Python “步行者不是”；“步行”；基于MCMC的模型拟合

我在执行模型的MCMC分析时遇到困难。我相信这与我在模型中有一个不完整的伽马函数有关

我试图最小化高斯对数似然，但似乎步行者被困在井里，而不是试图最小化似然函数。如下图所示，其中y轴为模型参数，x轴为步数。该图像显示了漫游者如何不探索参数空间。我添加了另一幅图像来演示如何正确探索参数空间

及

我在下面添加了一些代码来演示我在做什么，其中x、y和yerr是大约4000个点的数组。该代码适用于其他模型，但仅在该模型上中断，因此它必须是其他模型所没有的内在特性。其他模型最明显的变化是添加了不完全伽马函数，否则它的函数形式与其他模型非常相似

我正在安装的模型具有以下形式：

def singinhomobremss(freq,S_norm,alpha,p,freq_peak): # Single inhomogeneous free-free emission model
    return S_norm*(p+1)*(np.power((freq/freq_peak),(2.1*(p+1)-alpha))*special.gammainc((p+1),(freq/freq_peak)**(-2.1)))*special.gamma(p+1)

请注意，我使用的是python包emcee（我会发布链接，但我没有足够的声誉…）。我真的不明白为什么步行者拒绝为这个模型“步行”，而他们却为其他模型“步行”。非常感谢您的帮助，但我知道这是一个合理的利基领域

import numpy as np
import matplotlib.pyplot as plt
import scipy.optimize as opt
import scipy.special as special # For access to the incomplete gamma function.
import emcee
import triangle 
import inspect

def singinhomobremss(freq,S_norm,alpha,p,freq_peak): # Single inhomogeneous free-free emission model
    return S_norm*(p+1)*(np.power((freq/freq_peak),(2.1*(p+1)-alpha))*special.gammainc((p+1),(freq/freq_peak)**(-2.1)))*special.gamma(p+1)

# initial guess for a fit

p0guess = [7, 0.6, -0.5, 3.5]

# Defining log-likelihood function
def lnlike(theta,x,y,yerr):

    S_norm,alpha,p,freq_peak = theta
    model = singinhomobremss(x,S_norm,alpha,p,freq_peak)
    inv_sigma = 1.0/(yerr**2)

    return -0.5*(np.sum((y-model)**2*inv_sigma - np.log(inv_sigma))) 
    # Use the scipy.opt model to find the optimum of this likelihood function

nll = lambda *args: -lnlike(*args)
result = opt.fmin(nll,p0guess,args=(x,y,yerr),full_output='true')
S_norm_ml,alpha_ml,p_ml,freq_peak_ml = result[0]

# Defining priors
def lnprior(theta):
    S_norm,alpha,p,freq_peak = theta
    if S_norm_ml/100. < S_norm < S_norm_ml/100. and alpha_ml/100. < alpha < alpha_ml*100. and p_ml/100. < p < p_ml*100. and freq_peak_ml/100. < freq_peak < freq_peak_ml*100:
        return 0.00 
    return -np.inf      

# Combining this prior with the definition of the likelihood function, the probablity fucntion is:

def lnprob(theta, x, y, yerr):
    lp = lnprior(theta)
    if not np.isfinite(lp):
        return -np.inf
    return lp + lnlike(theta, x, y, yerr)

# Now implement emcee

ndim, nwalkers, nsteps = len(inspect.getargspec(singinhomobremss)[0])-1, 200, 2000 

# Initialising the walkers in a Gaussian ball around maximum likelihood result
#pos = [result['x'] + 1e-4*np.random.randn(ndim) for i in range(nwalkers)]
pos = [result[0] + 1e-4*np.random.randn(ndim) for i in range(nwalkers)]

sampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob, args = (x,y,yerr))
sampler.run_mcmc(pos, nsteps) # This is the workhorse step.

# Now plotting the walks of the walkers with each step, for each parameter. If they have converged on a good value they should have clumped together.

fig = plt.figure(2,figsize=(10, 10))
fig.clf()
for j in range(ndim):
    ax = fig.add_subplot(ndim,1,j+1)
    ax.plot(np.array([sampler.chain[:,i,j] for i in range(nsteps)]),"k", alpha = 0.3)
    ax.set_ylabel((r'$S_{norm}$',r'$\alpha$',r'$p$',r'$\nu_{peak}$')[j], fontsize = 15)
plt.xlabel('Steps', fontsize = 15)
fig.show()

# To me it looks like the burn in period is well and truly over by 400 steps. So I will exclude those. 
print 'The burnin applied was 400. Make sure the walkers have converged after that many steps.'
samples = sampler.chain[:,400:,:].reshape((-1,ndim))

# Plotting the histograms of the fit.
trifig = triangle.corner(samples, labels = [r'$S_{norm}$',r'$\alpha$',r'$p$',r'$\nu_{peak}$'])

# Finally to get the 1 sigma final uncertainties you do
S_norm_singinhomobremss_mcmc, alpha_singinhomobremss_mcmc, p_singinhomobremss_mcmc, freq_peak_singinhomobremss_mcmc = map(lambda v: (v[1], v[2]-v[1], v[1]-v[0]), zip(*np.percentile(samples,[16,50,84], axis = 0))) # Uncertainites based on the 16th, 50th and 84th percentile.

plt.figure()
plt.clf()
plt.errorbar(nu, flux, flux_err, marker = '.', color = 'gray', linestyle='none', label = 'Data',alpha=0.2)  
plt.loglog(nu_peaklong,singinhomobremss(nu_peaklong, *poptsinginhomobremss), 'saddlebrown',label="Best fit inhomogeneous model from least-square")
plt.loglog(nu_peaklong,singinhomobremss(nu_peaklong,S_norm_singinhomobremss_mcmc[0], alpha_singinhomobremss_mcmc[0], p_singinhomobremss_mcmc[0], freq_peak_singinhomobremss_mcmc[0]),color = 'r', linestyle='-', label="Best fit inhomogeneous free-free model.")
plt.title('PKS 1718-649 Epoch - '+filename, fontsize = 15)
minnu = np.array(min(nu))-0.05*np.array(min(nu))
plt.legend(loc='lower center', fontsize=10) # make a legend in the best location
plt.xlabel('Frequency (GHz)', fontsize = 15)
plt.ylabel('Flux Density (Jy)', fontsize = 15)
plt.axis([min(nu)-0.1*min(nu), max(nu)+0.1*max(nu), min(flux)-0.1*min(flux), max(flux)+0.1*max(flux)])
plt.rc('xtick',labelsize=15)
plt.rc('ytick',labelsize=15)
plt.xticks([2,4,6,8,10]) #Setting grid line positions.
plt.yticks([3,4,5])
plt.grid(True)
plt.show()

将numpy导入为np
将matplotlib.pyplot作为plt导入
导入scipy.optimize作为选项
将scipy.special导入为special#以访问不完整的gamma函数。
进口主持
导入三角形
进口检验
def singinhomobremss（频率、S_范数、α、p、频率峰值）：#单一非均匀自由-自由发射模型
返回S_范数*（p+1）*（np.功率（（频率/频率峰值），（2.1*（p+1）-alpha））*特殊.gammainc（（p+1），（频率/频率峰值）**（-2.1））*特殊.gamma（p+1）
#猜测是否合适
p0guess=[7,0.6，-0.5,3.5]
#对数似然函数的定义
def lnlike（θ，x，y，yerr）：
S_范数，α，p，频率峰值=θ
模型=singinhomobremss（x，S_范数，α，p，频率峰值）
投资西格玛=1.0/（年**2）
返回值-0.5*（np.总和（（y型）**2*库存西格玛-np.对数（库存西格玛）））
#使用scipy.opt模型找到该似然函数的最佳值
nll=lambda*args:-lnlike（*args）
结果=opt.fmin（nll，p0guess，args=（x，y，yerr），完整输出='true'）
S_norm_ml，alpha_ml，p_ml，freq_peak_ml=结果[0]
#定义先验
def lnprior（θ）：
S_范数，α，p，频率峰值=θ
如果S_norm_ml/100.

我已经解决了这个问题！只要在这里写下解决方案，以备将来有人遇到类似情况时参考。步行者没有步行，因为lnprob是-inf。解决方案实际上是补救性的，没有任何实际效果