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Lentil convexity: Difference between revisions
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===Lentil buying decisions are ''not'' independent=== | ===Lentil buying decisions are ''not'' independent=== | ||
[[File:Lentil distribution.png|300px|left|Lentil buying projections yesterday]]But in actual fact lentil purchasing decisions are ''not'' independent, as our new friend the coronavirus pandemic illustrates. They just ''seem'' like it, in most markets. 95 percent of the sample are connected by a general disposition ''not'' to buy lentils. But this disposition is a function of one’s apprehension of the proximity of [[apocalypse]]. Apocalypses are, in the main, rare, and one does not tend to form an opinion that one is nigh purely through ones’ own deductions from meteorological and astronomical data. Rather, we apprehend oblivion because someone said it on Twitter. Because Owen Jones wrote a thought-piece on it, or because — ''by Jupiter, everyone is suddenly buying lentils and there are hardly any left in Sainsbury’s''. | [[File:Lentil distribution.png|300px|left|Lentil buying projections in peacetime yesterday. The average per 100 persons is 15, and any more than 28 is genuinely unthinkable.|thumb]]But in actual fact lentil purchasing decisions are ''not'' independent, as our new friend the coronavirus pandemic illustrates. They just ''seem'' like it, in most markets. 95 percent of the sample are connected by a general disposition ''not'' to buy lentils. But this disposition is a function of one’s apprehension of the proximity of [[apocalypse]]. Apocalypses are, in the main, rare, and one does not tend to form an opinion that one is nigh purely through ones’ own deductions from meteorological and astronomical data. Rather, we apprehend oblivion because someone said it on Twitter. Because Owen Jones wrote a thought-piece on it, or because — ''by Jupiter, everyone is suddenly buying lentils and there are hardly any left in Sainsbury’s''. | ||
So, we get ''infected'' with the idea of Apocalypse'' by each other''. Our legume-buying habits ''are not independent after all''. They are all formed out of a collective consensus about the ''non-imminence'' of the second coming. While each person’s threshold for precautionary lentil purchase in the event of imminent apocalypse will differ, across the group, news of the unchecked spread of coronavirus will bring each person closer to that threshold, and some of them over it. As they walk past the tinned goods shelf, it only takes a small proportion of that 95% to pick up a tin to blow the grocer's expectations out the window. Let’s say 5 of the 95% decide to buy a in each.assume also that the hippies vegans and health food fanatics in the sample are also buying their regular quota, we can see that the grocers supply of lentils will quickly be depleted. | So, we get ''infected'' with the idea of Apocalypse ''by each other''. Our legume-buying habits ''are not independent after all''. They are all formed out of a collective consensus about the ''non-imminence'' of the second coming. While each person’s threshold for precautionary lentil purchase in the event of imminent apocalypse will differ, across the group, news of the unchecked spread of coronavirus will bring each person closer to that threshold, and some of them over it. As they walk past the tinned goods shelf, it only takes a small proportion of that 95% to pick up a tin to blow the grocer's expectations out the window. Let’s say 5 of the 95% decide to buy a in each.assume also that the hippies vegans and health food fanatics in the sample are also buying their regular quota, we can see that the grocers supply of lentils will quickly be depleted. | ||
And now a second order of dependence emerges. For imagine some of the 95% who have not crossed the the threshold for precautionary lentil purchase, but notice that the lentil shelf in the supermarket is nearly empty. This might prompt them to reconsider there apprehension of apocalypse. They will collect the remaining lentils. | And now a second order of dependence emerges. For imagine some of the 95% who have not crossed the the threshold for precautionary lentil purchase, but notice that the lentil shelf in the supermarket is nearly empty. This might prompt them to reconsider there apprehension of apocalypse. They will collect the remaining lentils. | ||
The supermarket management will now intervene, alarmed at this sudden run on lentils. Their immediate reaction will be to impose an item limit. It posts a sign on the shelf restricting customers to 3 tins each. Limiting most customers, who would normally be seen dead buying ''one'' tin, to three, has no effect on the problem, since our lentils-for-judgment-day-only types, would have bought one tin anyway: the problem is not that one or two people are bulk buying, but that all people are single-item buying. The three-items per customer (a) irritates the hippies who would ordinarily buy ten tins anyway, and (b) further validates the suspicion among non-hippies that we are indeed in desperate times. After all, one could hardly ask for a clearer sign of of imminent Armageddon than LENTIL RATIONING — I mean, panic buying hippy food: could it get any worse than that? The 95%, one one, up their demand from one tin to the maximum permissible three. | The supermarket management will now intervene, alarmed at this sudden run on lentils. Their immediate reaction will be to impose an item limit. It posts a sign on the shelf restricting customers to 3 tins each. Limiting most customers, who would normally be seen dead buying ''one'' tin, to three, has no effect on the problem, since our lentils-for-judgment-day-only types, would have bought one tin anyway: the problem is not that one or two people are bulk buying, but that all people are single-item buying. The three-items per customer (a) irritates the hippies who would ordinarily buy ten tins anyway, and (b) further validates the suspicion among non-hippies that we are indeed in desperate times. After all, one could hardly ask for a clearer sign of of imminent Armageddon than LENTIL RATIONING — I mean, panic buying hippy food: could it get any worse than that? The 95%, one one, up their demand from one tin to the maximum permissible three. | ||
Still: no panic hoarding, but the shelf is bare. And those latecomers, discovering it is now ''too late'' to stock up on lentils, head to borlotti beans and Mexican bean fiesta in brine. To their horror they discover these have been cleaned out as well. Every distant fear about a forthcoming final reckoning now confirmed, there is a stampede for the couscous and quinoa. | Still: no panic hoarding, but the shelf is bare. And those latecomers, discovering it is now ''too late'' to stock up on lentils, head to borlotti beans and Mexican bean fiesta in brine. To their horror they discover these have been cleaned out as well. Every distant fear about a forthcoming final reckoning now confirmed, there is a stampede for the couscous and quinoa. | ||
Note that the interconnectedness of these events is not stable or predictable. The [[correlation]] ''changes''. The very ''existence'' of those events, and third party reactions to those events, changes the interdependence between the events. | |||
===Implications=== | |||
[[File:Lentil convexity.png|300px|right|thumb|The fat tail of the distribution overlaid. In ordinary times, in the middle of the bell, the distributions look the same, but the tails are different. There is a material chance of demand a long way down the tail]]What are the implications of using a normal bell curve to model a secretly related events? Firstly, events which are genuinely independent ''stay'' independent whatever happens. The odds of flipping heads on a fair coin stays 0.5 however often you flip it, and whatever the previous results.<ref>Practical point though: the longer your sequence of heads, the greater the probability that the ''coin is not fair''.</ref> This makes the job of modelling independent events much, much easier. In fasct it makes it ''possible''. Modelling dependent events isn’t just a case of more complex maths. It isn’t possible. | |||