Bayesian Extraction of Jet Energy Loss Distributions in Heavy-Ion Collisions

1. 問題
   根據統計學模型來擬和[alpha,beta,gamma]這三個參數，在這裏使用的是貝葉斯分析方法以及MCMC方法來作爲模型。我們假設這三個參數的先驗分佈爲均勻分佈，介質能量損失爲均勻分佈。

2. 工作流程

   1. 數據準備
   2. 模型構造
   3. 擬和模型
   4. 得出擬和參數

具體的過程在代碼會非常清晰明瞭。

import numpy as np
import matplotlib.pyplot as plt
import pymc3 as pm
from pymc3 import Uniform, Normal
from scipy.special import gamma
from scipy.interpolate import interp1d
from scipy import optimize

def read_data():   //讀取數據
    dat = np.loadtxt("./RAA_2760.txt")
    return dat[:,0],dat[:,1]

pp_x,pp_y = read_data()
gala30x = np.array([
0.047407180540804851462, 0.249923916753160223994,
0.614833454392768284613, 1.143195825666100798284,
1.836454554622572291486, 2.69652187455721519578,
3.72581450777950894932, 4.92729376584988240966,
6.304515590965074522826, 7.861693293370260468756,
9.603775985479262079849, 11.5365465979561397008,
13.6667446930642362949, 16.00222118898106625462,
18.55213484014315012403, 21.32720432178312892793,
24.34003576453269340098, 27.60555479678096102758,
31.14158670111123581829, 34.96965200824906954359,
39.11608494906788912192, 43.61365290848482780666,
48.50398616380420042732, 53.84138540650750561747,
59.69912185923549547712, 66.18061779443848965169,
73.44123859555988223951, 81.73681050672768572223,
91.5564665225368382555, 104.1575244310588945118
])
gala30w = np.array([
0.121677895367261782329, 0.283556882734938525493,
0.446432426678773424704, 0.610532130075812839931,
0.77630347862220587813, 0.944233288641719426021,
1.11484470167521153566, 1.288705432832675646608,
1.466439137624214862, 1.64873949785431911827,
1.83638767787041103674, 2.03027425167718247131,
2.23142718244322451682, 2.44104811309112149776,
2.66056024337508997273, 2.89167264137737623469,
3.13646833382438459, 3.3975275957906408941,
3.67810472956067256012, 3.9823886239009637486,
4.315899244899456113557, 4.686114009126298021,
5.1035026514834184093, 5.58333298168755125349,
6.1490444086574353235, 6.8391305457947582711,
7.7229478770061872416, 8.9437424683710389523,
10.87187293837791147282, 15.026111628122932986
])

class RAA2ELoss():    
    def __init__(self,RAA_fname,pp_pt=None,pp_spectra=None):
        self.with_ppdata = False
        if not(pp_pt is None or pp_spectra is None):
            self.pp_fit = interp1d(pp_pt,pp_spectra,fill_value="extrapolate")
            self.with_ppdata = True
        RAA_data = np.loadtxt(RAA_fname)
        self.RAA_x = RAA_data[:,0]
        self.RAA_xerr = RAA_data[:,1]
        self.RAA_y = RAA_data[:,2]
        self.RAA_yerr = RAA_data[:,3]
    def mean_ptloss(self, pt, b, c):
        return b*pt**c*np.log(pt)
    def model(self):
        with pm.Model() as basic_model:
            a = Uniform('a', lower=0, upper=10)
            b = Uniform('b', lower=0, upper=10)
            c = Uniform('c', lower=0, upper=1)
            
            intg_res = np.zeros_like(self.RAA_x)
            for i, x in enumerate(gala30x):
                
                scale_fct = self.RAA_x / gala30x[-1]
                x = x * scale_fct
                shifted_pt = self.RAA_x + x
                mean_dpt = self.mean_ptloss(shifted_pt, b, c)
                alpha = a
                beta = a / mean_dpt
                pdpt = (beta**(alpha) / (alpha)) * np.exp(-beta*x) * x**(alpha-1)
              
                intg_res += scale_fct*gala30w[i]*pdpt*self.pp_fit(shifted_pt)
               #上面兩行代碼不是很懂
            mu = intg_res / self.pp_fit(self.RAA_x)
            likelihood_y = Normal('likelihood_y', mu=mu, observed=self.RAA_y)
            start = pm.find_MAP(fmin=optimize.fmin_powell)
            step = pm.Metropolis()
            trace = pm.sample(1200,start=start,step=step)

ps： 代碼和數據參考github，他上面的pymc的版本是1，如果使用pymc3的話將會報錯。

該GitHub的流程圖。省略了畫圖與保存數據的部分。