Likelihood of the cause, given the observations
As we have just seen, if we assume some value of q, where q is the p(heads),
we can calculate the probability of the observations HHTHT given q.
We can use this fact to help us work out the value of q when this is unknown.
Say I work out the probability of the observations, HHTHT, for every possible value of q
from 0 to 1 in steps of 0.01.
Let's try this using the pre-prepared MATLAB code in script 'UncertaintyTutorial1.m'.
Open the file 'UncertaintyTutorial1.m' in Matlab
Run section 1 only
You should have a plot a bit like this
From this plot, we can clearly see that the the observations, HHTHT, were most probable given
a certain value of q - what is that value?
You may recall from the lecture that there is a mathematical relationship between
- The probability of the data given the parameters of the environment,
in this case, p(HHTHT | q)
- The likelihood of the parameters, given the data we observed,
in this case p(q | HHTHT)
Namely: p(q | data) = p(data | q)