CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 51, 2016 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Tichun Wang, Hongyang Zhang, Lei Tian 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-43-3; ISSN 2283-9216 

Linear Bayesian Inference Theory Applied in Complex 
Analysis of Economic Forecasting and Management 

Chunyang Bai 
Henan Institute of Science and Technology, Henan, Xinxiang 453003, China 
happylover618@126.com 

This paper systematically studied the Bayesian theory and methods of modern economic management in the 
multivariate models, including three linear models: single equation model, multi-equation model systems and 
Bayesian vector VAR (p) model. Prediction tectonic theory and its application were more general Bayesian 
classification methods. Risk decision solutions random array parameters, Bayesian study single equation 
model estimation theory, including the design matrix singular linear model coefficients assumptions Bayesian 
test method. Random error sequence Bayesian diagnostic unit root test autocorrelation. At the same time, the 
Bayesian single equation model is applied to quality control, constructing Bayesian variance with mean and 
variance control chart the mean standard deviation unknown a Bayesian control charts. VAR model and 
Bayesian VAR model effects were compared, the results showed that the effect of prediction Bayesian VAR 
model is better than other types of predictive models. 

1. Introduction 

With the application of statistical theory and methods, Bayesian theory was welcomed and has been 
developing rapidly, especially in decision-making problems. It occupy an increasingly important position in 
statistical applications, and decision-making issues, first using the experience of knowledge is essential. 
Compared with purely theoretical problems, such problems are often subjective probabilities formulation more 
natural, while reflecting the degree of decision-makers access to information, and therefore easy to accept the 
Bayesian point of view (Miller et al., 2013). Bayesian that: prior distribution reflects the understanding of the 
overall parameters of the test before the distribution, after obtaining sample information, people of this 
understanding has changed (Buteler et al., 2009). The result is reflected in the posterior distribution is a 
combination of the posterior distribution prior distribution parameters and sample information. It can be seen 
that the frequency of school statistical inference is "from scratch" process before a trial on the case of 
unknown parameters is ignorant, and after the test is somewhat understood, but there is no common 
understanding of how much presentation methods in practice depends on the statistics used by the targeted 
(Kelly et al., 2013). Bayesian inference is not, it is a "from there to the" process, and the results clearly natural, 
in line with people's habits of mind obtained according to an information revise its previous opinion, not 
necessarily from scratch (Gruber et al., 2016). 
From highly theoretical point of view, we must pay attention to such a basic point: do not have the statistical 
inference is inference under incomplete information (Weaver et al., 2009). The information available is not 
enough to determine the unique solution of the problem, which provides the establishment of more possible 
kinds of theoretical system and methods (Krolzig et al., 2013). In fact, in addition to the frequency and 
Bayesian school, there are other methods based on theory and ideology of these have failed to develop into a 
school with two rival school, mainly due to apply too narrow, or fail to develop a set of common, there is 
inherent harmony of theory and square feet like (Wikle et al., 2015). However, these theories, in some cases 
also have successfully applied the two schools have their successes and deficiencies, there are broad 
prospects for development, it is practically complementary.  
This article will explore the structure of the model parameter prior distribution, including diffusion and 
conjugate prior distribution structure prior distribution. Risk decision solutions random array parameters, 
Bayesian study single equation model estimation theory, including the design matrix singular linear model 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1651157

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Bai C.Y., 2016, Linear bayesian inference theory applied in complex analysis of economic forecasting and 
management, Chemical Engineering Transactions, 51, 937-942  DOI:10.3303/CET1651157   

937



coefficients assumptions Bayesian test method, and random error sequence Bayesian diagnostic unit root test 
autocorrelation. At the same time, the Bayesian single equation model is applied to quality control, 
constructing Bayesian variance with mean and variance control chart the mean standard deviation unknown a 
Bayesian control charts. Bayesian theory constructed more general classification methods. Will be based on 
sufficient statistics to study various parameters generally follow a normal distribution, the prior distribution 
parameters were taken to diffuse prior distribution and normality when conjugate prior, how to use the forecast 
to be sentenced density function of the sample, after construction posterior probability ratio and the 
corresponding classification rule, treat accordingly sentenced to classify the sample. 

2. Bayesian inference theory 

2.1 Prior distribution 
The conjugate distribution Bayesian analysis commonly used in another type of prior distribution, its 
ideological foundation is the law of a priori and a posteriori laws consistent, concrete is the prior distribution 
and posterior distribution of this requirement to belong to the same distribution family (Lenk et al., 2013). For 
each specific distribution, it has its conjugate distribution, use the following formula like factoring and sufficient 
statistics and other natural function analysis method to construct the conjugate prior distribution required 
herein (D’Agostino et al., 2016). 

     1 2
1

, , , ,
n

i n n n

i

f Y g t h Y Y Y 


 ( ) /DD V D V   (1) 

If the definition of risk function 

           
^

, , ,
Y

R E L Y dP Y L Y dP Y
  

       
    (2) 

Then called      
^

,R R d     


  is Bayes risk. Bayesian decision function and Bayesian posterior 
decision function is equivalent to, even if the minimum posterior risk decision function also makes Bayes 
minimum risk (D'Agostino and Graefe, 2013). On the contrary, the minimum risk Bayes decision function also 
makes posteriori minimum risk, therefore, according to the posterior risk criteria to seek solutions to Bayesian 
decision (Capinera et al., 2015). 

2.2 The Bayesian estimation theory 

Assuming random variables x1, x2, ...,, xm follows a linear relationship exists between y and independent 
variables: 

 
1 0 1 1 2 2 1

2

1

,

~ . .  0, , 1, 2, ,

m m
y x x x

i i d N i n

    

 

     


 (3) 

The aforementioned single equation model using the following matrix form: 

 2 2, ( ) ~ 0, , 0nY X rank X N I       (4) 

In it: 

11 21 1

12 22 11

1 2

1

1

1

m

n n mn

x x x

x x x
X

x x x

 
 
 
 
 
 

1 0 1

2 1 2
, ,

n m n

y

y
Y

y

 

 
 

 

     
     
       
     
     
     

 
(5) 

Where increased matrix X column full rank condition; before making the following analysis, this paper first 
presents the definition required for the introduction of the use of multivariate t distribution (Knodel et al., 2009). 

2.3 The correlation coefficient of the posterior distribution 

Before the autocorrelation coefficient as Φ Bayesian analysis should be subject to a given prior distribution of 

unknown parameters in the model. Obviously, for a given β and σ, in the absence of a priori information, the 

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autocorrelation coefficients can be considered closed interval must obey uniform distribution [-1,1] on. Joint 
prior distribution density function is: 

     , , , , 1              (6) 

According to Bayes' theorem, parameter was Φ, β and σ. The joint posterior density function 

     

2

2 1 2 12

, , , , , ,

1 1
exp

2
n

Y X L

Y Y X X

         

   
 

 

 
     

 

 (7) 

Their specific expression is as follows: 

     

     

     

     

1 1 1 1 1

2 2 2 2 2

3 1 1 2 2

3 1
=

T

T

T

c Y X Y X

c Y X Y X

c Y X Y X

c c


  

  

  

   

   

   


  



 (8) 

3. Experiments and results 

3.1  Bayesian model simulation analysis 
In the prior distribution parameter settings, the principle is that when parameters are determined at a certain 
value (such as zero value), the model parameters close to this value to determine the orientation and not 
locked, as long as the necessary data to support, then this method can get a more accurate estimate. 
After the model parameters in order to obtain the Bayesian estimation, we can calculate the formula to get the 
exact model parameters conditional posterior distribution. In practice, we can use simulation methods to 
obtain approximate posterior distribution of their condition, which can avoid cumbersome and complex 
Bayesian calculations more quickly and easily obtain the parameters' Bayesian estimation. Here we use 
MCMC methods and WinBuGS packages using WinBUGS packages AR (2) model of Doodie Bayesian 
analysis (Figure 1) as follows: 
 

Y[i-1]

Y[i-2]

Ymean[i]

Y[i]

theta[1] theta[2]

tau

sigma

for(i IN 3:n)
 

Figure 1: Model Doodle 

During the analysis of the model, MCMC convergence diagnosis is very important, must not simply through a 
lot of iterations of iterations as the pre-simulation. Analyzing convergence in terms of, WinBuGs was through 
the chain of multi-parameter iterative analysis. Determine that the multi-input set of initial values to form a 
multilayer iterative chain, when the parameters of the model convergence, tend to coincide with the iteration 
graphical results. 
We use a time-series simulation MA to carry out our Bayesian analysis using SAS software to generate 500 
data from a model simulation of MA (2) model: 

1 2
0.5 0.3

t t t t
y u u u

 
    (9) 

Wherein ut-N (0,1) of data listed in Table 1. 

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Table 1: MA(2) model simulation 

obs y 

1 1.4837 1.3622 0.1142 -0.4396 0.4735 

6 -0.3714 -0.4030 -0.3972 -0.25687 0.3621 

11 0.1684 -0.3824 -0.1127 0.2107 1.1184 

16 1.1388 -0.4293 2.3779 1.2268 0.4808 

21 -0.8252 -1.7040 -0.7217 -0.6414 0.6162 

… … … … … … 

476 -1.5981 1.7895 -0.6281 0.6379 0.8349 

481 0.8521 -0.7006 -0.5161 -0.7638 0.6658 

486 0.3104 -0.5994 -1.7979 -2.6976 -0.6203 

491 -1.4563 0.4837 1.3188 1.1554 -0.1597 

496 1.2030 -0.4362 0.2017 0.42571 -0.19740 

3.2 Numerical examples 
Sims used the likelihood determine the optimal lag order model number ratio test statistic, model VAR (p1) 
model VAR (P2) of the likelihood ratio test statistic is: 

1 2

~ ^ ^

1 2
ln ln ,p pLR N N p p
  

       
  

 (10) 

Here are the Ώ model VAR (P1) and VAR (P2) residual covariance matrix, N is the sample size, N is the 

correction factor term, which is equal to VAR (p2) factor regression model number. Table 2 lists the VAR 
model 3rd order lag lag fifth-order hypothesis test results, test results in favor of shorter lag order; and the 
third-order lag lag order 4 hypothesis testing, test results in favor of 3 lags number, so after determining the 
VAR model of order 3. 

Table 2: The LR test result of optimal delay degree 

 H0:p=3          H1：=5 H0:p=3          H1：=4 

 LR Probability LR Probability 
GDP growth 
model 

10.89 0.281 29.57 0.002 

Inflation Model 9.21 0.423 34.92 0.000 

Import & Export 
Type model 

13.56 0.346 23.47 0.019 

 
Bayesian vector autoregression VAR (p) prediction model and its application in economic forecasting, 
including Tony Bayes restriction VAR (p) model predictive inference and non-limiting VAR (p) model prediction 
Bayesian inference, the structure Mirmesota total reel prior distribution and the prior distribution analysis 
Bayesian inference under VAR (p) forecasting model, combined with empirical analysis and forecasting of AR 
model, VAR model and Bayesian VAR model effects were compared, the results showed that: the effect of 
prediction Bayesian VAR model is better than other types of predictive models. 

3.3 Evaluation of the results and the accuracy of the prediction model 
After obtaining the estimated model parameters, you can use Bayesian VAR (p) model yt prediction analysis 
step of predicting a specific prediction step AR model and VAR model is basically the same, may be a step 
ahead forecast, prediction or two steps ahead step ahead forecast. Model prediction accuracy can be 
evaluated from many aspects, such as the absolute mean square error, mean absolute percentage error, U 
Theil statistic this paper to analyze the predictive accuracy of Bayesian VAR (p) model, U statistic is based on 

940



the root mean square error of prediction model research-based ratio of root mean square error of prediction 
random walk model, namely: 

 

 

2

,
^

1

2

,

1

1

1

N

M i i

i

N

RW t t

i

F A
N

U

F A
N














 (11) 

Where At represents the time series at the actual value t, where,, represents the predicted value based on the 
prediction method M, and where the foot is based on random walk model predictions. If the value of U is equal 
to 1.0, then the accuracy of the prediction method of prediction accuracy M random walk model is basically the 
same; if the value of U is less than 1.0, then the prediction accuracy of the prediction accuracy of the method 
M is more than the random walk model high: If the value of U is greater than 1.0, then the accuracy of the 
prediction method of material low prediction accuracy than the random walk, then the method cannot improve 
the prediction accuracy of time series, which is also not suitable for prediction purposes. Simulation of ɵ1and 
ɸ1 posterior distribution is shown in Figure 2. 
 

0.8 0.85 0.9 0.95 1

0

5

10

15

20

 

 

theta[1] sample:10000

 

0.4 0.6 0.8 1

0

1

2

3

4

phi[1] sample:10000

 

Figure 2: Simulation of ɵ1and ɸ1 posterior distribution 

4. Conclusions 

Risk decision solutions random array parameters, Bayesian study single equation model estimation theory, 
including the design matrix singular linear model coefficients assumptions Bayesian test method, and random 
error sequence Bayesian diagnostic unit root test autocorrelation. At the same time, the Bayesian single 
equation model is applied to quality control, constructing Bayesian variance with mean and variance control 
chart the mean standard deviation unknown a Bayesian control charts. Bayesian theory constructed more 
general classification methods. Will be based on sufficient statistics to study various parameters generally 
follow a normal distribution, the prior distribution parameters were taken to diffuse prior distribution and 
normality when conjugate prior, how to use the forecast to be sentenced density function of the sample, after 
construction posterior probability ratio and the corresponding classification rule, treat accordingly sentenced to 
classify the sample. 

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