Notes on Prologue

non-bayesians say bayesian is one way of doing things
bayesians say
- bayesian is the right way of doing things
- non-bayesian methods are approximations or an alternative when bayesian too hard to calculate

Notes on Introduction

we believe all sorts of things and do not believe others
even though we strictly dont have all information we need to be 100% certain
statistics is a tool when we cannot be 100% certain
in bayesian statistics probabilities are in the mind, not in the world
i.e. when we obtain more information the probabilities in our mind change
bayesian methods tell us exactly how to update our probabilities when we get new information

we start with prior probabilities and we update these with data to get posterior probabilities
Bayes box is a table that helps up calculate posterior probabilities more easily
we start by choosing values for prior probabilities and assigning to our competing hypotheses
bayes rule
bayes box is basically applying bayes rule lots of times, once for each hypothesis
phone example
we now get onto important equations

Almost any problem in statistics can be framed as a parameter estimation problem.
we move into probability of a good bus problem and use a discrete prior each with equal probability

we move away from a small set of possible values to a prior distribution
only analytical methods before computers
when maths doesnt work out nicely, MCMC and JAGS
we turn the bus prior into a continuous uniform distribution
if using a bayes box we would have infinite number of rows
parameter estimation form of bayes rule
\(posterior \propto prior \times likelihood\)
\(p(\theta | x) \propto p(\theta)p(x|\theta)\)
the proportional sign can save a bunch of work as can get rid of constant factors
the example shows uniform prior with binomial likelihood gives binomial posterior
which makes sense given prob denisity of unifor is 1
randome variable isnt best description as it is rather a measure of uncertainty in our beliefs
is a how likely a good bus is example we use a binomial for a sampling distribution and a few other options for priors: uniform, and 2 betas
first beta uses:
\(\theta^{1/2}(1 - \theta)^{1/2}\)
it is only proportional to though…
as you can see needs to be normalised…

first beta uses:
\(\theta^{100}(1 - \theta)^{100}\)
it is only proportional to though…
as you can see needs to be normalised…

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