Reading: Bayesian inference and MAP

In the section Formalizing Bayesian Statistics

In the material it is mentioned that samples comes from of the population with Bernouli distribution whereas the samples comes from the Binomial distribution.

Can someone explain why the samples come from Bernouli distribution.?

Hello @rajesh_rajendran
Bernoulli distribution models the outcome of a single binary event or trial, where the event can have two possible outcomes eg like when you flip a coin whereas samples coming from the “Binomial distribution,” it mostly means that we are dealing with a series of Bernoulli trials performed multiple times.

Hey @rajesh_rajendran,
To add to @Isaak_Kamau’s explanation, this is because how we have considered the random variable X. The random variable X is defined as a single coin toss, and for a single trial (or coin toss), trivially, we would choose Bernoulli distribution to model the random variable.

Now, from this random variable, we have sampled 10 observations, and that’s what forms our “observed sample vector”. In other words, 10 samples from the population following a Bernoulli Distribution.

However, had we defined the random variable X as 10 coin tosses, we would choose Binomial distribution to model the population, and instead of sampling 10 observations, we would only model 1 observation.

Additionally, note another key difference between the 2 distributions. In the case of Bernoulli, we don’t take the order of the tosses’ results in consideration, for instance, “HT” is considered the same as “TH”. This is because, each of the coin toss is an individual sample, and if you think about it, in none of the distributions, we take the order of samples into account.

However, in Binomial, the 10 coin tosses are considered as a single sample, and hence, in the single sample, we need to take into account the order of the results of the coin tosses and that’s why we have the Binomial coefficient as a part of the probability.

Lastly, even if you model the random variable X, in the later way (with the help of a Binomial distribution), then also the posterior probabilities won’t change, since the binomial coefficients would cancel out in the numerator and the denominator. For more information, you can check this video out.

I hope this helps you.

Cheers,
Elemento