uncorrect in my opinion, for two reasosn: odds cannot be multiplied (odds are not probability) and maybe the events are not statistically independent (for example if the man loves risky adventures)

uncorrect in my opinion, for two reasosn: odds cannot be multiplied (odds are not probability) and maybe the events are not statistically independent (for example if the man loves risky adventures)

It may be considered as Assumption subject to prove.

Bad use of and misunderstanding of statistics. If he doesn't get involved in any of these risky activities his chances are nil. I concur with Laura.

This is analogous to carrying a bomb on board a plane with the assumption that because the chances of 1 bomb on board a plane is (say) 1 in a million that the chances of 2 bombs is 1 in 1012...

There is no calculation given.

I think the words "odds" and "likelihood" are not used in ther correct technical meanings. As Laura stated, probabilities (or likelihoods) of independent events multiply, but odds are not probabilities, and hence they don't multiply.

I strongly doubt the assumption of independence. From what we do know about this world, we can not treat these events as independent, and we have much more additional information that is not used to judge the likelihood of this triple-event. First of all, all events involve the same person, and it may be well related to Dylan McWilliams personality and behaviour in the contact with wild animals to provoke attack. He could be considerably incautios and unwary.

I further suspect that Dylan McWilliams used the probabilities of these events refering to a "random person" on this planet, what is not ok, as we already know that he is an "outdoor enthusiast", and people spending time "outdoors" "in the wilderness are obviouesly "more likely" to be attacked by some wild animal. It is also known in what locations this happened and during what activities. The shark attack, for instance, happened in Hawaii, where attacks are much more likely than in Portugal, for instance. And the activity was bodyboarding, something that makes him look like a typical shark prey. If one takes the likelihood of bein attacked by a shark among people who do extensive bodyboarding in an area with a high shark population, one would be rather astonished if this person would not have been attacked. I don't have the stories for the other incidents, but it could be comparable. After all, it is possible an outdoor enthusiast with skills and personality similar to that of Dylan McWilliams, who has a similar spectrum of activities, will have an extremely high probability of facing three such incedents in three years.

What brings me to another point that presumably was not considered in the "calculation" of the given likelhood: Here, the question is about the probability to have one attack by a shark, one by a bear, and one by a rattlesnake. However, I think this is too narrow to ask for a probability of just that triple. There are many other "unlucky" combinations possible (there are many other dangerous wild animals out there, and what if we further would count events like being hit by a flash, a meteor, being buried by an avalanche etc), and to encounter one of all these possible (each quite unlikely) triples is much more likely, as the individual proabilities add up.

This would be very amusing to apply a Bayesian analysis to. The question as usual would be the prior assumptions. The sample size is small - 3 - so the use of frequentist methods is very debatable.

As noted earlier assuming that these events are independent - which is a requirement to multiply the odds/liklihoods/frequencies is a very strong assumption and one that in my view is very likely to be incorrect.

"This would be very amusing to apply a Bayesian analysis to"

Sorry, you have lost me here. However much I have tried, I really can't get my head round the concept of a Bayesian analysis. I think I have grasped the frequentist method, and I put up the original question as I suspected I understood it better than Burgess in the target article!

Actually, I wonder if any statistical calculation makes any sense in this situation. We can all agree Dylan was very unlucky, but maybe that is as far as it goes.

Bayesian is much harder to get used to. It can be difficult even for experts. Its the method of choice with small sample data like this. A statistcial calculation 'makes sense' only if the answers the question posed.

As I understand the question it was along the lines of what this chap especially unlucky? I dont think so but to answer that we would be a reasonable prior of the risk of being attacked in this fashion for all three settings.

"Bayesian is much harder to get used to. It can be difficult even for experts."

Thanks, that was the answer I was hoping for!

So, we now have two statistical opinions. One, that Dylan was not especially unlucky, two, that the odds of this sequence happening were 900 quadrillion to one based on multiplying the general prior probabilities of the 3 separate events as given in the original article. I may be a non-statistician, but it seems to me that these two estimates are significantly different. So, which statisticians, if any, should we believe?

This is a complex question. Frequentists and Bayesians may get different answers to the same question. Thats because they are actually answering different questions.

In this given case denominator is not well defined. There are few large studies done to estimate these risks so treating them as frequentists do is not a sensible method. Frequentists as they name suggest require large studies where there are multiple events. Bayesians also prefer lots of events but are much more reliable with small numbers.

@David,

"Frequentists and Bayesians may get different answers to the same question."

This is what people often say, but that it simply not the case. Statements like this are often severely misunderstood. You pin that point in your next sentence, but the statement above ramins "wrong". It would have been better to say:

Frequentists and Bayesians get different answers

and they do so simply because they ask different questions!

@Anthony,

"Bayesian is much harder to get used to. It can be difficult even for experts." (citation from David)

The problem is that frequentists are often not aware what the question is that they ask. They often think this question would be similar (or identical) to the questions Bayesians ask, but that is wrong. Bayesians ask a quite simple question ("what is my state of knowledge about the process under study?" or, possibly, "how much do I have to revise my knowledge, given the new set of data?"), but it is often technically difficult to answer this. Frequentists, in contrasts, actually ask tricky and indirect questions (like "what is probability of 'more extreme' test statistics assuming a particular model?") that are often technically easy to tackle, but they have a hard time to interpret the answers in any practically meaningful way. So for Bayesians, the difficulties are in the calculations, but the frequentists have similar difficulties in the interpretation of the numbers they calculate. There is no free lunch.

"So, which statisticians, if any, should we believe?"

Any kind of statistical analysis can be interpreted only when all the facts and assumptions used are known and understood. So you must look at the facts and assumptions and decide which statistician used more (and more relevant) facts and made the more reasonable assumptions.

"Frequentists and Bayesians may get different answers to the same question."

Here is an example where the same answer is obtained.

I got the correct answer quite quickly using the frequency format and a two-by-two table. Bayes theorem is too complex and difficult for my ageing brain, but why should I even bother with it at all?

"Science News from research organizations

Why don't we understand statistics? Fixed mindsets may be to blame

Date:October 12, 2018 Source:Frontiers ...

Unfavorable methods of teaching statistics in schools and universities may be to blame for people ignoring simple solutions to statistical problems, making them hard to solve. This can have serious consequences when applied to professional settings like court cases. Published in Frontiers in Psychology, the study shows for the first time that fixed mindsets -- potentially triggered by suboptimal education curricula -- lead to difficulties finding the simple solution to statistical problems....

"The same meta-analysis showed that when the task was presented in natural frequency format instead of probabilities, performance rates increased from 4% to 24%," says Weber. (See below for an example task.)...

"We want our findings to encourage curriculum designers to incorporate natural frequencies systematically into school mathematics and statistics. This would give students a helpful tool to understand the concept of uncertainty -- in addition to the 'standard' probabilities."

Example of a problem posed in probability and natural frequency format

Probability format: The probability of being addicted to heroin is 0.01% for a person randomly picked from a population (base rate). If a randomly picked person from this population is addicted to heroin, the probability is 100% that he or she will have fresh needle pricks (sensitivity). If a randomly picked person from this population is not addicted to heroin, the probability is 0.19% that he or she will still have fresh needle pricks (false alarm rate). What is the probability that a randomly picked person with fresh needle pricks is addicted to heroin (posterior probability)?

Solution: With the help of Bayes' theorem, the corresponding posterior probability P(H|N), with H denoting "person is addicted to heroin" and N denoting "person has fresh needle pricks," can be calculated:

P(H|N) = (P(N|H) x P(H)) / (P(N|H) x P(H) + P(N|¬H) x P(¬H)) = (100% x 0.01%) / (100% x 0.01% + 0.19% x 99.99%) = 5%

Natural frequencies format: 10 out of 100,000 people from a given population are addicted to heroin. 10 out of 10 people who are addicted to heroin will have fresh needle pricks. 190 out of 99,990 people who are not addicted to heroin will nevertheless have fresh needle pricks. What percentage of the people with fresh needle pricks is addicted to heroin?

Solution:

Number of heroin addicts: 10

Number of people with needle pricks: All the heroin addicts + 190 non-addicts = 200

Percentage of people with needle pricks who are addicts = 10/200 = 5%"

Anthony G Gordon, applying Bayes theorem is NOT making the analysis Bayesian. Particularily, the probability "that a randomly picked person with fresh needle pricks is addicted to heroin (posterior probability)" is NOT a posterior probability. It is a conditional probability. You change the reference set (the statistical population) from "people" to "people with fresh needle picks". This is still a completely frequentist analysis and interpretation and has nothing to do with the "Bayesian interpretation".

What is presented is simply the difficulty known as "base rate fallacy" that comes when probabilities (or relative frequenties) are related to a wrong population (or one has a wrong population in mind what talking about probabilities). The two solutions are both NOT involving any Bayesian thinking. One applies a theorem to get conditional probabilities directly, the other uses numerical examples to conclude frequencies in finite populations.

So there are no different questions involved here. It's one question, but two wasy to get the (same) answer.

Use of logistic regression is recommended in order to find out exp{B} results.

Dear Dr Ghoodjani

I think your remark would benefit from further clarification