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If the employee believes that continuing to work for the company is inconsistent with certain deeply-held moral convictions hiv infection rate per exposure purchase 400mg albendazole with visa, the violation of which would be a serious blow to his self-respect hiv infection diagnosis discount albendazole 400 mg otc, he should be permitted to quit and pay his employer damages for whatever loss the company suffers hiv infection rate russia buy albendazole 400 mg line. Perhaps the employee will regard even the payment of damages as a morally compromising act anti viral apps buy discount albendazole 400mg online, but he may well consider his continued employment a more personal and therefore more debasing contribution to a program he thinks immoral. To this extent, the bar against self-enslavement may be viewed as a compromise that protects employees from the special debasement sometimes associated with specific performance of an obligation, but leaves them exposed to the moral risks entailed by any contractual commitment made on the basis of incomplete information about the world. Some contracts, of course, are specifically enforceable, even though the parties have not included a provision to this effect. This may seem puzzling, however, for here, too, actual performance can be demoralizing in a way that compensation is not. Suppose, for example, that a seller of machine parts, unobtainable from any other source, discovers that his buyer intends to use them for a purpose the seller considers immoral. If the machine parts are unique, the seller may be ordered to perform despite his moral misgivings and a strong, ethically based preference for monetary compensation. Part of the explanation for this difference in treatment is that contracts of employment are more difficult to specifically enforce because they require greater cooperation by the defendant and therefore involve an element of moral hazard that contracts for the sale of goods do not. There is, however, another reason why courts are more willing to compel performance in the latter case. Long personal service under the direction of another and the payment of money are in reality poles of a continuum; one is the least, and the other the most impersonal way of meeting a contractual obligation. Many sales contracts do not require any personal cooperation or even contact between the parties and are therefore closer to the pole of maximum - 58 - impersonality represented by money damages. Since a waiver of this sort merely creates a continuing obligation on the part of the promisor to pay his prebankruptcy debts, it might not seem to pose the same threat to his integrity as other forms of self-enslavement, which require the performance of more personal services. Clearly, a large debt can be a source of regret even though its payment does not require a continuing personal relationship between the parties. One reason for giving the debtor a fresh start is to counteract the self-hatred he may feel, having mortgaged his entire future in a series of past decisions he now regrets. Whatever its macroeconomic function, the bankruptcy discharge has a moral purpose as well-to restore to the debtor some measure of confidence in his capacity to arrange his future as he wishes, free from the dead hand of the past. Without such confidence, the debtor may lose even that minimum of selfrespect that is a condition for his taking an interest in himself and his own life. To this extent, the right to a discharge in bankruptcy serves the same general goal as the right to depersonalize a contractual relationship by substituting damages for performance, and the inalienability of both can be explained on similar grounds. The rules of bankruptcy thus supplement, in a needed way, the other prohibitions against selfenslavement I have discussed. In re Baby M Supreme Court of New Jersey (1988) In this matter the Court is asked to determine the validity of a contract that purports to provide a new way of bringing children into a family. The wife is to adopt the child, and she and the natural father are to be regarded as its parents for all purposes. The contract providing for this is called a "surrogacy contract," the natural mother inappropriately called the "surrogate mother. While we recognize the depth of the yearning of infertile couples to have their own children, we find the payment of money to a "surrogate" mother illegal, perhaps criminal, and potentially degrading to women. We find no offense to our present laws where a woman voluntarily and without payment agrees to act as a "surrogate" mother, provided that she is not subject to a binding agreement to surrender her child. Moreover, our holding today does not preclude the Legislature from altering the current statutory scheme, within constitutional limits, so as to permit surrogacy contracts. Under current law, however, the surrogacy agreement before us is illegal and invalid. In February 1985, William Stern and Mary Beth Whitehead entered into a surrogacy contract. Whitehead would become pregnant, carry the child to term, bear it, deliver it to the Sterns, and thereafter do whatever was necessary to terminate her maternal rights so that Mrs. Whitehead promised to do all acts necessary to rebut the presumption of paternity under the Parentage Act. Stern was not a party to the surrogacy agreement, the contract gave her sole custody of the child in the event of Mr. Pursuant to the contract, it asked that the child be permanently placed in their custody, that Mrs. Our conclusion has two bases: direct conflict with existing statutes and conflict with the public policies of this state, as expressed in its - 60 - statutory and decisional law. The fact that the trial court remedied that aspect of the contract through the "best interests" phrase does not make the contractual provision any less offensive to the public policy of this state. The surrogacy contract guarantees permanent separation of the child from one of its natural parents. Our policy, however, has long been that to the extent possible, children should remain with and be brought up by both of their natural parents. While not so stated in the present adoption law, this purpose remains part of the public policy of this state. The impact of failure to follow that policy is nowhere better shown than in the results of this surrogacy contract. A child, instead of starting off its life with as much peace and security as possible, finds itself immediately in a tug-of-war between contending mother and father. The policies expressed in our comprehensive laws governing consent to the surrender of a child, stand in stark contrast to the surrogacy contract and what it implies. Here there is no counseling, independent or otherwise, of the natural mother, no evaluation, no warning. The only legal advice Mary Beth Whitehead received regarding the surrogacy contract was provided in connection with the contract that she previously entered into with another couple. His services consisted of spending one hour going through the contract with the Whiteheads, section by section, and answering their questions. Whitehead received no further legal advice prior to signing the contract with the Sterns. Whitehead was examined and psychologically evaluated, but if it was for her benefit, the record does not disclose that fact. The Sterns regarded the evaluation as important, particularly in connection with the question of whether she would change her mind. Yet they never asked to see it, and were content with the assumption that the Infertility Center had made an evaluation and had - 61 - concluded that there was no danger that the surrogate mother would change her mind. Although the evaluation was made, it was not put to any use, and understandably so, for the psychologist warned that Mrs. Whitehead demonstrated certain traits that might make surrender of the child difficult and that there should be further inquiry into this issue in connection with her surrogacy. Whitehead nor the Sterns were ever told of this fact, a fact that might have ended their surrogacy arrangement. Under the contract, the natural mother is irrevocably committed before she knows the strength of her bond with her child. Although the interest of the natural father and adoptive mother is certainly the predominant interest, realistically the only interest served, even they are left with less than what public policy requires. They know little about the natural mother, her genetic makeup, and her psychological and medical history. Moreover, not even a superficial attempt is made to determine their awareness of their responsibilities as parents. There is not the slightest suggestion that any inquiry will be made at any time to determine the fitness of the Sterns as custodial parents, of Mrs. Almost every evil that prompted the prohibition on the payment of money in connection with adoptions exists here. Nowhere, however, do we find any legal prohibition against surrogacy when the surrogate mother volunteers, without any payment, to act as a surrogate and is given the right to change her mind and to assert her parental rights. Moreover, the Legislature remains free to deal with this most sensitive issue as it sees fit, subject only to constitutional constraints. If the Legislature decides to address surrogacy, consideration of this case will highlight many of its potential harms. In addition to the inevitable confrontation with the ethical and moral issues involved, there is the question of the wisdom and effectiveness of regulating a matter so private, yet of such public interest. Legislative consideration of surrogacy may also provide the opportunity to begin to focus on the overall implications of the new reproductive biotechnology-in vitro - 62 - fertilization, preservation of sperm and eggs, embryo implantation and the like. The problem is how to enjoy the benefits of the technology-especially for infertile couples-while minimizing the risk of abuse. The problem can be addressed only when society decides what its values and objectives are in this troubling, yet promising, area. The judgment is affirmed in part, reversed in part, and remanded for further proceedings consistent with this opinion.
But minimizing this expression is the same as minimizing the sum of the squared residuals hiv infection cycle effective albendazole 400mg. Equivalently antiviral para que sirve generic 400 mg albendazole, we minimize over 00 and 01 the expression Note the similarity with the expression (Yi - 00 -xiOl) 2 anti viral hand foam norovirus purchase albendazole 400 mg visa, which is minimized in the earlier classical linear regression formulation what does hiv infection impairs buy albendazole 400mg on-line. Ll Bayesian Linear Regression · · Model: (a) We assume a linear relation Yi 80 + 8 1 xi + Wi. What is happening here is that the observations are too noisy and are essentially ignored, so that the estimates become the same as the prior means, which we assumed to be zero. When estimating 81, the values Yi of the observations Yi are weighted in proportion to the associated values Xi. This is intuitive: when Xi is large, the contribution of 81xi to Yi is relatively large, and therefore Yi contains useful information on 81. Conversely, if Xi is zero, the observation Yi is independent of 8 1 and can be ignored. Recall, however, that the Xi are treated as exogenous, non-random quanti ties, whereas the Yi are observed values of the random variables Yi. A A Multiple Linear Regression Our discussion of linear regression so far involved a single explanatory vari able, namely X, a special case known as simple regression. The objective was to build a model that explains the observed values Yi on the basis of the val ues Xi. For instance, suppose that our data consist of triples of the form (Xi, Yi, Zi) and that we wish to estimate the parameters j of a model of the form As an example, Yi may be the income, Xi the age, and Zi the years of education of the ith person in a random sample. We then seek to minimize the sum of the squared residuals L(Yi o - l Xi - 2 Zi) 2, i= 1 - n over all o, 1, and 2. More generally, there is no limit on the number of explanatory variables to be employed. Such a model would be appropriate if there are good reasons to expect a quadratic dependence of Yi on Xi. More generally, we may consider a model of the form m Y 00 + L Ojhj (x), j= 1 where the hj are functions that capture the general form of the anticipated de pendence of y on x. In particular, we assume that the variables x and Y obey a relation of the form Y h(x; 0), where h is a given function and 0 is a parameter to be estimated. However, fairly efficient computational methods are available for solving it in practice. Regression is used widely in many contexts, from engineering to the social sci ences. We discuss here some important issues that need to be kept in mind, and the main ways in which regression may fail to produce reliable estimates. For example, the variance of may be strongly affected by the value of (For a concrete example, suppose that is yearly income and is yearly consumption. It is natural to expect that the variance of the consumption of rich individuals is much larger than that of poorer individuals. An appropriate remedy is to consider a weighted least squares criterion of the form where the weights ;i are smaller for those i for which the variance of is large. Often a variable can explain the values of a variable y, but the effect is nonlinear. As already discussed, a regression model based on data pairs of the form may be more appropriate, with a suitably chosen function (c) Multicollinearity. Suppose that we use two explanatory variables X and z in a model that predicts another variable y. If the two variables and z bear a strong relation, the estimation procedure may be unable to distinguish reliably the relative effects of each explanatory variable. For an extreme example, suppose that the true relation is of the form = + 1 and that the relation z = always holds. Multiple regression with a large number of explanatory vari ables and a correspondingly large number of parameters to be estimated runs the danger of producing a model that fits the data well, but is oth erwise useless. A tight x has a causal on y but X i be child and Yi be t he wealth of the second child in the same family. Note these are not conditional probabilities, because the true hypothesis such as not treated as a random variable. Th e nul l of the set of aU observations into a set R and if the realized value of the For a of errors: (a) Rej ect r eg i o n this is called E a are two p ossible rejection, and happ en s with p robability o:(R) = even Ho even though Ho is I error, or a false Ho). On the other hand, before the experiment is carried out, the likelihood ratio is best viewed as a random variable, a function of the observation X, in which case it is denoted by L(X). The probability distribution of L(X) depends, of course, on which hypothesis is true. Because of this tradeoff, there is no single best way of choosing the critical value. Thus, we can find a value of for which the requirement P (L(X) >; Ho) = a is satisfied. If, however, L(X) is a discrete random variable, it may be impossible to satisfy the equality P (L(X) >; Ho) = a exactly, no matter how is chosen; cf. In such cases, there are several possibilities: (a) Strive for approximate equality. This variant (known as a "randomized likelihood ratio test") is of some theoretical interest. However, it is not sufficiently important in practice to deserve further discussion in this book. The purpose of this section is to introduce an approach to this more general class of problems. We caution, however, that a unique or uni versal methodology is not available, and that there is a significant element of judgment and art that comes into play. In all of the above casesl we are dealing with a phenomenon that involves uncertainty, presumably governed by a probabilistic model. In order to avoid obscuring the key ideas, we will mostly restrict the scope of our discussion to situations with the following characteristics. In reference to the motivating examples introduced earlier, notice that ex amples (i)-(ii) satisfy conditions (a)-(c) above. We have so far defined the shape of the rejection region R (the set of data vectors that lead to rejection of t he null hypothesis). We finally choose the critical value so t hat the probability of false rejection is equal to a given value 0: P(reject Ho: Ho) = 0, Typically. Under the null hypothesis, the random variable S is binomial with parameters n = 1000 and p = 1 / 2. Using the normal approximation to the binomial and the normal tables, we find that an appropriate choice is = 31. We can only assert that the observed value of S does not provide strong evidence against hypothesis Ho. Significance Testing Methodology A statistical test of a hypothesis "H0: = *" is to be performed, based on the observations Xl. Q of (d) Choose the critical value so that the probability of false re jection is equal (or approximately equal) to Q. Let us add some comments and interpretation for the various elements of the above methodology. In other cases, a meaningful choice of 5 involves a certain generalization of the likelihood ratio, to be touched upon later in this sec tion. Finally, in many situations, the primary consideration is whether 5 is simple enough to enable the calculations needed in step (d) of the above methodology. O f course, one wishes false rejections t o be rare, but in light of the tradeoff discussed in the context of simple binary hypotheses, a smaller value of Q makes it more difficult to reject a false hypothesis, i. Because 5 has a standard normal distribution, the value of corresponding to any particular value of u is easily found from the normal tables. In this case, the same statistic 5 can be used, but we will reject Ho if 5 >, where is chosen so that P(5 >) = u. Once more, since 5 has a standard normal distribution, the value of corresponding to any particular value of u is easily found from the normal tables. If n is relatively small, the t-tables should be used instead of the normal tables (cf. Our next example involves a composite null hypothesis Ho, i n the sense that there are multiple parameter choices that are compatible with Ho.
Wilder hiv infection and aids an overview generic albendazole 400 mg with mastercard, postmaster of Columbia antiviral lip cream albendazole 400mg discount, was commended in the Daily News antiviral for eyes 400 mg albendazole for sale, April 13 hiv infection treatment guidelines purchase albendazole 400 mg with visa, 1869, as a man "well-known" and "universally respected. Shrewsbury "an oppo- which he has kept nent of corruption," and declared that "he sustained a good reputation intact under great temptations" and that "he has exerted himself zealously and courageously to guard his people from compulsion and vengeance, and establish their claim to decency and respectability. Pike, a violent hater of Negroes, said that "all of the best speakers in the House are quite black" and added that Senator Beverly Nash 36 "has more native ability than half the white men in the Senate. There were, of course, illiterate and ignorant men among the Negro speakers, but, on the other hand, there were some of poise and eloquence, who spoke with ease and grace. These were the men and this the effort which have been endlessly blamed and reviled. There is that celebrated tirade by Pike: "The members of the Assembly issued forth from the State House. They were such a looking body of men as might pour out of a market-house or a courthouse at random in any Southern state. Every Negro type and physiognomy was here to be seen, from the genteel serving-man, to the rough-hewn customer from the rice or cotton field. There was the second-hand, black frockcoat of infirm gentility, glossy and threadbare. They have a genuine interest and a genuine earnestness in the business of the assembly which we are bound to recognize and respect. They have an earnest purpose, born of conviction that their conditions are not fully assured, which lends a sort of dignity to their t 37 proceedings. Take out the accusation of being black, which is still a crime in the United States, and there remains in such tirades as this only a protest against ignorance and poverty presuming to rule intelligence and wealth; and yet, under the circumstances, how else was the necessary economic and social revolution to be effected? The charge against the Negro legislators manifestly could not be simply the charge of being black. The answer summed up no doubt I believe that there can be deal of corruption did prevail much more than the ordinary measure was inevitable that it should be so under the circumstances; but to what degree it was so, it was very difficult 38 His conclusion is that the carpetbag rule did no permanent to tell. On the other hand, great as was the stress of wealth, the condition of the Negroes upon the former owners was infinitely worse. Outside of the three or four thousand free Negroes, he inherited no property, no tools, make a decent labor contract was about as small as could be imagined. A number worked living for the army and bought land; some earned a land furnished them. The of people best qualified to help and advise in the reconstruction when there was no legal barrier. They complained then and afterward that they were not asked to lead the Negroes; that they were not chosen to be leaders, when it was their clear duty to place themselves at the head of Negro groups and white groups and lead them aright. Nothing would have disgusted most of them more than to have a government, in which Negro slaves and Northern interlopers and poor whites participated, succeed. They had there, therefore, every motive for making progress difficult, and for using charges of failure for propaganda in the North. The wilder charges have all the stigmata of propaganda and are in of the state refused even some respects intrinsically unbelievable. It is impossible to be con- vinced that the people tution, who gave South Carolina so excellent a constiwho founded good social legislation, a new system of public schools, and who were orderly and earnest in their general demeanor, all and breaking every law of decency. Yet the accusers in the case of South Carolina reconstruction attacked everybody, and when one Reynolds runs out cases could at the same time in be stealing, carousing of accusations in attacking the character of a leading Negro statesman, he turns around and without adducing a single line of proof, calls his wife a "strumpet. As two of the younger and and more honest students of the situation frankly admit, the accusations do not sound true. If this report is to be believed in its entirety, then the people of South Carolina were the most extraordinary set of thieves in the United States; and this applied mainly to the native white South Carolinians, belonging both to the old aristocracy and the poor whites; next to the carpetbaggers, necessarily limited in numbers, but large in influence; and least to the Negroes to the Negroes in small measure as actual recipients of money, but in larger responsibility as dupes and victims of their white leaders. This is not only a falsification of history; it is not even a fair interinterpretation that has identify the scalawags the pretation of the Fraud Report. Sir George Campbell said: "In South Carolina I was given the report of the committee of Investigation disclosing terrible things, and said to be most impartial and conclusive. It was the bounden duty of this legislature to prove that their action was justified. No considerations of human life, character or desert, had deterred them from this bloody revolution, and it is not conceivable that any considerations of exact truth or fidelity to fact would deter them from defending it to such an extent that the Federal government should not interfere. The men who made the report had in their hands all of the governmental records and documents to use or suppress as they wished. They could call as witnesses persons upon whom they were able to put the severof South Carolina, just after their party, by force est pressure. Yet their testimony was given full credence, with the curious assumption that such thieves would not lie, when it was to their distinct advantage to deceive. The report piled charge upon charge; It it grouped together sworn as facts the testimony, gossip and suspicion. It accepted as proof of artiand supplies furnished, the lists and statements of those who sold them, and who profited by the sale and bribed the purchasers. This committee, as a matter of fact, constituted itself judge and jury in an indictment which nobody since has had opportunity to scrutinize and criticize carefully. No court in Christendom would, without furcles men who ther data, receive the fraud report of South Carolina as the exact truth. There was nothing in their general conduct during this time to leave any doubt that men would go to any limit of deception in order to prove that Negroes were not fit to vote and that all Northern men in the state were thieves. The whole story of this era has not been revealed nor studied with impartial and scientific accuracy. In South Carolina, the charges of stealing were primarily sixty thousand dollars in bribery to pass the phosphate bill; forty thousand dolat this late lars to elect John J. Patterson to the United States Senate; $200,000 in four years for furnishing the capital; $200,000 as appropriations for state printing; large sums for supplies; the issue of fraudulent and excessive pay certificates to members of the legislature; the increase of needless clerks; a saloon in the State House and fraud in the sale of land to the state. In none of these charges do colored men appear as principals accused except, possibly, in the case of Jones, a member of the printing ring, upon whose own testimony some of the charges are based. In the case of the phosphate bill, there was, doubtless, general bribery of both colored and white members of the legislature, but it was to establish an industry which the state sorely needed, and which it seemed able to get only by granting a monopoly to Southern white men. In the case of the Patterson election, the graft was dispensed by a white man in order to defeat his colored opponent, Eliott, who refused a $10,000 bribe to withdraw. White Northerners who owned the two leading dailies got contracts for the public printing, but, later, clerks of the two Houses, one of whom was colored, got in on this graft and shared at least a part of it. In the case of the land commission, an excellent and needed movement to furnish small farmers land at reasonable prices was turned into a theft by which white land-holders were the chief gainers. The lands I have purchased have been paid for, through orders of the 40 State treasurer, approved by the chairman of the advisory board. The greater portion of the farms bought were already occupied, and numbers of thrifty and industrious farmers, white and black, were eagerly securing homes. It was said that the legislative sessions were unduly prolonged; that unnecessary clerks were employed; that a liquor saloon was maintained, and that under the head of supplies, all sorts of personal things were furnished individual members of the legislature, and charged to the state. But it is not usually added that merchants got the contracts some Northern, some Southern. They furnished committees and members of the legislature in order to secure for themselves the right to charge taxpayers outrageous prices for shoddy materials. They were doing no more in this case than business men of New York and Philadelphia; but, also, it is perfectly clear, they were doing no less. The state got a capitol decked out in the flamboyant taste of the day, but we must not forget that for the first time in their drab life, representatives of black and white for these furnishings, the money to bribe labor, toiling in the fields and swamps and living in the unpaved slums of the towns, saw something that meant to them beauty and luxury saw it and touched it, and owned it. And somehow, I have more respect for the golden spittoons of freed Negro lawmakers in 1872, than for the chaste elegance of the colonial mansions of slave- - drivers in i860. Graft and bribery spread in the state, but the "worst feature of corruption in South Carolina is that members of both parties and men of all classes are involved in which is it, and that public abhorrence of corruption, the safeguard of popular government, seems wanting or dormant. Smith of Spartanburg County, an old citizen and Demomember of the legislature, testified that he could see no wrong and compared the transaction to the purchase of a mule. Warley of Darlington County, an old citizen of high standing, spoke as follows: "As I said on yesterday, public frauds would not exist were it not for private individuals who act the part of corruptors. Were none of these engaged in bribing members of the legislature, we would hear nothing of such frauds, as the one I have endeavored to expose. President, one prominent feature in this transaction is the part which native Carolinians have played in it; and it is to this feature that I ask to be allowed to address myself in closing. I say, sir, and I say it in sorrow, that some of our own household, men whom the state in the past has delighted to honor, but whose honors have been withered by the atmosphere of corruption that they breathe, are involved in this swindle. To give them the legislature, "A slaves, has been - privilege, by law, of plundering the property holders of the state, now almost bankrupt labor. Probably one cause contributing to produce this result is the condition of civil war which has prevailed in the state, in which the power has been almost exclusively in the hands of one class, and the property in the hands of the other.
This implies that R of the P(Yn = j I Yo = i) > 0 antivirus software for mac discount albendazole 400 mg with amex, Problems 395 for every n 11 hiv infection kissing purchase 400 mg albendazole with visa, every i early stage hiv infection symptoms generic albendazole 400mg amex, and every recurrent class hiv infection rate chart buy albendazole 400 mg overnight delivery, which is aperiodic. The n-step transition probabilities of the process Yn are of the form rij (ln), and therefore also converge to the same limits 7rj. This establishes that the 7rj are the steady-state probabilities of the process Yn. Thus, the state in the new chain can be associated with the last transition in the original chain. Since the Markov chain Xn has a single recurrent class which is aperiodic, P (Xn- 1 = i) converges to the steady-state probability 7ri, for every i. It follows that p ((Xn - 1, Xn) = (i, j)) converges to 7riPij, which is therefore the steady-state probability of (i, j). There are m classes offered by a particular department, and each year, the students rank each class from 1 to m, in order of difficulty, with rank m being the highest. In fact, any given class is equally likely to receive any given rank on a given year (two classes may not receive the same rank). A certain professor chooses to remember only the highest ranking his class has ever gotten. Show that the prob abilities a i of eventually reaching s starting from a state i are the unique solution to the equations as = 1, ai = 0, = ai L Pij aj, j= l m for all absorbing i t= s, for all transient i. Hint: If there are two solutions, find a system of equations that is satisfied by their Solution. The fact that the a t satisfy these equations was established in the text, using the total probability theorem. Denoting by A the set of absorbing states and using the fact t5j = 0 for all j E A, we have t5i = L Pijt5j L Pijt5j, j= l = m for all transient i. Consider a Markov chain that has more that one recurrent class, as well as some transient states. The transient states correspond to the transient states of the original, while the absorbing states correspond to the recurrent classes of the original. The transition probabilities Pij of the new chain are as follows: if i and j are transient, Pij = Pij; if i is a transient state and k corresponds to a recurrent class, Pik is the sum of the transition probabilities from i to states in the recurrent class in the original Markov chain. The desired probabilities a i (k) are the absorption probabilities in the new Markov chain and are given by the corresponding formulas: ai (k) = Pik + j: t ransient L Pijaj (k), for all transient i. If i and j are recurrent but belong to the same class, Tij (n) converges to the steady-state probability of j in a chain consisting of only this particular recurrent class. Finally, if i is transient and j is recurrent, then Tij (n) converges to the product of two probabilities: (1) the probability that starting from i we will reach a state in the recurrent class of j, and (2) the steady-state probability of j conditioned on the initial state being in the class of j. Show that the system of equations ts = 0, m ti = 1 + L Pijtj, j=l for all i f: s, satisfied by the mean first passage times, has a unique solution. Hint: If there are two solutions, find a system of equations that is satisfied by their difference, and look for its solutions. Then we have for all i =I- s ti = 1 + L Pij tj, j#s 8i = Ii = 1 + L PijIj, j#s and by subtraction, we obtain L Pij8j, j# s where 8i = Ii - ti. By applying 8i = m successive times this relation, if follows that jm # s iI # s L PiiI L PiI h · · · L Pjm - I im. This is because starting from any state there is positive probability that s is reached in m steps. Consider a Markov chain with a single recurrent class, and let s be a fixed recurrent state. For any state i, let Pt = E [N umber of visits to i between two successive visits to s], ps = where by convention, 1. The main idea is to break the process into "cycles," with a new cycle starting each time that the recurrent state is visited. The steady-state probability of can be interpreted as the long-term expected frequency of visits to state which is inversely proportional to the average time between consecutive visits (the mean recurrence time); cf. Furthermore, if a state i is expected to be visited, say, twice as often as some other state j during a typical cycle, it is plausible that the long-term expected frequency 7ri of state i will be twice as large as 7rj. Thus, the steady-state probabilities 7ri should be proportional to the expected number of visits Pi during a cycle; cf. Note: this problem not only provides an alternative proof of the existence and unique s s, s Xn, initialized with Xo s. Then, the number of visits to state i before the next visit to state is equal to 2::= 1 In. Then, P(T -+ Since the mean recurrence time: t = t) is finite, the limit, as k 00 of t = t) is equal to zero. If we start at a transient state i, the probability of being at that state at time n diminishes to as n - 00. Consider a finite-state Markov chain in which all states belong to a single recurrent class which is aperiodic. Thus, the random variables Yk+ l - Yk, for k 2:: 1, correspond to the time between successive visits to state s. Because of the Markov property (the past is independent of the future, given the present), the process "starts fresh" at each revisit to state s and, therefore, the random variables Yk+ l - Yk are independent and identically distributed, with mean equal to the mean recurrence time t;. Problems 403 n between the time of the kth visit to state s and the time just before the next visit to that state: For every n in this range, we have Vn Yk and Vn be the values of the random variables Yk and Vn, respectively. Furthermore, let us assume that the sequence Yk / k converges to t;; according to the result of part (a), the set of trajectories with this property has probability 1. This implies the intuitive result that Vn/n converges to 1rs, with probability 1. Note: It is tempting to try to establish the convergence of Vn/n to 1r1J by combining the facts that Vn/n converges [part (b)] together with the fact that E[Vnl/n converges to 1rs (cf. This is because it is generally possible for a sequence Yn of random variables to converge with probability 1 to a constant, while the expected val ues converge to a different constant. Therefore, the sequence vn/n is between two sequences both of which converge to l/t;, which implies that the sequence vn/n converges to l/t; as well. Since this happens for every trajectory in a set of trajectories that has probability equal to 1, we conclude that Vn/n converges to l/t;, with probability 1. Empty taxis pass by a street corner at a Poisson rate of two per minute and pick up a passenger if one is waiting there. Passengers arrive at the street corner at a Poisson rate of one per minute and wait for a taxi only if there are less than four persons waiting; otherwise they leave and never return. Each user al ternates between "thinking" intervals whose durations are independent exponentially distributed with parameter A, and an "active" mode that starts by submitting a ser vice request. The server can only serve one request at a time, and will serve a request completely before serving other requests. The service times of different requests are independent exponentially distributed random variables with parameter J. Construct a Markov chain model and derive the steady-state distribution of the number of pending requests, including the one presently served, if any. The sequence is a sequence of independent exponentially distributed time intervals with param n eter v. Therefore they can be associated with the arrival times of a Poisson process with rate v. These are the balance equations for the embedded Markov chain, which have a unique solution since the embedded Markov chain has a single recurrent class, which is aperi odic. In this chapter and the next one we aim to: (a) Develop an appreciation of the main two approaches (Bayesian and classi cal), their differences, and similarities. Probability versus Statistics Statistical inference differs from probability theory in some fundamental ways. Probability is a self-contained mathematical subject, based on the axioms in troduced in Chapter 1. In probabilistic reasoning, we assume a fully specified probabilistic model that obeys these axioms. We then use mathematical meth ods to quantify the consequences of this model or answer various questions of interest. In particular, every unambiguous question has a unique correct answer, even if this answer is sometimes hard to find. The model is taken for granted and, in principle, it need not bear any resemblance to reality (although for the model to be useful, this would better be the case). For any particular problem, there may be several reasonable methods, yielding different answers. In general, there is no principled way for selecting the "best" method, unless one makes several assumptions and imposes additional constraints on the inference problem.
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