Hello, today's topic is important sampling and important sampling is a sampling procedure

which is here. Last time we talked about how to sample uniform distributions and Gaussian

distributions and more complicated things if we had some kind of procedure

such we could compute the inverse of the CDF and that is usually not the case

only for nice densities which have a good analytic form and it's it's already

hard for the normal distribution. So what can we do if we can't do any of the above

here and the first approximate Monte Carlo sampling method is so-called

importance sampling and one thing upfront important sampling does not

generate samples so that's it's a bit weird that it's called sampling but you

might recall that our main goal for the generations of samples was that we can

compute those integrals here and that is what important sampling does do so it

doesn't generate samples per se but we can then compute those integrals so we

can compute expectations and variances and higher moments and things like that

now how does it work whoops the first thing that we need is a second measure

new this measure new we will call a reference measure and this reference

measure does well has to fulfill two conditions the first condition is mu

needs to be absolutely continuous with respect to this measure mu new so mu

absolutely continuous with respect to new that means what we can't have is if

mu looks like that and new looks like that and this is forbidden so if that's

you know zero here the density of new is zero here because that violates this

condition if you type for example take set a which is here this set a has zero

mass with respect to the reference measure new because integrating that

means integrating the density over this set a and this density is

identically equal to zero here so new of a zero but mu our measure of interest

as non-zero measure on that set because you know this is a non-zero density here

so that it's not allowed so this is invalid but has to look I know if mu is

zero here and then it goes up and does something that's mu then new has so I

can it can be zero here where it mu is zero as well but or else it has to be

positive now of course the absolute quantity the absolute width of those

lines that's not correct because of course every density has to have area

equal to one under the graph so this usually this has to be lower than that of

course or this has to be higher than that but we disregarding the absolute

height of that okay so that's the first requirement the second requirement is

that we can sample efficiently from new so usually then new is something like a

Gaussian that's the most easy case or of course a uniform distribution okay how

does it work well it's quite easy actually well we'll look at this

integral this is our quantity of interest and we write d mu of X as D

sorry R of X times D nu of X and then we just sample from this measure new so we

apply the ideas from from last time or time before that so if we can sample

with respect to this measure here then we just have to take the average of

plugging those samples in the function which we're integrating over we just

write one over n times the sum of all f of X I times R of X I this is true

because of the weak law of large numbers so this will converge in probability to

the integral f d mu okay so what we do is we actually compute an integral over

new the reference measure but we have some way of translating between mu and

nu and this is exactly this rather nicodin density that we have here now

three examples which all have kind of a different flavor to them first is if new

is the Lebesgue measure now in order for the Lebesgue measure to be valid in that case we

have to be on the bounded state space because it's impossible to choose a

random number on the whole real line because which real number could you pick