According to the font of all truthful and accurate knowledge - Wikipedia - "Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief."
So what does that mean, the YouTube video below gives a good visual explanation.
Here is another explanation:
According to Wikipedia dizygotic (fraternal) twins usually occur when two fertilized eggs are implanted in the uterus wall at the same time while monozygotic (identical) twins occur when a single egg is fertilized to form one zygote (hence, "monozygotic") which then divides into two separate embryos.
Fraternal twins can be mm, mf, fm or ff (where m = male and f = female), identical twins can only be mm, or ff.
For the sake of this example let's say the probability of each option is equal, so P(mm) = P(mf) = P(fm) = P(ff) = 0.25 for Fraternal twins and P(mm) = P(ff) = 0.5 for identical twins. The probability that twins are identical is P(I) = 0.1 so P(F) = 0.9 (probability of Fraternal), assuming twins must be either identical or fraternal (not strictly true but let's not make things too complicated).
If we have two brothers who are twins what is the probability that they are identical twins?
The non-Baysean answer might be 0.1 or 10% because I said above that 10% of twins are identical. However the Bayesian approach gives a different answer:
The probability of identical twins given that both twins are brothers written as P(I|B) = P(B|I)P(I)/P(B)
and since we are assuming twins must be either identical or fraternal then: P(B) = P(B|I)P(I) + P(B|F)P(F)
substituting this into the above gives: P(I|B) = P(B|I)P(I)/P(B|I)P(I) + P(B|F)P(F)
then putting in the numbers gives (0.5 x 0.1)/((0.5 x 0.1) + (0.25 x 0.9)) = 2/11 (about 18.2%) - so the knowledge that both twins are male makes the probability they are identical higher.
The video below makes a good point about the advantages and at least one disadvantage of a Bayesian approach.