Nine are enough.
You have a set of samples and you are interested in learning something about the probability distribution from which they are drawn. That something is the parameter of interest. It might be the mean. If you do something to the samples, add them together, for example, then you might lose some piece of information that they contain regarding the parameter. But you also might not. Whether you lose information or not by manipulating your samples depends on what you do to them.
For example, if you are sampling from a binomial distribution for which success has value 1 and failure value 0, then adding up the results of the samples won’t destroy information about the mean of the distribution (i.e., the probability of success). That’s because the mean is expressed in the number of successes, rather than their order. You know just as much about the mean of the distribution if your first nine samples are successes and your tenth a failure as if your first is a failure and the next 9 successes. In other words, when you add up the results, you lose information on the order with which the successes occurred, but the mean does not determine that order, and so you don’t lose any information relevant to determining the mean.
When the mean increases, the sample results change because you end up with more successes. So a statistic that counts successes changes too. Both the sample and the statistic change in the same way. That is what happens when a statistic is “sufficient.”
That’s why for a sufficient statistic the probability of drawing a particular sample, conditional on a particular result for the statistic, is independent of the parameter. As the parameter changes, both the sample and the statistic change in the same way. So their relationship to each other remains constant regardless of what happens to the parameter. In a sense, the sufficient statistic transforms the sample, instead of altering it. So any change to the parameter doesn’t change the relationship between the samples and the statistic. Sample and statistic are just different ways of expressing the same thing about the parameter.
The sample conditional on the statistic is just the ratio of the probability of the sample to that of the statistic. This means that if the statistic is sufficient, the probabilities of the sample and the statistic must both be products of the parameter, so that the parameter will cancel out and therefore have no effect on this conditional probability.
Raise taxes until philanthropy disappears. Why should the unelected rich decide how your taxes are spent on public projects?
Saying that adverse selection in insurance is a problem to be eliminated because it frustrates marginal cost pricing is like saying that R&D fixed costs leading to innovation and product improvement are a problem to be eliminated because they frustrate marginal cost pricing.
If you are going to put monumental plant sculptures in the botanical garden they ought to be made entirely of plants. That includes the skeleton. It might take 50 years to raise a tree to look like a cobra and grow appropriate plants upon it in a symbiotic mix. But so much the more for wonder.
And the sculptures themselves should not be stereotyped representations of gorillas and frogs and such. They should be beautiful and original. Imaginary Worlds is disappointing: unremarkable sculptures papered over with plants.
(Note: I don’t prefer topiary; pruning a bush to look like a squirrel is just as superficial an art as wallpapering a squirrel statue with herbs.)
And if you’re going to give me Arcimboldo’s vegetables portraits in the flesh, they’d better be made with real vegetables. Anything else is a monument to disappointment.
I’d much rather not have these tasteless distractions in what is an otherwise wonderful garden.
I do, though, like the Chihuly.
The price system, like queuing, is just another way of rationing access to resources. And it’s perfectly reasonable for the government to prefer to impose a queuing system over a price system in certain circumstances. For example, if you don’t want the rich to take all of a fixed resource.
So it’s perfectly reasonable for San Francisco to prefer that parking spaces go to the lucky, persistent, or fleet, rather than to the highest willingness to pay.
And perfectly reasonable too for San Francisco to put a stop to attempts to circumvent its preferred rationing system. Attempts like this.