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So far, we've been discussing statistical inference from a particular perspective,

which is the frequentist perspective.

And so frequentists are concerned with the probability of seeing a particular

data sample given the null hypothesis and that's what the P value gives you.

But there's another perspective to statistic called the Bayesian Approach,

which is concerned with the probability of having a certain outcome given

the data that we've already seen and so

I wanna talk a little bit about Bayesian approaches and give an example of power.

What are the differences here?

Well, you can think about the differences in terms of what is fixed.

So the frequentist perspective is that data are repeatable random sample,

you're always allowed to go do another experiment the same way.

And in fact, you need to at least virtually,

reason about the probabilities that get produced by these methods.

There is some sort of frequency you can reason about,

frequency of achieving a certain outcome and the underlying parameters

of the population remain constant during this repeatable process.

Your population stays fixed and you run experiments.

Meanwhile with the Bayesian approach, the data observed from a realized sample and

the parameters of the population are unknown, but

can be described probabilistically.

So they're not sort of fixed values.

However, the data are fixed.

You don't think about going back and sampling more data,

you just have the observations you have.

The Bayesian approach is 100% concerned with the application of Bayes' rule,

which is this and so this what Thomas Bayes did was relate

these conditional probabilities with the prior beliefs.

So, it allows you to take your belief about

the probabilities of certain events happening and

update them when you more data is collected and so here's what it says.

So the probability of a vent A happening given that the vent b has already happened

is equal to the probability that B happening given that

A has already happened multiplied by the probability of A happening across

the board divided by the probability of B happening across the board.

And if that's not clear, that's okay, we're gonna go into more detail here.

So right now though, recognize that the key benefit here is the ability to

incorporate prior knowledge, which is not the case with the frequentist approach.

Now, a key weakness here is that you need to incorporate prior knowledge.

So there's a couple of problems with this.

One is if you don't know anything about the population you're modeling,

then there's not a good way to model this, the distributions.

Now there are some techniques to sort of derive so-called

uninformative priors that try not to influence things too much, but

give you a plugin to be able to apply the rule, but still that is an issue, fine.

Perhaps more insidiously and the reason why the Bayesian approach was popular and

then fell out of favor in the earlier part of the 20th century

was that you can use this rule to kind of do anything you wan to confirm or

deny the effective need for evidence, just by plugging in your own prior belief.

So given a fixed set of data, two different people could come up with

different conclusions about the data, because they had different prior beliefs.

They modeled the prior distributions differently and

this was seen as a major flaw and gave rise to the frequentist approach,

which looks more or less objectively at the data itself.

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