决策理论与方法2_（Decisiontheoryandmethod2_）

1、决策理论与方法 2_ (decision theory and method 2_)the second chapter is subjective probability and prior distributionsubjective, probability, and, prior, distributionthe main references in this chap ter are: 60, 52, how god throws dicethe basic concept of 2-1probability (probability)1. frequencyfn (a) =na/n

2、p (a) = fn (a). definition of classical probabilitythe definition of 2. laplace in the theoretical analysis of probability (1812)p (a) =k/nin the formula, k is the basic event number contained in a,n is the total number of basic eventsapplicable conditions 1. basic events limited2. each basic event,

3、 etc.3. axiomatic definitione is a random test, and s is the sample space of e for each event of e, a corresponds to a definite real number p (a) if it satisfies:non negative: 0 二 p (a 二 1)normative: p (s) =1the countable additivity of 22 incompatible events (k=l, 2 ak (ai) aj=.)p (ak) = sigma p (ak

4、)the p (a) is called the probability that the event a occurstwo, subjective probability (subjective, probability, likelihood)1. why introduce subjective probabilities?.some natural states cannot be repeatedwill it rain tomorrow?how's the new product going?what is the rate of national economic gr

5、owth next year?can you take a phd?.the cost of the experiment is too expensive and too costlyexample: continental missile hit ratean estimate of the enemy，s next move in a war2. subjective probability definition: a measure of rational beliefa possible measure of what happens to a particular eventthe

6、 extent to which he believed (or believed) the possibility of an event to occurthis degree of belief is a belief, subjective, but based on experience, knowledge, and knowledgeobjective analysis, reasoning, and synthetic judgment (assignment) is different from subjective conjectureexample: doctoral c

7、andidates, flip coins, flip pinsthree the mathematical definition of probabilitythe non empty elements of omega omega, omega 二omega, that is, f is a subset of a omega sigma omega epsilon f domain (i. e.;if a is a, f, f;if ai 二 f i二 1,2, . then, ai, f)if p (a) is the real valued set function set on f

8、, it satisfiesnon negative p (a) = 0normative p (omega) =1the countable additivityp (a) is a straight (principal or objective) probability measure, referred to as probabilityomega is the basic eventa for eventsthe $three population (omega, f, p) is called probability spacenote: subjective probabilit

9、y and objective probability(objective, probability) have the same definitionfour. comparison of subjective and objective probabilities(1) basic attributes:0: the inherent objective nature of a system, the limits of frequency passing under repeated tests in the same conditionss: probability is the na

10、ture of the observer rather than the system, which is the degree to which the observer is trustingin the system(two) coin toss: positive upward, with a probability of 1 / 20: as long as the coin is uniform, the throwing method is similar, and the number is enough the positive upward probability is 1

11、/2, which is simpledefinition。s: that's the definition dmer thinks the coins are homogeneous. the likelihood of the positive and the negative is the same (1)2 is a subjective quantity(three) the probability of a positive coin next time is 1 / 20: this statement is wrong. without repeated tests,

12、the probability is out of the questions: for dmer,next time a positive or negative is possible. buthe does notmean thatthe coin itself is fair,it isthere may be a bias, as far as his knowledge is concerned, there is no reason to predict that one side may appear greater than the other, but repeatedly

13、 throw itthe observation of the throw may change his belief.0: s: will the next coin be positive or negative, but i know?:either the front or the back.the 2-2 distribution (prior distribution) and setin decision analysis, the information that is collected when the state information has not been test

14、ed is called a priori information, which is determined by a priori informationthe determined probability distribution is called a prior distribution.setting a priori distribution is the need for bayesean analysisa few assumptions about the prior distribution1. connectivity (connectivity), also known

15、 as comparabilitythat is, the likelihood b of events a and likelihood can be compared:a > l, b or a, l, b or b > l a, there must be one and only one* * a > l b read as a, the likelihood of occurrence is greater than the likelihood of b occurrence,the likelihood of occurrence of a l b as a i

16、s comparable to the likelihood of b occurrence2. transitivity (transitivity)for events a, b, c, a, > l, b, b, > l, c, then a > l cthe 3. part is less than the a b: if b l a?example: setting the rate of national economic growth next year:a:8 to 11%, b:12 to 15%, c:15 to 20%if a > l, b, b,

17、 > l, c, then a > l ca:811%, d:810%, there must be d > l atwo, the setting of the prior distribution of discrete random variables1. compare the events to determine the relative likelihood rateexample l doctoral candidatese: pass e: do not takethe examif p (e)二2p (e), then p (e),二2/3, p (e)二

18、 1/3example 2. climate condition: theta 2, flood theta 3normal year, theta 1,droughtnormal and famine ratio: 3: 2(p. 1) =0.6water drought ratio 1 to 1 p (theta 2) =p (theta 3)二 0.2the method is suitable for occasions with small number of states2. beta e event occurs when the income of p (0 < p &l

19、t; 1) and e / c (p=1)adjust p, so that the decision makers feel no difference between the two so far, then: p (e)二pthree, the setting of the prior distribution of continuous rv1. histogram methodthis method applies to the case where theta is a certain interval of the real axissteps: 1. delimit the m

20、olecular interval theta idiscretizationsetting the likelihood ratio (i) for each sub region. assignmenttransform to probability density curvefor example: the growth rate of the national economy next year disadvantages: the division of sub intervals is not standard assignment is not easythe tail erro

21、r is too large 2. relative likelihood rate methodscope of application: same as 1steps: discretizationassignment: the relative odds of each interval likelihood are givenstandardization:for example: same as 1a. relative likelihood r likelihood ratio pi (a)sub intervals 8 、 9% 1010 / sigma r7899 / sigm

22、a r910, 7. 5, 7. 5 / sigma rb. decision makers give a proportional relationship betweeneach of the two state likelihoodsaij二 pi/pj (1)dueaij二 1/aji (2)aij=aikakj (3)when the (3) formula is not satisfied, the real subjective probability distribution in the mind of the decision-maker can be estimated

23、by the least square method, pi, i=l, nprogramming problemminsigma sigma (aijpj - pi)s. t. pi二 1, pi 二 0* using lagrange multiplier method to construct lagrange function二 lupper pair, i=l, 2. n calculates partial derivatives and makes them 0:l1, 2, n.and join, form n+1 order homogeneous equation grou

24、p, obtain pi, i 二 1,.n.3. interval bisection methodscope of application: can be open intervalsteps: 1. seeking the middle positionthe upper and lower four loci (quartile, fractile) were determineddue to the accumulation of errors, the maximum eight loci (eighth, fractile) were determinedexample: sal

25、es of products (expected next year)disadvantages: poor precision4. matches a given form of distribution functionthis is the most commonly used and often misused methodstep 1. select the best function that matches the prior informationsuch as normal, poisson, beta, e-cauchy distribution and so onexam

26、ple: a) occurs at constant average rate per unit time, and the number of times that the event occurs obeys the unit length of tpoisson，s distribution2-4(b) if there are many factors that influence a random variable, and the role of each factor is not significant, the variable follows a normal scorec

27、loth for example, the measurement error, the landing point, the measurement of human physiological characteristics, crop yield and so on all obey the normal distribution.(c) the probability of occurrence of the event a is p, and the n independent test appears r times the probability of a (p, r, n, b

28、)二that is, obey the two distributionparameter estimation:a. moment method: n (mu, sigma) be (alpha),beta)disadvantages: the tail is not accurate, but the impact on the moment is greatb. quantile: utilizes several quantile loci and readily available probability densitiesfunction quantile table, estim

29、ate parameters, and test5. probability disk method (dart)the fan section indicates the lottery event, which is used by western managersnote: the probability or probability distribution of states is neither and should not be set by decision analysts, but should be made by policy makers andexpert ques

30、tions provide basic information.reason：section 2-3 noninformative prior distributionwhy study information free priors?the bayesean method requires a priori distribution, and the simplicity of the bayes method makes it desirable to use it without information.two, how to set a priori distribution of no information1. position parameterrandom variables, the probability density function of x like f