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Then the world produces observations IID according to N mu,1. After some number n of observations, we can apply Bayes law to construct a posterior on mu. From this posterior, we can cut out an interval with measure 0. My impression is that all Bayesian confidence intervals are mathematically equivalent to the above sequence of actions with different choices of priors and observations plugged in.

I believe all disagreements lie in interpretation of the prior. In this situation, applying Bayes law to process observations is the right thing to do even though the constant might be formally fixed at the time the posterior is computed. Everything becomes murky of course when the distribution for the constant is unknown and disagreed on.

Some of us see this as one of the major advantages of the Bayesian approach! In the situation where different analysts use the same likelihood but obtain substantially different posteriors because they are using different priors one can say that the data are not conclusive. Rational people can genuinely disagree without being incoherent.

Observed Confidence Levels: Theory and Application by Alan M. Polansky - Semantic Scholar

For example, the drug company might be using a prior that displays a degree of scepticism that is acceptable to. This is the ideal outcome and it would be good to know. Here we need to debate the priors and get more data if we can. Powered by WordPress. You are confident about events which have a large probability. This meaning of confidence is inadequate in many applications because we want to reason about how much more information we have, how much more is needed, and where to get it.

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As an example, a learning algorithm might predict that the probability of an event is 0. Classical Confidence Intervals. These are common in learning theory. The essential idea is that world has some true-but-hidden value, such as the error rate of a classifier.

Any procedure whose Bayesian properties have not been explored can have properties that make it unsuitable for post-data inference. Procedures whose properties have not been adequately studied are inappropriate for general use. Warn readers if the confidence procedure does not correspond to a Bayesian procedure. Authors using confidence intervals have a responsibility to keep their readers from invalid inferences if they choose to use them, and it is almost sure that readers will misinterpret them without a warning Hoekstra et al, Never report a confidence interval without noting the procedure and the corresponding statistics.

Observed Confidence Levels

As we have described, there are many different ways to construct confidence intervals, and they will have different properties. Some will have better frequentist properties than others; some will correspond to credible intervals, and others will not.


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It is unfortunately common for authors to report confidence intervals without noting how they were constructed. A narrow interval could correspond to very precise information or very imprecise information depending on which procedure was used. Not knowing which procedure was used could lead to very poor inferences. In addition, enough information should be presented so that any reader can compute a different confidence interval or credible interval. In most cases, this is covered by standard reporting practices, but in other cases more information may need to be given.

Consider reporting likelihoods or posteriors instead. An interval provides fairly impoverished information. Just as proponents of confidence intervals argue that CIs provide more information than a significance test although this is debatable for many CIs , a likelihood or a posterior provides much more information than an interval.

With modern scientific graphics so easy to create, along with the fact that likelihoods are often approximately normal, we see no reason why likelihoods and posteriors cannot replace intervals in most circumstances. Recall from our discussion in Section 4. If needed, read Section 1. Try to imagine all the pennies being used in the United States in One way to compute this value would be to gather up all pennies being used in the US, record the year, and compute the average.

However, this would be near impossible! An image of these 50 pennies can be seen in Figure 8. Based on these 50 sampled pennies, what can we say about all US pennies in ? Since year is a numerical variable, we use a histogram in Figure 8. Observe a slightly left-skewed distribution, since most pennies fall in between and with only a few pennies older than What is the average year for the 50 sampled pennies? Eyeballing the histogram it appears to be around In other words, around This should all start sounding similar to what we did previously in Chapter 7!

We summarize the correspondence between the sampling bowl exercise in Chapter 7 and our pennies exercise in Table 8. Going back to our 50 sampled pennies in Figure 8. Recall that we also saw in Chapter 7 that such estimates are prone to sampling variation. For example, in this particular sample in Figure 8.

If we sampled another 50 pennies, would we observe exactly three pennies with the year of again? More than likely not. We might observe none, or one, or two, or maybe even all 50! The same can be said for the other 26 unique years that are represented in our sample of 50 pennies. To study the effects of sampling variation in Chapter 7 we took many samples, something we could easily do with our shovel.

Observation Theory: Estimating the Unknown

In our case with pennies however, how would we obtain another sample? By going to the bank and getting another roll of 50 pennies. How can we study the effects of sampling variation using our single sample. Step 2 : Put the 50 slips of paper into a hat or tuque as seen in Figure 8. Record the year. Step 4 : Put the slip of paper back in the hat! In other words, replace it as seen in Figure 8. What we just performed was a resampling of the original sample of 50 pennies.

We are not sampling 50 pennies from the population of all US pennies as we did in our trip to the bank. Instead, we are mimicking this act by resampling 50 pennies from our original sample of 50 pennies. Now ask yourselves, why did we replace our resampled slip of paper back into the hat in Step 4? Because if we left the slip of paper out of the hat each time we performed Step 4, we would end up with the same 50 original pennies! In other words, replacing the slips of paper induces sampling variation. Being more precise with our terminology, we just performed a resampling with replacement from the original sample of 50 pennies.

Had we left the slip of paper out of the hat each time we performed Step 4, this would be resampling without replacement. Note that the 50 values you resample will almost certainly not be the same as ours given the inherent randomness. We display the 50 resampled pennies in Figure 8. Observe in Figure 8. Recall from the previous section that the sample mean of the original sample of 50 pennies from the bank was What about for our resample?

Any guesses? We obtained a different mean year of This variation is induced by resampling with replacement we performed earlier. What if we repeated this resampling exercise many times? Would we obtain the same mean year each time? In other words, would our guess at the mean year of all pennies in the US in be exactly We recorded these values in a shared spreadsheet with 50 rows plus a header row and 35 columns.

We display a snapshot of the first 10 rows and 5 columns of this shared spreadsheet in Figure 8. What did each of our 35 friends obtain as the mean year?

Estimation and Confidence Intervals

Once again, dplyr to the rescue! After grouping the rows by name , we summarize each group of 50 rows by their mean year :. Also observe how the distribution is roughly centered at , which is the sample mean of What we just demonstrated in this activity is the statistical procedure known as bootstrap resampling with replacement. We used resampling to mimic the sampling variation we studied in Chapter 7 on sampling. However in this case, we did so using only a single sample from the population.

In fact, the histogram of sample means from 35 resamples in Figure 8. It is an approximation to the sampling distribution of the sample mean, in the sense that both distributions will have a similar shape and similar spread. In fact in the upcoming Section 8. Using this bootstrap distribution, we can study the effect of sampling variation on our estimates. In Section 8. Lastly, observe also that the size argument is set to match the original sample size of 50 pennies.

As we saw when we did our tactile resampling exercise, the resulting mean year is different than the mean year of our 50 originally sampled pennies of Thus, we want to repeat the resampling with the replacement of 50 pennies 35 times. Observe how they are somewhat similar, but not identical.

Furthermore, they are an approximation to the sampling distribution of the sample mean, a concept you saw in Chapter 7 on sampling. These distributions allow us to study the effect of sampling variation on our estimates of the true population mean, in this case the true mean year for all US pennies. However, unlike in Chapter 7 where took multiple samples something one would never do in practice , bootstrap distributions are constructed by taking multiple resamples from a single sample.

In this case the 50 original pennies from the bank. Remember that one of the goals of resampling with replacement is to construct the bootstrap distribution, which is an approximation of the sampling distribution. However, the bootstrap distribution in Figure 8. In Figure 8. Note here that the bell shape is starting to become much more apparent. We now have a general sense for the range of values that the sample mean may take on.


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But where is this histogram centered? The mean of these means is This is the case since each of the resamples are based on the original sample of 50 pennies. Say you are trying to catch a fish. On the one hand, you could use a spear, while on the other you could use a net. Using the net will probably allow you to catch more fish!

Based on our sample of 50 pennies from the bank, the sample mean was Look at the bootstrap distribution in Figure 8. While this question is somewhat subjective, saying that most sample means lie between and would not be unreasonable. Our proposed interval of to was constructed by eye and was thus somewhat subjective. We now introduce two methods for constructing such intervals in a more exact fashion: the percentile method and the standard error method. Both methods for confidence interval construction share some commonalities. First, they are both constructed from a bootstrap distribution, as you constructed in Subsection 8.

Second, they both require you to specify the confidence level. All other things being equal, higher confidence levels correspond to wider confidence intervals and lower confidence levels correspond to narrower confidence intervals. We can do this by computing the 2.

What does it mean if a result is statistically significant*?

This is known as the percentile method for constructing confidence intervals. Interval marked by vertical lines. Recall in Appendix A. Given that our bootstrap distribution based on resamples with replacement in Figure 8. First, recall the bootstrap distribution has a mean equal to What is this value? Recall that the bootstrap distribution is an approximation to the sampling distribution. Recall also that the standard deviation of a sampling distribution has a special name: the standard error.

Putting these two facts together, we can say that 2. However, recall that we can only use the standard error rule when the bootstrap distribution is roughly normally-shaped. Describe what changes are needed to make this happen. Hint: we suggest you look at Appendix A. Recall that the process of resampling with a replacement we performed by hand in Section 8. Or more precisely, constructing an approximation to the sampling distribution using only one sample. To perform this resampling with replacement virtually in Section 8.