Vacation I’m away until the 6th, so except for tomorrow and Tuesday and except possibly for a Stream interlude, favorite and classic posts will be run. The comment filter has been set to Unreasonably High. We are all ladies and gentlemen here.
This post originally ran 3 November 2014.
Reader Professor Doctor Moritz Heene writes:
I read your post on CIs with great interest, especially this one: http://wmbriggs.com/blog/?p=11862, see “Thinner is better”: “Frequentists prefer thinner, which is to say, narrower intervals over wide, assuming that, ceteris paribus, narrow intervals are more precise. For example, larger samples result in narrower intervals than small samples. But since all you can say is your interval either contains the true value or it doesn’t, its width does not matter. The temptation to interpret the width of an interval in the Bayesian fashion is so overwhelming that I have never seen it passed up.”
However, a colleague, with whom I discussed this issue sent me the following lines and I wonder what you think of it. I think he made a reasonable point: “For me a confidence interval is a summary of the effects I would [have] rejected if submitted to a hypothesis test (and we don’t need to think discretely here, we can think of the p-value as the continuous measure that it is of inconsistency of data with null, so I have stronger evidence against effects closer to the end of the confidence interval).
So a tight confidence interval is one that rejects many effects I may find interesting to know are rejected. A wide confidence interval is one that does not reject many effects I may find interesting.”
Regards,
Moritz
Your colleague didn’t pass up the Bayesian interpretation, either. He can’t really be blamed. The official frequentist meaning is too perplexing to keep in mind, its consequences intolerable, that relief is sought.
To repeat the official definition. Observe data, posit a parameterized probability model to “explain” that data, and construct a confidence interval (for one of these parameters). Now repeat the “experiment” such that the repetition is exactly the same as the first run, except that it is “randomly” different. Reconstruct the confidence interval. Repeat the “experiment” a third time, giving a third confidence interval. Repeat again. Then again. Then again forever.
When you reach forever, 95% of your collection of confidence intervals will overlap the “true” value of the parameter.
But what, you ask, about the confidence interval you have in hand? What does it mean? Well, it means just what I said, and nothing more. The only thing you can say about the confidence interval before you—regardless of its width—is that either the true value of the parameter lies within it or it doesn’t.
Suppose your interval is [a, b]. Either the true value of the parameters is in the interval or it isn’t. Introduce hypothesis testing: form a “null” which says the true value of the parameter is some number c. The frequentist then checks whether c is outside [a, b]. If so, he “rejects” the null.
Rejects is a word more apt than you think. For, as Neyman the man who invented confidence intervals tells us, rejecting a “null” is a pure act of will, just as is assigning the “null” a value of c. When the “null” is rejected, because all we know is that the true value of the parameter is in [a,b] or it isn’t, which is a tautology and true for any interval, there is no basis besides “I want”.
Your colleague says he would reject “nulls” where c is anywhere not in a to b. Well, he might. But he does so—on the official frequentist theory—with no basis. We are not entitled to say that the true value of the parameter has any probability to be c nor any probability to be in the interval [a, b]. We are not entitled to say that any finite collection of confidence intervals will be “well behaved” either. Only once we have an infinite collection are we allowed to speak—but only because we have observed everything that can ever be observed.
It is a Bayesian interpretation to say that the parameter “probably” or “likely” lies in [a, b]. It is a Bayesian interpretation that the parameter “could very well be” c. If you decide to reject the “null”, or to “fail to reject” it, with any kind of sureness or conviction or hope (the word “confidence” is lost to us here) then you have used a Bayesian interpretation.
Of course, this Bayesian interpretation is not a formal one, where the priors have been set up in the official fashion and so forth, but assigning any kind of probability, quantified or in the form of a “feeling”, to a parameter just is to be Bayesian.
If this is confusing, well, so it is. But that’s frequentism for you. A bizarre idea that you only know a probability at the end of all time.
Stream — Prediction: Supreme Court Will Punt In Fake Marriage Cake Case.
A frisson of delight ran through the Christianity community when it was announced Monday the Supreme Court will hear the case of Masterpiece Cakeshop v. Colorado Civil Rights Commission.
In 2012, Masterpiece Cakeshop owner Jack Phillips was asked to bake a fake marriage (or gmarriage) cake for two men. He refused. He told the men, “I’m sorry, but I can’t promote messages that violate my beliefs, though I’d be happy to sell you anything else.”
For his faith and via the machinations of the ACLU — Anti-Christian Liberals Union? — he was hauled in front of a tribunal and told he must bake the cake.
According to the Alliance Defending Freedom, the judge ordering the forced fondant “held that coercing Jack to participate in a same-sex ceremony did not ‘unduly abridge [his] right to the free exercise of religion.'”
It was a long legal rode from there to the Supreme Court, which said it will decide whether Phillips discriminated or not.
This is being interpreted as a kind of small victory because, of course, there is the chance the Court sees reason and does the right thing.
Don’t count on it. My prediction is the Court will punt.
The Court punts
When the two same-sex attracted men came to Phillips and demanded a cake, fake marriage was not legal in Colorado, and of course not yet in the United States as a whole. The homosexuals were going to go through a fake marriage ceremony in Massachusetts, where gmarriage was legal. And then the men were going to use the cake made by Philips in a reception to take place in Colorado.
There’s the out.
One guess is the Court will say Phillips didn’t really “discriminate” because fake marriage was not yet legal in Colorado. He was still, they might argue, within his rights to refuse to knead the dough. Why?
The cake was not legally a fake marriage cake, just an ordinary cake, and by law bakers are allowed to refuse to bake ordinary cakes.This kind of decision will not be satisfactory. What Christians — and Muslims and others of orthodox faiths — want to know is if they are allowed to discriminate now.
Discrimination is necessary
Make no mistake: Phillips did discriminate against the two men, and God bless him for it. Religion by definition is discrimination. A religion eschews one belief and accepts another. A person following his religion acts on these defined beliefs. In other words, a religious man must discriminate if he wants to remain faithful.
Phillips knew the difference between real marriage and fake. His religion does not allow him to participate actively in fake marriages. He discriminated. He should have been allowed to discriminate.
[…]
Here’s how the concept of Judicial Review came to be. There was this guy named Marbury, and he had been appointed as a Justice of the Peace in the closing hours of the Adams Presidency. But the new government of Thomas Jefferson (under Secretary of State Madison) refused to deliver the title to the appointment. So Marbury sued the Federal government. The facts were on his side, and so was the law. The law in this case was the Judiciary Act of 1789, which gave the courts the power to issue a writ of mandamus. What’s that? It’s an order from a court to an inferior government official ordering the government official to properly fulfill their official duties or correct an abuse of discretion.
Here was the magical part. By claiming that this power (granted by Congress) in the Judiciary Act of 1789 wasn’t specified in the Constitution, the Supremes (led by Chief Justice John Marshall) ruled that they didn’t have the Constitutional authority to issue that same writ of mandamus! And in the confusion that followed, several mistakes were made by everyone.
First, the Supremes said they did not have the power to order the Presidency around. And Constitutionally speaking, they were right, in a way. But since the Legislative branch had given this power to the Judiciary subsequent to the ratification of the Constitution, it makes sense that the Courts could properly order an underling of the President to obey the law passed by Congress. If the Court didn’t fulfill this duty given them by Congress, then that would vault the Presidency above Congress. And none of the Founding Deists (who were all still alive) wanted that. That would be too much like a King! After all, it wasn’t like the Courts were making new law on their own—no, they were simply staying within the literal bounds of the Constitution. As it was on Day One. And not one minute past then.
Let’s think about this. It was definitely the Supreme Court’s job to rule when a law had been broken. But if the law had indeed been broken, there had to be some mechanism to force the law-breaker into compliance. Even if that law-breaker was the President! And that mechanism was the Judiciary Act of 1789. The tool of that power was the writ of mandamus. It had been granted to the Courts, in a legally and constitutionally binding manner in the same year that the Constitution was enacted. So, if this power-grant by Congress was so damned un-constitutional, why didn’t the Supreme Court ‘review’ that same law in 1789? It’s not like they had a backlog of cases, right?
Here’s why they didn’t. It’s because they had no motive to do so. Official or otherwise. But soon enough, a motive would arise. A personal and political motive. Not a constitutional one. So let’s look closely at how this thing shook out. On the surface, we had a situation wherein the Judiciary seemed to be demurring the acceptance of additional power. Sounds humble, right? Sure. Except this demurral created a de facto Kingship in the Presidency. The President could, evidently, ignore any law Congress passed if the President felt like it, and even if the Courts ruled he did that illegally, there would be no way to force him to comply.
We now know the real outcome of this little shell game. Because the real result of this scam was not to defer to the Presidency (thus increasing its power), but rather to diminish the power of both the Legislature and the Presidency. After all, if the Courts could now say in any case before it that ‘the law itself was invalid’, then guess who just won the jackpot? No, it wasn’t the Presidency. Why not? Because this same power of nullification could (and would) be used against the Presidency. And always against the Legislature. And the States as well.
It’s simple. Let’s say Congress passes a law, and the Supremes let it stand. But when the Executive branch tries to enforce it (which after all, is its job), the Supremes rule that the Executive’s methods of enforcement are unconstitutional! Pretty slick, eh? The Supremes can either rule a law itself is unconstitutional, or the particular enforcement of it is unconstitutional. Check and mate, mate. Get the picture? No wonder they are called the Supremes.
I’m sure a lot of folks are going to start barking about my suspicions of human intentions, conflating them with conspiracy theories. They will claim (with no proof) that there was no such attempt to counteract the framers’ Original Intentions, as the strict constructionists would say. And you know what? At the purely human level, they are probably right. I don’t think there was such a (human) intention to derail the entire invention of American constitutional law and jurisprudence. But that doesn’t mean that the venal acts of a few can’t have large-scale effects that they never foresaw. And it doesn’t mean it wasn’t foreseen by somebody else. I’ve said it before, so I’ll say it again, to the consternation of those who can’t see anything they literally can’t see: there’s more to this game that meets the eye. Never forget the gods!
Yes, the gods could see the logical outcome of such a human judicial finding. A perfect opportunity to derail the train at any point in the future. Who wouldn’t go for this kind of chit in the original game of Judicial Warfare? These guys aren’t stupid, you know. But human surely are. And greedy too. And that’s how this whole thing came to pass.
What am I talking about? I’m talking about why the Human Supremes made this convoluted ruling. There probably was no vast overarching plan to overturn the Constitution. Nope. But there was a conspiracy. A rather small one. Just two people, and they didn’t even have to actively plan it. All they had to do was act in their own personal interests, which happened to coincide. It was a little bit of brotherly love, my friend.
Bear with me here. Justice Marshall was acting at this very same time as both the Chief Justice and the Secretary of State under the new Jefferson administration (as delegated by Madison, the actual Secretary of State). So it was a case of Marbury vs Marshall, if we really want to be accurate. Guess who it was that was originally tasked with the duty to deliver this contested commission of the office of Justice of the Peace to Mr. Marbury? Well, whadd’ya know, it was the brother of Chief Justice-Secretary of State Marshall, James Marshall. And guess who might be legally liable for not delivering this same said commission? Yep, little brother Jimmy.
That little personal family problem also happened to intersect with the political life of the Marshall clan. More to the point, Chief Justice Marshall was acting in political lock-step with the Jefferson administration when he decided to refuse to command his brother to deliver the commission. But he did all of this while also wearing his Constitutional robe, deciding in favor of his family and party and against Marbury and The People. Shazzam.
The rest is history. Bad history. This total inversion of the original intention of the Constitutional structure of American government set off a power struggle that continues to this day. Here’s the funny thing to me. Why hasn’t any President ever declared some ruling of the Court to be ‘unconstitutional’? And why hasn’t Congress ever done the same, using the power of appellate review I mentioned earlier? Either action would have at least as much legality behind it, or more, than the whole-cloth invention of Marbury vs Madison. Has everyone been asleep for 213 years? Am I the only one that can see this?
To be fair, historically speaking, there has been lots of action between 1804 and today. The most important came with the accession of President Lincoln to the Pantheon of Caesar. Lincoln did his part to subdue the courts by suspending Habeas Corpus, and attempting to arrest the Chief Justice. The Court seems to have gotten the message. Lincoln then set about neutering Congress by ignoring anything they said. And the stupid Southerners made the fatal mistake of allowing those silly cadets to fire the first shots at Ft Sumter. That took the leash off the dogs of war.
That left Lincoln with the only chips that really count at times like those, the Army. Which he gladly used in his mad pursuit of total power. While Julius committed the gravest sin possible against the Republic of Rome by illegally bringing armed troops across the Rubicon, inside the actual district of Rome, without the consent of the Senate, Lincoln did the converse. He illegally sent troops across the Potomac, outside the Capital district into the sovereign States without the explicit consent of those States. Game on.
A final bit of history. Nobody ever bothers to read the original Constitution. Yes, the Articles of Confederation. The first American Constitution. If they did, they would know that it explicitly stated that this union of sovereign states would be ‘a perpetual union‘. Yet this union was dissolved peaceably by these same states as they withdrew from the Articles of Confederation when they adopted the new Constitution. The new Constitution made absolutely so no such claim to indissolubility. And in fact, a majority of states ratifying this new Constitution did so with the explicit statement that they reserved their sovereign right to secede at any time in the future. After all, that’s what sovereignty is all about, right Komrade? Well, no, not according to Caesar Abraham. He wasn’t going to preside over some rump state nation. Hell no. After all, he was (or would soon be) Caesar! We all know the rest of the story. Glorious Lincoln. And every American President pays him his first homage, whether he be Republican or Democrat. Until Emperor Donald!
Anyway, once Dishonest Abe decided he was King, it was lights out for the sovereign States. Think of the Civil War as the Big C-Section that gave birth to The Empire. The result is now we’re all equal. Equally enslaved. The final bit of evil, the last thrust of the dagger into the body of the Republic, came in 1913 with the re-establishment of the Federal Reserve system (which Jackson had abolished), the imposition of the hated Income Tax (which the Supremes had ruled unconstitutional several times years before), and the final coup de grâce, the destruction of the meaning of the States themselves. Yes, the imposition of the direct election of Senators by The People (versus their appointment by the States) made the concept of State Sovereignty a moot point. The last checks on the powers of the new Federalized, unitary, centralized, and voracious State were gone. Out of the cloud of confusion surrounding this magic trick known as Marbury vs Madison emerged The Empire. Hail Caesar!
What is my point? It’s pretty simple my friend. The Constitution is dead. Has been for a long time. There’s no turning back, because all of the bridges have been burned by Caesar. We’re all prisoners now. Equally enslaved. All of this has been given to us by factional political actions. Self-interested groups that have no interest in anything beyond their own self-interest. But hey, this isn’t an exclusively American thing. No, this is how it always goes. Once you get rid of the King, this is what happens. Everybody starts jockeying for position, place, and, most of all, for power. It’s human nature. Fallen human nature. And it will never stop until the end. The end? The end of what? Time, silly. Till the final iteration of The Empire, in all its gory glory. It seems that we’re closing in on that point in time. But Hell, it could take another 100 years. Seriously. I don’t see the technological grasp matching the Imperial reach for at least another twenty-five years or so. A lot can happen in that time, including setbacks to the hubris of the gods, and men. But eventually, we (or our children) will get there. And won’t that be swell, eh?
In the meantime, what is the real point of this missive? It’s simple, Komrade. No matter how hard men try to set up some kind of alternative to the original Mosaic religious/political system that is pure and just, it is bound to fail. Why? Because of the players, silly. Forget Artificial Intelligence. We don’t even have real intelligence. All we have is our continuous rebellions against every form of government dreamed up by anyone but ourselves. It’s not hard to see why we rebel. Because each system that has replaced the original Monarchy of God is bound to fall short of perfection. And the resulting injustices that rebellion produced brings forth the call for another ‘reform’.
Well, if I’m so damned smart, what do I think we should do? After all, I’ve painted a pretty bleak picture, haven’t I? I’ve said the Republic is dead, and the rot has irreversibly set in, correct? So, what could anyone possibly do to reverse this course? Nothing, actually. As far as bringing the corpse back to life, that is. After all, we’re not dealing with a religious question here. At least, not a Christian religious question. The Messiah never said he would visit three times. Twice will be plenty, according to His word. And I take Him at His word. That’s crucial to my understanding of where we are now, and what remains to be accomplished. Because until we, as Christians, finally understand the difference between Holy Rome and Imperial Rome, we will continue to be fooled by The Lie sold to Sgt. Alvin York. The lie that says our foreign forays are the moral equivalent of the Crusades. And that we must invest our blood for the saving of Caesar. And that the fate of the entire world depends upon this secular task.
And actually, it does. But that is a bad fate, an evil one, for a bad world, an evil one. The world of Caesar. But if we want to regain our sanity, we must distinguish between the Catacombs and the Pantheon. They are not the same, and we have no means of entering the latter. Why would we want to? Do we really want to advance the cause of the gods? Because that is exactly what we are doing as we fail to see that our secular problems have no secular answers. And that the world rejects the real answers to our real problems.
Am I saying all is lost, and that there is absolutely nothing that can be done, here and now, to benefit our country and her citizens? No, I’m not saying that at all. What I am saying is this: we have to realize that the natural progression of man is in a downward slide. Evolution of man, whether from a scientific, artistic, intellectual and yes, even (and especially) the political standpoint, has been shown to be a fraud. A fraud as dead as the supposedly-living Republic. And this downward slide will continue until the final confrontation between the Acolytes of God and the Mammonites of Caesar.
That doesn’t mean we have to give up. If we can reject The Empire, we can still work for Our Country. We can still have a positive effect on the rate of decline, the rate of the rot that is decomposing the body politic. Because God can always squeeze some good out of our bad past. Here is how. We must still participate in the electoral process. Yes, it is imperfect. Just look at Donald. But there is still a lot that good people can accomplish, especially at the local level of the political spectrum. That is what we must do. Why? Because the power to influence the course of events, however minimally, at whatever level, is a gift that we must not waste. At least, not if we value our children. And theirs.
Yes, there is another thing that we can do that will be of lasting benefit for ourselves and others that we may have an influence on. I think we all might be smart to brush up on our manners. Eh? You know, our courtly manners. You know, bows and curtseys. Genuflections, too! After all, regardless of whether you believe in Heaven or Hell, the visible manifestation of all the final concentration of Imperial human power, in all its eventual global and gory glory, is coming soon. Because Caesar is growing ever stronger. And Caesar wants his due. Even Jesus says to give him his due. So get ready to kneel. Whether it’s Donald or Vlad (or both, in tandem), it’s coming soon. Maybe within our children’s lifetimes.
For those who can see the unseen, here’s another interesting thought that might make my advice about courtly manners more apropos. Heaven is not going to be a republic. And neither is Hell.
We began our first predictive analysis, and spent a lot of time with it. But we still haven’t got to the main question!
And that is how it should be.
Since the predictive method separates probability from decision and emphasizes decision, we should spend most of our time with defining the decision. The probability part will be easy, and is just math. So, as they say on the planes, but here I mean it, sit back, relax, and enjoy the flight.
When we left off, we were exploring CGPA. If a person was only taking one class, there were 14 possible CGPAs (0, 0.33, …, 4.33). Now, if all we knew was the scoring (grading) system and that person was taking just one class, then we deduce the probability (from the symmetry of logical constants leading to the statistical syllogism) of a CGPA of, e.g., 4 as 1/14—and the same for the other possibilities.
Because all probability is conditional on a specified list of premises, and only on that list, it’s well to be explicit. The probability CGPA equals, say, 0, needs givens. Those assumptions, premises, givens, truths, are the list itself, (0, 0.33, …, 4.33), the implicit premise that CGPA must be one of these; or the explicit premise there is only one class and explicit rules of the scoring system which together imply the list. Notice we do not allow an “incomplete”. Why not? Why not indeed? It is as assumption on our part and nothing else. If we assumed an incomplete, the probability changes (homework: how?). Remember: the all in “if all we knew…” is as rigorous as can be. We calculate the probability on these premises and none other. Probability is not subjective, except in the sense that we choose the premises: after the premises are chosen, probability is deduced.
If the person were taking 2 classes, there are 196 different possible grades (from 14^(number of classes); see this document on permutations), of which only 42 are unique (0, 0.165, 0.330, 0.335, …, 4.33). If all we knew were the scoring system and that there were 2 classes, the chance of a CGPA = 3 is 9/196 = 0.046. Use this self-explanatory R code to play (but don’t push r much beyond 5!; install gtools if you don’t have it; this code is not meant to be efficient, but explicative; if you can’t follow the code, don’t worry, just use it).
require(gtools)
# possible grades; a premise set by us
s = c(0,.33, .67, 1, 1.33, 1.67, 2, 2.33, 2.67, 3, 3.33, 3.67, 4, 4.33)
r=2 # number of classes; another premise
result = as.matrix(expand.grid(lapply(numeric(r), function(x) s)), ncol=r)
cgpa = apply(result,1,function(x) sum(x)/r)
table(cgpa)
The table(cgpa) gives a count of possibilities for each CGPA; i.e. with r = 2 there is only 1 way to get 0, 2 ways to get 0.165, and so on.
Again, the “all” in the “all you know” cannot be stressed too highly. All probability is conditional on the information assumed, and only on that information, so the probabilities above are only valid assuming just the premises given and none other. In particular, it does not matter what you might know about a person and their study habits, or the school, or anything else. The probabilities are true given the premises. Whether these are the right premises for the question we want to answer is another question which we’ll explore — in depth — later.
Now, what if all we knew were the scoring system and that the person were going to take 1 or 2 classes? Suppose we’re interested in a CGPA of 3 again. If 1 class, the probability is 1/4; if 2 classes, it’s 9/196. And since we don’t know if 1 or 2 classes, we apply the statistical syllogism again, and deduce 1/2 * 1/4 + 1/2 * 9/196 = 0.148.
You can see that we in principle can derive exact answers—though the counting will grow difficult. For a “full load” of 12 classes, there are 14^12 possible grades (5.7e13), of which only about 1,000 are unique.
Two of these possibilities are 1.860833 and 1.861667. We could, of course, compute the probability of these CGPAs given the by-now usual premises. But is that what we want? Is this the decision? Compute the probability of barely distinguishable grade points?
It could be that we care about such small differences. If we do, then we have the apparatus to solve the problem. Not for incorporating SAT or HGPA or past observations yet, but for our “naked” premises. We’ll come to that other information in time. But let’s be clear what we’re trying to do first or we risk making all the usual mistakes.
Now, I do not care about such small differences, and neither do most people. I just do not want to differentiate (though I could if I wanted) between, e.g. 1.860833 and 1.861667. To the nearest, say, tenth place is good enough for the decision I want to make about CGPA. Yet small differences are important if our goal is ordering; if, say, we want to predict who has the highest or lowest CGPA and that kind of thing. We’re not doing that there. Our decision is quantifying uncertainty in CGPA for individual people and accuracy to the 6th decimal place isn’t that interesting to me — to you it might be.
We have a decision about our decision to make: keep the small differences, which carry computational burdens and produces not very interesting answers, or make an approximation. Pay attention here. Tradition (classical methods) approximates the finite discrete CGPA as a continuous number, usually on the real line, a.k.a. the continuum. This approximation is so common that few pause to think it is an approximation! But, of course, it is, and a crude one.
If this most important point has not sunk in, then stop and think on it.1
One difficulty with the traditional approximation is that it says the probability of any caCGPA (the “ca” prefix is for the continuous approximation) is 0, which is dissatisfying (the continuum is a strange place!). The benefit is that all sorts of canned software is ready for use, and the math is much easier. Whether these benefits are worth it is the point in question and cannot be assumed true in all problems.
Besides the continuum, another approximation is to compress CGPA. It is already finite and discrete: we keep that nature, but further reduce the level of detail. I don’t care about the differences between 1.860833 and 1.861667, but suppose I do care about the difference between 1 and 2, and between 2 and 3, and 3 and 4.
That is, one compression is to put CGPA on the set (0, 1, 2, 3, 4). There are no computational difficulties with such a small set; all probability statements based on it are readily calculated. Number of classes has much less effect on this set, too.
It’s a crude compression, true. Still, that doesn’t mean a useless one. It depends—as all things do—on the decisions I want to make with CGPA. If I’m a Dean of some sort (Heaven forfend), this compression may be perfect, and I can even consider going cruder, say, (0-2, 3-4).
Or again, it may be too crude at that. Maybe every tenth is more what I’m looking for, especially if I’m considering eligibility of some scholarship.
We’ll see what these approximations do next time.
I’ll answer all pertinent questions, but please look elsewhere on the site (or in Uncertainty) for criticisms of classical methods. Non-pertinent objections will be ignored.
1You may argue that CGPA is embedded (in some mathematical sense) in an infinite sequence, and thus CGPA would live on the continuum, and thus the continuous is no longer an approximation. Since probability is conditional, accepting this condition works in the math. But, of course, CGPA is not embedded in any infinite sequence. Nothing is, because nothing contingent is infinite. So we’re back to the continuous as an approximation.
