[MUSIC]. 
Okay, so, so far we've been discussing 
statistical inference from a particular 
perspective, which is the frequentest 
perspective. 
And so frequentists are concerned with 
the Probability of seeing a particular 
data sample, given the null hypothesis. 
Okay, and that's what the p-value gives 
you. 
But there's another perspective to 
statistics called the Bayesian approach, 
which is concerned with the probability 
of Having a certain outcome given the 
data we've already seen. 
Okay, and so I want to talk a little bit 
about Bayesian approaches and give an 
example of their power. 
Okay. 
What are the differences here? 
Well, you can think about the differences 
In terms of what is fixed. 
So the frequency's perspective is that 
data are a repeatable random sample, 
right? 
You're always allowed to go do another 
experiment the same way, and in fact you 
need to at least virtually, to reason 
about, the probabilities that get 
produced by these methods. 
Okay. 
So there is some sort of frequency you 
can reason about, frequency of of 
achieving a certain outcome. 
Okay. 
And the underlying parameters of the 
population remain constant during this 
repeatable process. 
Right, your, your, the population stays 
fixed and you run experiments to 
determine, you know likelihoods. 
Okay, meanwhile, with the Bayesian 
approach, the data observed from a 
realized sample, and the parameters A, of 
the population or unknown, but can be 
described probabilistically, right. 
So there not sort of, fixed values. 
However, the data are fixed, right, you 
don't think about going back and sampling 
more data, you just have the observations 
you have, okay. 
And so, the Bayesian approach is 100% 
concerned with the application of 
Bayesian rule which is this. 
Okay, so this, what Thomas Bayesian did 
was relate these conditional 
probabilities with the prior beliefs. 
Okay, so it allows you to take your- 
Belief about the probabilities of certain 
events happening and update them when 
more data is collected, okay. 
And so here's what it says, it says the 
probability of event A happening, given 
that event B has already happened. 
Is equal to the probability of b 
happening given that a has already 
happened, multiplied by the probability 
of a happening across the board, divided 
by the probability of be happening across 
the board. 
And if that's not clear, that's okay. 
We're, we're going to 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 frequency approach. 
Now, a key weakness here is that you need 
to incorporate prior knowledge So if you, 
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 
is not much you can do with the prior, 
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 to much 
but- Give you a plugin to be able to 
rule. 
But still, that is an issue. 
fine. 
Perhaps more insideously and the reason 
why Bayes' the Bayesian approach, was 
popular and then it fell out of favor in 
the earlier part of the twentieth 
century. 
Was that, you can use this rule to kind 
of, do anything you want, to confirm or 
deny the affect of any sort of evidence, 
right? 
Just by plugging in your own prior 
belief. 
Okay. 
So, given a fixed set of data, two 
different people can come up with two 
different conclusion about the data 
because they had prior beliefs. 
They modeled the, the, the prior 
distributions differently, okay. 
And this was seen as a major flaw, and 
gave rise to the frequency approach, 
which. 
Looks more or less objectively at the 
data itself. 
Okay. 
Right. 
And so that's what is spelled out here in 
a nice essay by Matthews 1998, is that 
different people could use Bayesian 
Theorem and get different results. 
And so, faced with some, some 
experimental evidence per se ESP true 
believers could uses Bayes's Theorem to 
sh show that the new results confirmed 
it, while skeptics could use it to show 
that it the ESP didn't exist. 
Okay, and both views are possible because 
Bayesian Theorem only shows how to alter 
one's prior level of belief. 
And different people can start out with 
different opinions. 
Okay, so that's the issue. 
However the frequency oriented, the 
frequentist approach So, the frequency 
approach, laid out by Fisher and others, 
were able to achieve what was thought to 
be impossible. 
Which is a way of judging the 
significance of experimental data 
independent of any prior beliefs. 
Okay. 
So he'd found a way that anyone could use 
to show that a result was too impressive, 
too statistically significant to be 
dismissed as a fluke. 
And, you know, all you had to do was 
convert your raw data into this thing 
called a P-value. 
Alright. 
Now, you have to have some sort of 
threshold to measure the significance of 
this P-value, and that was written up as 
0.05 in by Fisher. 
You know, in a, in a, in a original 
paper. 
So what were the insights that led to 
this particular value of 0.05? 
Well, Fisher admitted that there weren't 
any. 
He simply decided 0.05 because it was 
mathematically convenient. 
Okay? 
And you know, in the many people have 
pointed out, say from the 60s on, there's 
been periods of time where there's been a 
significant amount of work showing that 
this approach, with these particular 
thresholds are able to produce a lot of 
incorrect conclusions. 
So James Berger Purdue wrote his entire 
series of papers warning about the quote 
astonishing tendency of Fisher's p-values 
to exaggerate significance. 
Findings that met the 120 standard can 
actually arise when the data provide 
little, very little of no evidence in 
favor of an effect. 
Okay, so that's the problem, so now 
perhaps the, we can say that the pendulum 
is in some sense swinging back towards 
the Bayesian approach. 
One more problem with the Bayesian 
approach that I'll bring up that I don't 
necessarily have a slide on, is that, 
which we'll see in a little bit of detail 
in. 
Perhaps the next segment, is that these 
conditional probabilities and these prior 
probabilities end up forcing you to, to 
model a situation mathematically, and 
that situation may be very difficult to 
model mathematically. 
Okay. 
So what you end up with is these, these 
chains of complicated conditional 
probabilities that need to be integrated 
in order to apply Bayes' rule, and so 
this was computationally intractable, and 
there were a norm, a huge number of 
tricks and simplications and, and ideas 
used to make this more tractable. 
Okay. 
But, thanks to the development of methods 
that allow you sample these complicated 
distributions computationally these, this 
approach is becoming increasingly 
popular. 
Okay. 
And there's a lot of thinkers in this 
space that believe that, you know, for 
the 21st century, a combination of 
frequent and Bayesian approaches is going 
to be dominant. 
Okay, so for these reasons we're going to 
make sure we talk about the Bayesian 
approach at a very introductory level and 
leading up to machine learning algorithm 
called naive bayes.