应该预先为样本分配内存并将它们存储为数组。像samples = np.zeros(n_samples)这样的东西比在每次迭代时附加到列表更有效。
您已经提到过，您试图通过不记录老化样本来提高效率。这是一个好主意。您也可以通过仅记录每个第m个样本来进行细化，因为您无论如何都会在np.array(sample_list)[::m]的return语句中丢弃这些。您可以通过更改：
# do not add burn-in samples
if n_sampled > burn_in:
sample_list.append(z)
至
# Only keep iterations after burn-in and for every m-th iteration
if n_sampled > burn_in and n_sampled % m == 0:
samples[(n_sampled - burn_in) // m] = z
值得注意的是，你不需要计算min(1, p(cand) / p(z))，只能计算p(cand) / p(z)。我意识到正式的min是必要的（以确保概率在0和1之间）。但是，从计算上来说，我们不需要min，因为如果p(cand) / p(z) > 1那么p(cand) / p(z)总是大于u。
把所有这些放在一起以及预先计算随机创新，接受概率u并且只计算你真正需要时的可能性：
def my_Metropolis_Gaussian(p, z0, sigma, n_samples=100, burn_in=0, m=1):
# Pre-allocate memory for samples (much more efficient than using append)
samples = np.zeros(n_samples)
# Store initial value
samples[0] = z0
z = z0
# Compute the current likelihood
l_cur = p(z)
# Counter
iter = 0
# Total number of iterations to make to achieve desired number of samples
iters = (n_samples * m) + burn_in
# Sample outside the for loop
innov = np.random.normal(loc=0, scale=sigma, size=iters)
u = np.random.rand(iters)
while iter < iters:
# Random walk innovation on z
cand = z + innov[iter]
# Compute candidate likelihood
l_cand = p(cand)
# Accept or reject candidate
if l_cand / l_cur > u[iter]:
z = cand
l_cur = l_cand
# Only keep iterations after burn-in and for every m-th iteration
if iter > burn_in and iter % m == 0:
samples[(iter - burn_in) // m] = z
iter += 1
return samples
如果我们看一下性能，我们发现这个实现比原来快2倍，这对于一些小的改动来说并不坏。
In [1]: %timeit Metropolis_Gaussian(pdf_t, 3, 1, n_samples=100, burn_in=100, m=10)
205 ms ± 2.16 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
In [2]: %timeit my_Metropolis_Gaussian(pdf_t, 3, 1, n_samples=100, burn_in=100, m=10)
102 ms ± 1.12 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)