# Bayesian Inference of Ammunition Consumption Based on Normal-Inverse Gamma Distribution

Authors: Haobang Liu; Xianming Shi; Xiaojuan Chen; Yuan Li; Mei Zhao; Yongchao Jiang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

To address the problems of high cost of new ammunition experiment, few data of field test and low accuracy of consumption prediction, this article proposes a Bayesian estimation method of ammunition consumption based on normal-inverse gamma distribution, and estimates the hyperparameters in the prior distribution through the prior information from the consumption of ammunition under different damage degrees of point targets, based on the normal distribution phenomenon of ammunition consumption at each damage degree. It is to establish a Bayesian estimation model for ammunition consumption under different damage degrees according to field test data based on Bayesian formula and solve for its posterior distribution. The example proves that the estimation results of ammunition consumption for point target with different damage degrees based on this method is more scientific and reasonable according to various prior information.


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## Body

## 1. Introduction

Ammunition support capability is a key factor affecting the combat effectiveness of the army. Ammunition consumption prediction is expected as a basic task of ammunition support, and its accuracy is an important embodiment of ammunition support capabilities. Through accurate prediction of ammunition consumption, ammunition support actions can be planned scientifically and rationally, and its supplement can be effectively implemented [1]. Therefore, more and more scholars around the world have begun to study the methods of ammunition consumption prediction.At present, due to the development of weapons and equipment, all kinds of new ammunition begin to be deployed in troops, which requires higher capability of ammunition support [2, 3].In addition, it is increased the difficulty of estimating the consumption of some new types ammo because of the limitations of the experimental cost and experimental environment, the small number of field shooting experiments, and the irregularity of ammo consumption. Therefore, it is a key problem of ammunition support to improve the accuracy of ammunition consumption prediction under limited conditions.In recent years, the United States (US) military has made some progress in the area of ammunition consumption prediction. The US Army has developed the Defense Standard Ammunition Computer System (DSACS) to manage ammunition [4]. The US Navy has established a real-time ammunition management information support system that directly transfers data between computers, and determines the demand for conventional ammunition in accordance with the combat objectives in the combat plan. The US Air Force developed the Combat Ammunition System (CAS). The US military has also developed the Munitions Requirements Decision Support System (MRDSS) to help users determine the best ammunition requirements for various types of ammunition [5]. In addition, the US military adopts the “Level of Effort” (LOE) method, which is mainly based on ammunition effectiveness in ammunition consumption prediction, to further improve the accuracy of ammunition consumption prediction [6]. The above research shows that the US military attaches great importance to the study of ammunition consumption prediction, and has achieved some research results worthy of reference.Domestic scholars have also made some new progress in ammunition consumption prediction, mainly including the forecast of ammunition demand based on target damage and the forecast of ammunition demand based on intelligent algorithms, military operations research, and statistical theories and so on. The forecasting method of ammunition demand based on target damage takes into account the performance of ammunition and the effect of target damage, improves the accuracy of prediction, and provides a theoretical basis for the accurate guarantee of ammunition [7–11]. The prediction of ammunition consumption based on intelligent algorithms mainly refers to the application of algorithms such as artificial neural networks, support vector machines, and particle swarm optimization to the field of ammunition consumption prediction. Li et al. [12] established a fuzzy logic model and integrated many factors that affect ammunition consumption to predict ammunition consumption. Tian et al. [13] combined the neighborhood rough set and support vector machine to predict aviation ammunition consumption. Liang et al. [14] combined genetic algorithm and neural network to study the consumption rule of air-to-air missiles. Ammunition consumption prediction methods based on military operations research mainly use Lanchester equation, planning optimization, combat simulation and other military operations research methods to predict ammunition consumption under wartime conditions. Li et al. [15] divided the battle process into multiple stages and took multiple factors into consideration to predict ammunition consumption. Qi [16] explored the effects of target characteristics, ammunition performance, and ammunition power on the results of ammunition consumption calculations.The above methods can improve the accuracy of ammunition consumption estimation to a certain extent, but these methods do not specifically solve the problem of ammunition consumption estimation under the condition of insufficient test samples. In order to overcome the problem of insufficient sample size, Zhao et al. [17] made continuous treatment of ammunition consumption when reaching various damage degrees, starting from the relationship between ammunition strikes and damage changes, combined with Bayes’ theorem, a method for forecasting ammunition consumption based on the Gompertz distribution is established. Li et al. [18] proposed a Bayesian estimation method of ammunition consumption based on the phenomenon that damage degrees caused by ammunition hitting targets obey multinomial distribution. In addition, Bayesian methods are also widely used in engineering predictions in other fields. Wang et al. [19] used Bayesian method to predict the soil-water characteristic curve. Wang et al. [20] used the Bayesian method to study the probabilistic characterization of soil Young’s modulus, and used the Markov chain Monte Carlo (MCMC) method for simulation. Han et al. [21] used Bayesian MCMC method to measure the uncertainty of the statistical characteristics of geological properties. The Bayesian method is used to fuse information from different sources, increase information sources, and improve the accuracy of ammunition consumption prediction [22]. However, the ammunition consumption under different damage degree obtained by these studies is a certain value. In fact, ammunition consumption should follow a certain probability distribution, such as the normal distribution. In this regard, it is considered that the consumption of ammunition at each damage degree shows a normal distribution phenomenon as the research basis. Based on the advantage that Bayes’ theorem can overcome the problem of insufficient sample number, the conjugate prior distribution of the normal distribution, the normal-inverse gamma distribution, is selected as the prior distribution of the consumption parameter [23]. Through the processing of prior information such as simulation information, historical information, and expert information, the values of hyperparameters in the prior distribution are determined. Then use Bayesian formula to solve the posterior distribution. Compared with only relying on the information of ammunition shooting samples in the field, the information used in the posterior distribution is more comprehensive, and the ammunition consumption calculated based on the posterior distribution is more scientific and reasonable.


## 2. Bayesian Inference of Ammunition Consumption under Different Damage Degrees

Compared with the classical statistical theory which only uses sample information, Bayesian inference can comprehensively use various sources of prior information such as simulation information and historical information, and has certain adaptability to the problem of parameter estimation with insufficient sample size.


### 2.1. Overall Information of Ammunition Consumption at Different Damage Degrees

As the ammunition continues to strike, the combat effectiveness of the target continues to decreases. Therefore, the damage degree of the point target can be judged according to the loss of combat effectiveness of the point target. Under normal circumstances, the damage degree of the target is divided into 5° [24], and the determination criteria are shown in Table 1.Table 1
Damage degrees classification of point targets.
i Point target damage degrees(0) Zero damage: the target is slightly damaged or undamaged, and the combat effectiveness loss within 5%.(1) Mild damage: target performance suffers a slight loss. If repairs are not carried out in time, combat performance will be affected, and combat effectiveness will be lost by 5% to 20%.(2) Moderate damage: the target is seriously damaged and requires replacement parts for repair, resulting in a 20% to 50% loss of combat effectiveness.(3) Severe damage: the target is seriously damaged and needs to be returned to the factory for overhaul, and the repair takes a long time, and the combat effectiveness is lost by 50% to 80%.(4) Destroy the annihilation: the target is completely destroyed and cannot be repaired, and the combat effectiveness loss is more than 80%.This research is based on the normal distribution phenomenon of ammunition consumption when the point target reaches each damage degree. Suppose that the overall ammunition consumptionqi under the damage degree i of the point target obeys the normal distribution with the parameter θi,σi2 [25]. Thus, the probability density function of ammunition consumption at damage degree i can be obtained:(1)fqi,θi,σi2=12πσie−qi−θi2/2σi2;i=0,1,2,3,4.In the formula:θi and σi2, respectively, represent the mean and variance of ammunition consumption under damage degree i. Determining the value of θi and σi2 will determine the probability distribution of ammunition consumption under point target damage degree i. How to use Bayesian theory to fuse prior information, overall information and sample information to estimate these two parameters is the focus of this article.


### 2.2. Sample Information of Ammunition Consumption Parameters

Under normal circumstances, in order to determine the amount of ammunition required to strike a target to different degrees of damage, it is necessary to conduct an outfield shooting test. According to the classical statistical theory, the sample information of ammunition consumption is used as the source of parameter estimation, the ammunition quantityqij, i=0,1,2,3,4.j=1,2,…n required by n point targets when they hit the damage level i was counted respectively. Then the estimated value of the normal distribution parameter θi,σi2 can be expressed by the mean q¯i and variance Si2 of the field sample:(2)θ^i=q¯i=1n∑j=1nqij,σ^i2=Si2=1n−1∑j=1nqij−q¯i2,where θ^i and σ^i2 are the unbiased estimates of θi and σi2, respectively.In order to measure the degree of error of the estimated valuesθ^i and σ^i2, the mean square error is used to express(3)MSEθ^i=Eθ^i−θi2=varθ^i,MSEσ^i2=Eσ^i2−σi22=varσ^i2.Because(4)θ^i∼Nθi,σi2n,n−1σ^i2σi2∼χ2n−1,then the mean square error values of θ^i and σ^i2 can be calculated:(5)MSEθ^i=σi2n,MSEσ^i2=2σi4n−1.It can be seen from the mean square error value formula that when the number of point targets in the shooting test is small, the error of the estimated value of the ammunition consumption measurement parameter is larger. Normally, for live-fire strike tests under field conditions, due to the limitations of test costs and test environment, the number of test samples is small, and the error of parameter estimates obtained by classical statistical methods that only rely on sample information is large.


### 2.3. Prior Distribution of Ammunition Consumption Parameters

In order to overcome the problem of insufficient number of test samples under external field conditions, the idea of Bayesian statistics can be considered for research, using Bayesian theory to introduce prior information and increase the information amount of parameter statistics. The Bayesian statistical method regards the normal distribution parameterθi,σi2 of ammunition consumption as a random variable, which is different from the concept that parameters are fixed values in classical statistics. Then mark θi,σi2∼πθi,σi2 as the joint prior distribution of parameter θi,σi2. Considering the mutual influence between θi and σi2, the joint prior distribution is the product of the two distributions. The joint prior distribution of θi,σi2 can be expressed as(6)πθi,σi2=πθi|σi2πσi2,where πσi2 is the prior distribution of σi2 and πθi|σi2 is the prior distribution of θi under condition σi2.There are many methods to determine the prior distribution in Bayesian theory, including the conjugate prior distribution method, the maximum entropy method, and the noninformation prior distribution. In order to facilitate the solution and take into account the available prior information, the conjugate prior distribution method is selected to determine the prior distribution of the hit accuracy parameter. By consulting the family of conjugate prior distributions, it can be known that the conjugate prior distribution under the condition that the characteristic parametersθi and σi2 of the normal distribution are both unknown is the normal-inverse gamma distribution [26–28]. Therefore, the normal distribution is selected as the prior distribution of θi, and the inverse gamma distribution is selected as the prior distribution of σi2, namely:(7)θi|σi2∼Nθ0i,σi2k0i.(8)σi2∼IGav0i2,v0iσ0i22.The normal-inverse gamma distribution is the product of formula (7) and formula (8):(9)πθi,σi2=πθi|σi2πσi2=Nθ0i,σi2k0iIGav0i2,v0iσ0i22,where θ0i, k0i, v0i, and σ0i2 are hyperparameters in the prior distribution2.4. The Posterior Distribution of Ammunition Consumption ParametersThe posterior distribution of ammunition consumption can be obtained by integrating the overall information, sample information and prior information of ammunition consumption by Bayesian formula. Compared with the classical statistical method based only on sample information, the posterior distribution obtained by Bayesian formula contains more information and is more suitable for parameter estimation problems with insufficient sample size. According to Bayes’ theorem, the posterior density function of the ammunition consumption parameterθi,σi2 is as follows:(10)πθi,σi2|qi=Lqi|θi,σi2πθi|σi2πσi2mqi,(11)mqi=∬Lqi|θi,σi2πθi|σi2πσi2dθidσi2,where Lqi|θi,σi2 is the likelihood function of the ammunition consumption parameter field sample, mqi is the marginal distribution or unconditional distribution of field sample and qi=qi1,qi2,…,qin is the i th field sample. It can be seen from formula (10) that there is a scale factor mqi difference between the posterior distribution of parameter on the left side of the equal sign and the product of the sample likelihood function and the parameter prior distribution on the numerator of the right side of the equal sign. Since mqi has nothing to do with the parameters θi,σi2, it does not contain any information about θi,σi2, when looking for the form of the posterior distribution of the parameters, mqi can be ignored. Then the posterior distribution can be expressed as(12)πθi,σi2|qi∝σi2−vni/2+3/2e−vniσni2+kniθi−θni2/2σi2,where(13)kni=k0i+n,vni=v0i+n,θni=k0ik0i+nθ0i+nk0i+nq¯i,vniσni2=v0iσ0i2+n−1Si2+k0ink0i+nθ0i−q¯i2,where Si2 is the variance of the field sample.The following studies how to use prior information such as historical data and simulation data to determine the prior distribution hyperparameters, and to determine the posterior distribution of the parameters according to the Bayesian statistical formula to provide a basis for the estimation of ammunition consumption under different damage degrees.


## 2.1. Overall Information of Ammunition Consumption at Different Damage Degrees

As the ammunition continues to strike, the combat effectiveness of the target continues to decreases. Therefore, the damage degree of the point target can be judged according to the loss of combat effectiveness of the point target. Under normal circumstances, the damage degree of the target is divided into 5° [24], and the determination criteria are shown in Table 1.Table 1
Damage degrees classification of point targets.
i Point target damage degrees(0) Zero damage: the target is slightly damaged or undamaged, and the combat effectiveness loss within 5%.(1) Mild damage: target performance suffers a slight loss. If repairs are not carried out in time, combat performance will be affected, and combat effectiveness will be lost by 5% to 20%.(2) Moderate damage: the target is seriously damaged and requires replacement parts for repair, resulting in a 20% to 50% loss of combat effectiveness.(3) Severe damage: the target is seriously damaged and needs to be returned to the factory for overhaul, and the repair takes a long time, and the combat effectiveness is lost by 50% to 80%.(4) Destroy the annihilation: the target is completely destroyed and cannot be repaired, and the combat effectiveness loss is more than 80%.This research is based on the normal distribution phenomenon of ammunition consumption when the point target reaches each damage degree. Suppose that the overall ammunition consumptionqi under the damage degree i of the point target obeys the normal distribution with the parameter θi,σi2 [25]. Thus, the probability density function of ammunition consumption at damage degree i can be obtained:(1)fqi,θi,σi2=12πσie−qi−θi2/2σi2;i=0,1,2,3,4.In the formula:θi and σi2, respectively, represent the mean and variance of ammunition consumption under damage degree i. Determining the value of θi and σi2 will determine the probability distribution of ammunition consumption under point target damage degree i. How to use Bayesian theory to fuse prior information, overall information and sample information to estimate these two parameters is the focus of this article.


## 2.2. Sample Information of Ammunition Consumption Parameters

Under normal circumstances, in order to determine the amount of ammunition required to strike a target to different degrees of damage, it is necessary to conduct an outfield shooting test. According to the classical statistical theory, the sample information of ammunition consumption is used as the source of parameter estimation, the ammunition quantityqij, i=0,1,2,3,4.j=1,2,…n required by n point targets when they hit the damage level i was counted respectively. Then the estimated value of the normal distribution parameter θi,σi2 can be expressed by the mean q¯i and variance Si2 of the field sample:(2)θ^i=q¯i=1n∑j=1nqij,σ^i2=Si2=1n−1∑j=1nqij−q¯i2,where θ^i and σ^i2 are the unbiased estimates of θi and σi2, respectively.In order to measure the degree of error of the estimated valuesθ^i and σ^i2, the mean square error is used to express(3)MSEθ^i=Eθ^i−θi2=varθ^i,MSEσ^i2=Eσ^i2−σi22=varσ^i2.Because(4)θ^i∼Nθi,σi2n,n−1σ^i2σi2∼χ2n−1,then the mean square error values of θ^i and σ^i2 can be calculated:(5)MSEθ^i=σi2n,MSEσ^i2=2σi4n−1.It can be seen from the mean square error value formula that when the number of point targets in the shooting test is small, the error of the estimated value of the ammunition consumption measurement parameter is larger. Normally, for live-fire strike tests under field conditions, due to the limitations of test costs and test environment, the number of test samples is small, and the error of parameter estimates obtained by classical statistical methods that only rely on sample information is large.


## 2.3. Prior Distribution of Ammunition Consumption Parameters

In order to overcome the problem of insufficient number of test samples under external field conditions, the idea of Bayesian statistics can be considered for research, using Bayesian theory to introduce prior information and increase the information amount of parameter statistics. The Bayesian statistical method regards the normal distribution parameterθi,σi2 of ammunition consumption as a random variable, which is different from the concept that parameters are fixed values in classical statistics. Then mark θi,σi2∼πθi,σi2 as the joint prior distribution of parameter θi,σi2. Considering the mutual influence between θi and σi2, the joint prior distribution is the product of the two distributions. The joint prior distribution of θi,σi2 can be expressed as(6)πθi,σi2=πθi|σi2πσi2,where πσi2 is the prior distribution of σi2 and πθi|σi2 is the prior distribution of θi under condition σi2.There are many methods to determine the prior distribution in Bayesian theory, including the conjugate prior distribution method, the maximum entropy method, and the noninformation prior distribution. In order to facilitate the solution and take into account the available prior information, the conjugate prior distribution method is selected to determine the prior distribution of the hit accuracy parameter. By consulting the family of conjugate prior distributions, it can be known that the conjugate prior distribution under the condition that the characteristic parametersθi and σi2 of the normal distribution are both unknown is the normal-inverse gamma distribution [26–28]. Therefore, the normal distribution is selected as the prior distribution of θi, and the inverse gamma distribution is selected as the prior distribution of σi2, namely:(7)θi|σi2∼Nθ0i,σi2k0i.(8)σi2∼IGav0i2,v0iσ0i22.The normal-inverse gamma distribution is the product of formula (7) and formula (8):(9)πθi,σi2=πθi|σi2πσi2=Nθ0i,σi2k0iIGav0i2,v0iσ0i22,where θ0i, k0i, v0i, and σ0i2 are hyperparameters in the prior distribution2.4. The Posterior Distribution of Ammunition Consumption ParametersThe posterior distribution of ammunition consumption can be obtained by integrating the overall information, sample information and prior information of ammunition consumption by Bayesian formula. Compared with the classical statistical method based only on sample information, the posterior distribution obtained by Bayesian formula contains more information and is more suitable for parameter estimation problems with insufficient sample size. According to Bayes’ theorem, the posterior density function of the ammunition consumption parameterθi,σi2 is as follows:(10)πθi,σi2|qi=Lqi|θi,σi2πθi|σi2πσi2mqi,(11)mqi=∬Lqi|θi,σi2πθi|σi2πσi2dθidσi2,where Lqi|θi,σi2 is the likelihood function of the ammunition consumption parameter field sample, mqi is the marginal distribution or unconditional distribution of field sample and qi=qi1,qi2,…,qin is the i th field sample. It can be seen from formula (10) that there is a scale factor mqi difference between the posterior distribution of parameter on the left side of the equal sign and the product of the sample likelihood function and the parameter prior distribution on the numerator of the right side of the equal sign. Since mqi has nothing to do with the parameters θi,σi2, it does not contain any information about θi,σi2, when looking for the form of the posterior distribution of the parameters, mqi can be ignored. Then the posterior distribution can be expressed as(12)πθi,σi2|qi∝σi2−vni/2+3/2e−vniσni2+kniθi−θni2/2σi2,where(13)kni=k0i+n,vni=v0i+n,θni=k0ik0i+nθ0i+nk0i+nq¯i,vniσni2=v0iσ0i2+n−1Si2+k0ink0i+nθ0i−q¯i2,where Si2 is the variance of the field sample.The following studies how to use prior information such as historical data and simulation data to determine the prior distribution hyperparameters, and to determine the posterior distribution of the parameters according to the Bayesian statistical formula to provide a basis for the estimation of ammunition consumption under different damage degrees.


## 3. Determination of Hyperparameters of the Prior Distribution of Ammunition Consumption

From the previous derivation, it can be known that the ammunition consumption parameterθi,σi2 under the point target damage degree i obeys the normal-inverse gamma distribution, and the posterior distribution under the on-site sample condition is deduced. Before determining the posterior distribution, it is necessary to process the prior information that has been mastered, and to determine the value of the hyperparameter in the prior distribution.


### 3.1. Priori Information Processing of Ammunition Consumption

A certain amount of prior information such as simulation information, historical information, expert information, etc. will be collected before the test of new ammunition hitting point targets in the field conditions. As important information of ammunition consumption, these prior information can provide help for parameter estimation with a small number of samples. In the past, all prior information was directly used as prior samples for parameter statistics, and then the parameter prior distribution was obtained. The premise of this was that all prior information samples came from the same distribution. Since simulation information, historical information, expert information, etc. Are affected by random factors, the test environment and test conditions considered will be different. As a result, the probability of these information satisfying the same distribution becomes smaller. To solve this problem, a new processing method is needed to obtain the prior distribution. Bayesian theory treats parameters as random variables, and the prior distribution refers to the probability distribution of the parameters. The prior distribution in this study is aimed at the joint distribution of the ammunition consumption distribution parameterθi,σi2. Normally, the collected prior information is the ammunition consumption qi under point target damage degree i, which is not a direct sample of the parameter θi,σi2. However, these information can be transformed into the direct sample information of the parameter θi,σi2 through secondary processing, and the prior distribution of the parameter can be obtained finally.The collected simulation information, historical information, expert information, etc. Are processed in groups, forming a total of 30 sets of sample data. Calculate the mean and variance of ammunition consumptionqi under each group of point target damage degree i respectively. According to the judgment of the damage degree of the point target, the damage degree is divided into 4°, and the parameter samples forming the prior information are shown in Table 2.Table 2
Parameter sample data formed by prior information.
mMild damageModerate damageSevere damageDestroy the annihilationθ1σ12θ2σ22θ3σ32θ4σ4215.4761.86417.8942.54136.2944.01039.7294.12725.6350.61516.3944.94335.7815.38341.6915.47636.5750.60916.7930.75235.7674.52540.2832.19944.7992.29415.0381.04435.0420.93140.8381.78857.2011.96015.4631.15636.7683.23141.7972.60765.8540.70815.6041.58335.6733.02042.3224.96575.2840.52116.5912.49335.1210.77339.9931.89785.0100.29915.6001.26437.0842.61042.5623.90596.8250.56816.3052.03635.5861.12041.9134.775105.4850.47414.3231.49735.2402.28041.1271.574115.3850.57315.5452.22035.5213.29640.9892.308126.1980.44115.2070.77634.9942.17340.9650.897134.9390.59114.6650.80034.7362.27240.8731.619145.5930.75116.1661.92638.2001.93641.2252.916155.4281.01216.0031.18537.3734.20041.2551.851165.4681.00315.8242.48136.0921.66742.0923.171175.4860.48914.8400.95934.6062.13042.8491.670185.2910.67816.6121.61834.9784.56741.7041.620194.6050.53816.0521.66035.6323.91640.9746.849204.8750.31215.5850.66336.5523.72041.8751.519215.4100.75915.8167.04134.5351.68741.3981.077224.9220.20215.6111.10733.5873.42940.0883.874235.9320.91014.5400.33834.4231.63742.2251.864243.8270.78715.5941.54736.1174.46541.5322.114255.7320.74115.2013.65336.1721.21841.3461.334266.2910.83615.0950.85636.2291.92141.7562.383275.1100.46814.9670.88335.6761.87141.4823.165285.5650.49115.4161.92935.9752.27040.1831.764296.3120.73114.3581.09435.3480.96441.0252.349305.3320.58416.4941.11236.6192.24141.0424.951According to the data in Table2, draw the frequency distribution histogram of the parameter sample, as shown in Figure 1. The mean and variance of the ammunition consumption parameter θi,σi2 are calculated, as shown in Table 3.Figure 1
Frequency histogram of prior sample parameters.Table 3
The mean and variance of prior sample parameters.
Statisticsθ1σ12θ2σ22θ3σ32θ4σ42Mean5.6160.76015.801.77235.8732.64941.2382.754Variance0.4230.2270.7261.8930.9501.5801.0832.181


### 3.2. Determination of Hyperparameters of Prior Distribution of Ammunition Consumption

In Bayesian theory, the unknown parameters in the prior distribution are called hyperparameters, and the prior distribution of parameters can be determined if the value of the hyperparameters is determined. The following study is on how to use the prior information to determine the hyperparameters.According to the parameter sample information processed in Table2 and Table 3, and through its frequency distribution histogram, it can be found that the distributions of parameters θ1, θ2, θ3, θ4 are approximately normal distributions, and the distributions of parameters σ12, σ22, σ32, σ42 are approximately inverse gamma distributions, this also conforms to the inference of prior distribution types of θi and σi2 in the above equation. Estimate the hyperparameters according to the respective distribution types of θi and σi2. Among them, the maximum likelihood estimation and the moment estimation of the normal distribution are coincident, while the maximum likelihood estimation and the moment estimation of the inverse gamma distribution are not coincident. When the number of samples is small, the maximum likelihood estimation of the inverse gamma distribution is better than the moment estimation.According to the preceding formula, we can see thatσi2∼IGav0i/2,v0iσ0i2/2. Let α^i be the estimator of v0i/2, and β^i be the estimator of v0iσ0i2/2. According to the parameter sample data in Table 2 to solve the maximum likelihood estimation of hyperparameters α^i and β^i, the solving steps are as follows.(1)
Establish the likelihood function of the inverse gamma distribution:(14)Lαi,βi=βiαimΓmαi∏j=1mσij2−αi+1·exp−∑j=1mβiσij2.(2)
Take logarithm of the likelihood functionLαi,βi:(15)lnLαi,βi=αimlnβi−mlnΓαi−αi+1ln∏j=1mσij2−∑j=1mβiσij2.(3)
Take the first-order partial derivative of the functionlnLαi,βi:(16)∂lnLαi,βi∂αi=mlnβi−mΓαi′Γαi−∑j=1mlnσij2,∂lnLαi,βi∂βi=αimβi−∑j=1m1σij2.(4)
Let the first-order partial derivative be 0 and establish the log-likelihood equation:(17)mlnβi−mΓαi′Γαi−∑j=1mlnσij2=0,αimβi−∑j=1m1σij2=0.(5)
Simultaneous equations, substituting the prior parameter sample data to obtain the maximum likelihood estimate of parameterαi,βi:(18)lnα^iMLE−Γα^iMLEΓα^iMLE=∑j=1mlnσij2m+ln∑j=1m1σij2−lnm,β^iMLE=mα^iMLE∑j=1m1/σij2.In the formula,σij2 is the variance value of ammunition consumption at damage degree i corresponding to the j th group of parameter sample data in the prior information; m is the number of prior samples; α^iMLE is the maximum likelihood estimator of αi; β^iMLE is the maximum likelihood estimator of βi, Γαi is the gamma function, and(19)∂lnΓαi∂αi=Γαi′Γαi=∫0+∞e−ttαi−1lntdt∫0+∞e−ttαi−1dt.Take the mean value of the parameter sampleσij2 as the estimated value of σi2:(20)σ^i2=∑j=1mσij2m.Andθi|σi2∼Nθ0i,σi2/k0i can be known from the above formula, and then the estimated value of parameter θ0i, σi2/k0i can be obtained according to the data in Table 2.(21)θ^0i=∑j=1mθijm,(22)σ^i2k^0i=∑j=1mθij−∑j=1mθij/m2m.Simultaneous formula (21) and formula (22) can find the value of parameter k0i.(23)k^0i=σ^i2σ^i2/k^0i=∑j=1mσij2∑j=1mθij−∑j=1mθij/m2.From the above derivation, the estimated value of the hyperparameter in the prior distribution corresponding to the ammunition consumption under different damage degrees can be obtained. Kolmogorov Smirnov (KS) method was used to conduct goodness of fit test, and the test degree was 0.05. If the distribution of the corresponding parameters are consistent with the sample,p is required to be greater than 0.05, otherwise it indicates that the parameters are inconsistent with the sample. The test results are shown in Table 4. It can be seen that all p-values are greater than 0.05, which proves that it is reasonable to adopt the parameter estimation value calculated above as the assumption of parameter prior distribution.Table 4
Hyperparameters of prior distribution.
ParameterIGaαi,βiNθ0i,σi2/k0iα^iβ^iKS inspectionθ^0iσ^i2/k^0iσ^i2k^0iKS inspectionStatisticp valueStatisticp valueπθ1,σ122.5631.4660.2240.2095.6160.4230.7601.7970.1120.370πθ2,σ222.8723.3530.1780.37415.8000.7261.7722.4410.1370.244πθ3,σ323.1065.4760.2110.40335.8730.9502.6492.7880.1620.109πθ4,σ423.2566.9370.1270.16941.2381.0832.7542.5430.2780.347


## 3.1. Priori Information Processing of Ammunition Consumption

A certain amount of prior information such as simulation information, historical information, expert information, etc. will be collected before the test of new ammunition hitting point targets in the field conditions. As important information of ammunition consumption, these prior information can provide help for parameter estimation with a small number of samples. In the past, all prior information was directly used as prior samples for parameter statistics, and then the parameter prior distribution was obtained. The premise of this was that all prior information samples came from the same distribution. Since simulation information, historical information, expert information, etc. Are affected by random factors, the test environment and test conditions considered will be different. As a result, the probability of these information satisfying the same distribution becomes smaller. To solve this problem, a new processing method is needed to obtain the prior distribution. Bayesian theory treats parameters as random variables, and the prior distribution refers to the probability distribution of the parameters. The prior distribution in this study is aimed at the joint distribution of the ammunition consumption distribution parameterθi,σi2. Normally, the collected prior information is the ammunition consumption qi under point target damage degree i, which is not a direct sample of the parameter θi,σi2. However, these information can be transformed into the direct sample information of the parameter θi,σi2 through secondary processing, and the prior distribution of the parameter can be obtained finally.The collected simulation information, historical information, expert information, etc. Are processed in groups, forming a total of 30 sets of sample data. Calculate the mean and variance of ammunition consumptionqi under each group of point target damage degree i respectively. According to the judgment of the damage degree of the point target, the damage degree is divided into 4°, and the parameter samples forming the prior information are shown in Table 2.Table 2
Parameter sample data formed by prior information.
mMild damageModerate damageSevere damageDestroy the annihilationθ1σ12θ2σ22θ3σ32θ4σ4215.4761.86417.8942.54136.2944.01039.7294.12725.6350.61516.3944.94335.7815.38341.6915.47636.5750.60916.7930.75235.7674.52540.2832.19944.7992.29415.0381.04435.0420.93140.8381.78857.2011.96015.4631.15636.7683.23141.7972.60765.8540.70815.6041.58335.6733.02042.3224.96575.2840.52116.5912.49335.1210.77339.9931.89785.0100.29915.6001.26437.0842.61042.5623.90596.8250.56816.3052.03635.5861.12041.9134.775105.4850.47414.3231.49735.2402.28041.1271.574115.3850.57315.5452.22035.5213.29640.9892.308126.1980.44115.2070.77634.9942.17340.9650.897134.9390.59114.6650.80034.7362.27240.8731.619145.5930.75116.1661.92638.2001.93641.2252.916155.4281.01216.0031.18537.3734.20041.2551.851165.4681.00315.8242.48136.0921.66742.0923.171175.4860.48914.8400.95934.6062.13042.8491.670185.2910.67816.6121.61834.9784.56741.7041.620194.6050.53816.0521.66035.6323.91640.9746.849204.8750.31215.5850.66336.5523.72041.8751.519215.4100.75915.8167.04134.5351.68741.3981.077224.9220.20215.6111.10733.5873.42940.0883.874235.9320.91014.5400.33834.4231.63742.2251.864243.8270.78715.5941.54736.1174.46541.5322.114255.7320.74115.2013.65336.1721.21841.3461.334266.2910.83615.0950.85636.2291.92141.7562.383275.1100.46814.9670.88335.6761.87141.4823.165285.5650.49115.4161.92935.9752.27040.1831.764296.3120.73114.3581.09435.3480.96441.0252.349305.3320.58416.4941.11236.6192.24141.0424.951According to the data in Table2, draw the frequency distribution histogram of the parameter sample, as shown in Figure 1. The mean and variance of the ammunition consumption parameter θi,σi2 are calculated, as shown in Table 3.Figure 1
Frequency histogram of prior sample parameters.Table 3
The mean and variance of prior sample parameters.
Statisticsθ1σ12θ2σ22θ3σ32θ4σ42Mean5.6160.76015.801.77235.8732.64941.2382.754Variance0.4230.2270.7261.8930.9501.5801.0832.181


## 3.2. Determination of Hyperparameters of Prior Distribution of Ammunition Consumption

In Bayesian theory, the unknown parameters in the prior distribution are called hyperparameters, and the prior distribution of parameters can be determined if the value of the hyperparameters is determined. The following study is on how to use the prior information to determine the hyperparameters.According to the parameter sample information processed in Table2 and Table 3, and through its frequency distribution histogram, it can be found that the distributions of parameters θ1, θ2, θ3, θ4 are approximately normal distributions, and the distributions of parameters σ12, σ22, σ32, σ42 are approximately inverse gamma distributions, this also conforms to the inference of prior distribution types of θi and σi2 in the above equation. Estimate the hyperparameters according to the respective distribution types of θi and σi2. Among them, the maximum likelihood estimation and the moment estimation of the normal distribution are coincident, while the maximum likelihood estimation and the moment estimation of the inverse gamma distribution are not coincident. When the number of samples is small, the maximum likelihood estimation of the inverse gamma distribution is better than the moment estimation.According to the preceding formula, we can see thatσi2∼IGav0i/2,v0iσ0i2/2. Let α^i be the estimator of v0i/2, and β^i be the estimator of v0iσ0i2/2. According to the parameter sample data in Table 2 to solve the maximum likelihood estimation of hyperparameters α^i and β^i, the solving steps are as follows.(1)
Establish the likelihood function of the inverse gamma distribution:(14)Lαi,βi=βiαimΓmαi∏j=1mσij2−αi+1·exp−∑j=1mβiσij2.(2)
Take logarithm of the likelihood functionLαi,βi:(15)lnLαi,βi=αimlnβi−mlnΓαi−αi+1ln∏j=1mσij2−∑j=1mβiσij2.(3)
Take the first-order partial derivative of the functionlnLαi,βi:(16)∂lnLαi,βi∂αi=mlnβi−mΓαi′Γαi−∑j=1mlnσij2,∂lnLαi,βi∂βi=αimβi−∑j=1m1σij2.(4)
Let the first-order partial derivative be 0 and establish the log-likelihood equation:(17)mlnβi−mΓαi′Γαi−∑j=1mlnσij2=0,αimβi−∑j=1m1σij2=0.(5)
Simultaneous equations, substituting the prior parameter sample data to obtain the maximum likelihood estimate of parameterαi,βi:(18)lnα^iMLE−Γα^iMLEΓα^iMLE=∑j=1mlnσij2m+ln∑j=1m1σij2−lnm,β^iMLE=mα^iMLE∑j=1m1/σij2.In the formula,σij2 is the variance value of ammunition consumption at damage degree i corresponding to the j th group of parameter sample data in the prior information; m is the number of prior samples; α^iMLE is the maximum likelihood estimator of αi; β^iMLE is the maximum likelihood estimator of βi, Γαi is the gamma function, and(19)∂lnΓαi∂αi=Γαi′Γαi=∫0+∞e−ttαi−1lntdt∫0+∞e−ttαi−1dt.Take the mean value of the parameter sampleσij2 as the estimated value of σi2:(20)σ^i2=∑j=1mσij2m.Andθi|σi2∼Nθ0i,σi2/k0i can be known from the above formula, and then the estimated value of parameter θ0i, σi2/k0i can be obtained according to the data in Table 2.(21)θ^0i=∑j=1mθijm,(22)σ^i2k^0i=∑j=1mθij−∑j=1mθij/m2m.Simultaneous formula (21) and formula (22) can find the value of parameter k0i.(23)k^0i=σ^i2σ^i2/k^0i=∑j=1mσij2∑j=1mθij−∑j=1mθij/m2.From the above derivation, the estimated value of the hyperparameter in the prior distribution corresponding to the ammunition consumption under different damage degrees can be obtained. Kolmogorov Smirnov (KS) method was used to conduct goodness of fit test, and the test degree was 0.05. If the distribution of the corresponding parameters are consistent with the sample,p is required to be greater than 0.05, otherwise it indicates that the parameters are inconsistent with the sample. The test results are shown in Table 4. It can be seen that all p-values are greater than 0.05, which proves that it is reasonable to adopt the parameter estimation value calculated above as the assumption of parameter prior distribution.Table 4
Hyperparameters of prior distribution.
ParameterIGaαi,βiNθ0i,σi2/k0iα^iβ^iKS inspectionθ^0iσ^i2/k^0iσ^i2k^0iKS inspectionStatisticp valueStatisticp valueπθ1,σ122.5631.4660.2240.2095.6160.4230.7601.7970.1120.370πθ2,σ222.8723.3530.1780.37415.8000.7261.7722.4410.1370.244πθ3,σ323.1065.4760.2110.40335.8730.9502.6492.7880.1620.109πθ4,σ423.2566.9370.1270.16941.2381.0832.7542.5430.2780.347


## 4. Maximum Posteriori Estimation of Ammunition Consumption Parameters

The posterior distribution obtained by the Bayesian formula integrates sample information, prior information, and overall information, and contains all possible information that can be used for the parameters. Therefore, parameter estimation can be carried out based on the posterior distribution of the parameters. In accordance with the principle of maximum occurrence probability of ammunition consumption parameters, the posterior mode estimate of parameterθi,σi2 is selected as the parameter estimation value of this study. Therefore, it is necessary to solve the bivariate extreme value problem of the posterior probability density function πθi,σi2|qi. Let(24)pθi,σi2=σi2−vni/2+3/2e−vniσni2+kniθi−θni2/2σi2.Then,(25)∂pθi,σi2∂θi=0⟹θi−θni=0,∂pθi,σi2∂σi2=0⟹vniσni2σi2−vni−3=0.Hence,(26)θiMD=θniσiMD2=vniσni2vni+3.From the combination of formula (13) and formula (26), we can get(27)θiMD=k0ik0i+nθ0i+nk0i+nqi¯,σiMD2=v0iσ0i2+n−1si2+k0in/k0i+nθ0i−q¯i2v0i+n+3.The above formula is the calculation formula of the posterior mode estimation of the ammunition consumption parameterθi,σi2 expressed in the normal-inverse gamma distribution under the condition of obtaining the shooting samples in the field.


## 5. Example Analysis

In order to measure the ammunition consumption of point targets under different damage degrees, six armor targets were tested with live ammunition in the field. Count the ammunition consumption of each armored target under damage degreei, and the statistical data is shown in Table 5. Using the Bayesian method proposed above to estimate the probability characteristic parameters of ammunition consumption, the calculation process is as follows:(1)
According to the distribution of ammunition consumption at different damage degrees to targets at the hitting point, it is assumed that the ammunition consumptionqi under the damage degree i follows the normal distribution qi∼Nθi,σi2. The conjugate prior distribution normal-inverse gamma distribution of parameter θi,σi2 is selected as the prior distribution πθi,σi2, and the values of the hyperparameters in the prior distribution are shown in Table 4.(2)
According to the Bayesian formula, the posterior distributionπθi,σi2|qi of the parameter θi,σi2 is obtained by integrating the sample information, prior information, and overall information. The values of the posterior distribution parameters are shown in Table 6.(3)
The posterior mode estimation of parameterθi,σi2 was obtained according to the posterior distribution of parameter θi,σi2, and the posterior mode estimation of parameter was taken as the parameter condition of the normal distribution of ammunition consumption under damage grade i.Table 5
Ammunition consumption data under different damage degrees of armor targets.
Target equipment serial numberq1q2q3q4E15143338E26143237E36133439E45153336E55163337E64153237Table 6
Bayesian research results.
Damage degreesniq¯isi2k^0ik^niv^0iv^niθ^0iθ^niv^0iσ^0i2v^niσ^ni2θiMDσiMD2i=165.160.561.797.795.1211.125.615.272.936.045.270.42i=2614.501.102.448.445.7411.7415.8014.876.7015.1314.871.02i=3632.830.562.788.786.2112.2135.8733.7910.9531.3733.792.06i=4637.331.062.548.546.5112.5141.2338.4913.8746.4438.492.99It can be seen from the results of the above analysis.The Bayesian results of this study can be obtained by solving the posterior distribution of parameters based on the above Bayesian theory, as shown in Table6. Plot the joint prior distribution and joint posterior distribution probability density function graph of parameter θi,σi2, as shown in Figure 2.(1)
Both the prior distribution and the posterior distribution obey the normal-inverse gamma distribution. From the results shown in Table6, it can be seen that the mean ammunition consumption value q¯i of on-site shooting is smaller than the priori mean θ^0i and the posterior mean θ^ni. The posterior mean lies between the sample mean and the prior mean and is more inclined to the sample mean, indicating that the posterior distribution is the modification of the prior distribution after acquiring the outfield shooting sample data.(2)
It can be seen from the expression of formulaθni that n and k0i determine the weight of sample information and prior information in the posterior distribution. The larger the number of on-site samples, the greater the weight of the sample information. And it can be seen from the formula that the value of k0i has little relation with the number of prior information samples, but is related to the variance of the prior information. When the variance of ammunition consumption under each damage degree in the prior information is larger, k0i will be larger. When the variance of the consumption mean is larger, k0i will be smaller.(3)
It can also be seen from Table6 that the mean and variance of ammunition consumption of different damage degrees are different, and the variance of consumption increases as the damage degree increases. This shows that there is a certain relationship between θi and σi2. Therefore, the two parameters should be considered together, which also proves the correctness of the research idea from the joint distribution of parameters.Figure 2
Joint prior distribution and joint posterior distribution of parameters.Draw the ammunition consumption distribution diagram of sample information, prior information, and posterior information, as shown in Figure3. From the figure, the adjustment process of the prior information to the sample information can be more intuitively shown, and finally the posterior distribution under the Bayesian solution can be obtained. Taking the use of this new type of ammunition to hit an armor target as an example, the damage requirement is severe damage, and the possibility of achieving the expected damage effect is not less than 95%. In this regard, the upper limit of the one-sided confidence interval for ammunition consumption can be solved.(28)∫0Qsfqs,θs,σs2dqs=0.95,s=1,2,3.Figure 3
The ammunition consumption distribution diagram of sample information, prior information, and posterior information.In the formula,s=1,2,3 represents the consumption information of three types of ammunition for severely damage: sample, a priori, and a posteriori respectively. The solution can be: Q1  = 34.07, Q2  = 38.55, Q3  = 36.16. It can be seen that the posterior information consumption is between the sample information consumption and the prior information consumption, and it is the fusion of the two information. We adopt the ammunition consumption of the posterior distribution of 36.16 rounds as the ammunition consumption of the target of this strike point.


## 6. Conclusion

Bayesian statistics can fuse various prior information, and has certain advantages in solving the problem of ammunition consumption estimation under small sample conditions. Therefore, based on the normal distribution of ammunition consumption under different damage degrees, this paper proposes a Bayesian inference method for ammunition consumption based on normal-inverse gamma distribution. This method regards the ammunition consumption under different damage degrees as subject to a certain probability distribution, rather than a fixed value. This is more consistent with ammunition consumption situation, and the research results are more adaptable.



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*Source: 6365712-2022-04-14.xml*