1.算法描述

1.1先验概率的推导

根据贝叶斯概率论可知，某一事件的后验概率可以根据先验概率来获得，因此，这里首先对事件的先验概率分布进行理论的推导。假设测量的腐蚀数据服从gamma分布，其概率密度函数可以通过如下表达式表示：

根据参考文献1和参考文献2的理论推导可知，采用反gamma分布，可以作为腐蚀数据的先验分布，即:

公式3为公式2的自然指数形式，公式3中，x表示腐蚀数据，参数a和b分别表示反gamma分布的参数估计值。

从公式7可知，此时后验概率值则取决于最后一次测量结果.根据上述推导过程，完备集的后验概率可以通过如下公式计算得到：

但是完备集下的后验概率所满足的公式3条件和公式4条件，在实际中往往不太可能发生，因此需要考虑非完备集下的后验概率计算方法。

1.2.共轭条件下的非完备集的后验概率的推导

完备集下的后验概率不太适用于实际情况，因此，对于实际情况，需要考虑非完备集下的后验概率的计算。非完备集下的后验概率是关于随机事件的条件概率，是在相关证据给定并纳入考虑之后的条件概率。后验概率和先验概率满足如下关系式：

从公式可知，后验概率等同于先验函数和似然函数的乘积，这里先验函数根据本文公式2获得，下面主要对似然函数进行公式推导，根据参考文献5的相关推导过程可知，后验概率的基本计算公式如下：

根据本文上述章节的介绍，参数A和B满足如下关系式：

因此，似然函数可以通过如下表达式表示：

2.仿真效果预览

matlab2022a仿真结果如下：

3.MATLAB核心程序

 
K_d        = length(dt(:,:,kk1)); %total number of d
K_l        = length(Lt(:,:,kk1)); %total number of l
 
for i = 1:K_d
    if Nn2(i) == 1
       dt1(i,:,kk1) = dt1(i,:,kk1); 
    else
       dt1(i,:,kk1) = 5.39 + 0.19*dt1(i,:,kk1) - 0.02*Lt(i,:,kk1) + 0.35*Nn2(i);
    end
end
%m->mm
dt1        = 1000*dt1;
%to obtaion a average number of do_rate and Lo_rate
do_rate    = sum(dt1(:,:,kk1))/K_d;  
Lo_rate    = sum(Lt(:,:,kk1))/K_l; 
% Q = sqrt(1+0.31*power(Lo_rate/sqrt(D/t),2)); 
% Q--length of correction factor
Q1         =(Lo_rate/sqrt(D_t))^2;
Q          = sqrt(1+0.31*Q1);
% pf_rate=(2*t*sigma_u*(1-do_rate/t))/(D-t)/(1-(do_rate/t)/Q);
% pf -- failure pressure
pf_rate_1  = 2*t*sigma_u*(1-do_rate/t);
pf_rate_2  =(D-t)*(1-do_rate/t/Q);
pf_rate    = pf_rate_1/pf_rate_2;
grid_dist  = 0.1/20; % in order to get the obvious result on the plot
x          = grid_dist:grid_dist:pf_rate*0.015;
%fit the contineous inverted gamma density to the data
par        = invgamafit(0.1); % change pf_rate from mPa to kPa, in order to get the obvious result on the plot
a          = par(1);
b          = 1/par(2);
%Examining inverted gamma distributed prior
prior     = exp(a*log(b)-gammaln(a)+(-a-1)*log(x)-b./x);
load r2.mat
prior     = post_imp_prior';
%Examination of inverted gamma post prior after perfect inspection
A         = a + dt1(K_d)/pf_rate^2;
B         = b +  Lt(K_l)/pf_rate^2;
postprior = exp(A*log(B)-gammaln(A)-(A+1)*log(x)-B./x);
%***********************************************************************************
% % %***********************************************************************************
% %定义likelyhood
% likeliprod = likelihoods(x,t,dt(:,:,kk1),Lt(:,:,kk1),Nn2);
%***********************************************************************************
%这个部分和之前的不一样了，修改后的如下所示：
%***********************************************************************************
%对prior参数进行随机化构造
m = 10;
for ijk = 1:m
    ijk
    %***********************************************************************************
    %***********************************************************************************
    %Calaulate the depth change rate and length change rate with time 
    for kk1 =1:(kk -1);
        drate1 = normrnd(drate,drateS, nsamples,1, kk1); % Measured defect depth @ time T 
        Lrate1 = normrnd(Lrate,LrateS, nsamples,1, kk1); % Measured defect length @ time T    
        if kk1 == 1
           dt(:,:,kk1) = do1(:,:,kk1) + drate1(:,:,kk1)*(delT) ; 
           dt1(:,:,kk1) = dt(:,:,kk1);
           Lt(:,:,kk1) = Lo1(:,:,kk1) + Lrate1(:,:,kk1)*(delT) ;    
        else 
           dt(:,:,kk1) = dt(:,:,kk1-1)   + drate1(:,:,kk1)*(delT);
           dt1(:,:,kk1) = dt(:,:,kk1) ;
           Lt(:,:,kk1) = Lt(:,:,kk1-1) + Lrate1(:,:,kk1)*(delT); 
        end  
    end
    K_d        = length(dt(:,:,kk1)); %total number of d
    K_l        = length(Lt(:,:,kk1)); %total number of l
    for i = 1:K_d
        if Nn2(i) == 1
           dt1(i,:,kk1) = dt1(i,:,kk1); 
        else
           dt1(i,:,kk1) = 5.39 + 0.19*dt1(i,:,kk1) - 0.02*Lt(i,:,kk1) + 0.35*Nn2(i);
        end
    end
    %m->mm
    dt1        = 1000*dt1;
    %to obtaion a average number of do_rate and Lo_rate
    do_rate    = sum(dt1(:,:,kk1))/K_d;  
    Lo_rate    = sum(Lt(:,:,kk1))/K_l; 
    % Q = sqrt(1+0.31*power(Lo_rate/sqrt(D/t),2)); 
    % Q--length of correction factor
    Q1         =(Lo_rate/sqrt(D_t))^2;
    Q          = sqrt(1+0.31*Q1);
    % pf_rate=(2*t*sigma_u*(1-do_rate/t))/(D-t)/(1-(do_rate/t)/Q);
    % pf -- failure pressure
    pf_rate_1  = 2*t*sigma_u*(1-do_rate/t);
    pf_rate_2  =(D-t)*(1-do_rate/t/Q);
    pf_rate    = pf_rate_1/pf_rate_2;
    grid_dist  = 0.1/20; % in order to get the obvious result on the plot
    x          = grid_dist:grid_dist:pf_rate*0.015;
    %fit the contineous inverted gamma density to the data
    par        = invgamafit(0.1); % change pf_rate from mPa to kPa, in order to get the obvious result on the plot
    as(1,ijk)  = par(1);
    bs(1,ijk)  = 1/par(2);
    %***********************************************************************************
    %***********************************************************************************
end
 
npar     = m;             % dimension of the target
drscale  = m;             % DR shrink factor
adascale = 2.4/sqrt(npar); % scale for adaptation
nsimu    = 5e5;            % number of simulations
 
c        = 10;           % cond number of the target covariance 
a        = ones(npar,1); % 1. direction
[Sig,Lam]= covcond(c,a); % covariance and its inverse
mu       = 1.35*as;% center point
model.ssfun      = inline('(x-d.mu)*d.Lam*(x-d.mu)''','x','d');
params.par0      = mu+0.1; % initial value
params.bounds    = (ones(npar,2)*diag([0,Inf]))';
data             = struct('mu',mu,'Lam',Lam);
options.nsimu    = nsimu;
options.adaptint = 100;
options.qcov     = Sig.*2.4^2./npar;
options.drscale  = drscale;
options.adascale = adascale; % default is 2.4/sqrt(npar) ;
options.printint = 100;
[Aresults,Achain]= dramrun(model,data,params,options);
mu       = bs;% center point
model.ssfun      = inline('(x-d.mu)*d.Lam*(x-d.mu)''','x','d');
params.par0      = mu+0.1; % initial value
params.bounds    = (ones(npar,2)*diag([0,Inf]))';
data             = struct('mu',mu,'Lam',Lam);
options.nsimu    = nsimu;
options.adaptint = 100;
options.qcov     = Sig.*2.4^2./npar;
options.drscale  = drscale;
options.adascale = adascale; % default is 2.4/sqrt(npar) ;
options.printint = 100;
[Bresults,Bchain]= dramrun(model,data,params,options);
%选择值最集中的最为最终的值； 
for i = 1:m
    [Na,Xa] = hist(Achain(:,i));
    [V,I]   = max(Na);
    A1(i)   = Xa(I);
    [V,I]   = min(Na);
    A2(i)   = Xa(I);    
    
    [Nb,Xb] = hist(Bchain(:,i));
    [V,I]   = max(Nb);
    B1(i)   = Xb(I);
    [V,I]   = min(Nb);
    B2(i)   = Xb(I); 
end
As                 = mean(A1);
Bs                 = mean(B1);
post_imp_prior     = exp(As*log(Bs)-gammaln(As)-(As+1)*log(x)-Bs./x);
post_imp_prior_CDF = cumsum(post_imp_prior)*grid_dist;
A137