I’m sorry that the terminology of random variables caused confusion! If it helps, you can basically ignore the formalism of random variables and instead simply talk about the probability of certain events. For a random variable X with values in X and density p(x), an event is (up to technicalities that you shouldn’t care about) any subset A⊆X. Its probability is given by the integral
P(A):=∫x∈Ap(x).
In the case that X is discrete and not continuous (e.g., in the case that it is the set of all possible human DNA sequences), one would take a sum instead of an integral:
P(A):=∑x∈Ap(x).
The connection to reality is that if we sample x∈X from the random variable X, then its probability of being in the event A is modeled as being precisely P(A). I think with these definitions, it should be possible to read the post again without getting into the technicalities of what a random variable is.
I think this post would be much easier to learn from if it was a jupyter notebook with python code intermixed or R markdown.
In the end of the article I link to this piece of code of how to do the twin study analysis. I hope that’s somewhat helpful.
FYI Likelihood refers to a function of parameters given the observed data.
L(θ)=P(x∣θ).
Likelihood being larger supports a particular choice of parameter estimate, ergo one may write some hypothesis is likely (in response to the observation of one or more events).
The likelihood of a hypothesis is distinct from the probability of a hypothesis under both bayesianism and frequentism.
Likelihood is not a probability: it does not integrate to unity over the parameter space, and scaling it up to a monotonic transformation does not change its usage or meaning.
I digress, the main point is there is no such thing as the likelihood of an event. Again, Likelihood is a function of the parameter viz. the hypothesis. Every hypothesis has a likelihood (and a probability, presuming you are a bayesian). Every event has a probability, but not a likelihood.
I’m sorry that the terminology of random variables caused confusion!
P(A):=∫x∈Ap(x).If it helps, you can basically ignore the formalism of random variables and instead simply talk about the probability of certain events. For a random variable X with values in X and density p(x), an event is (up to technicalities that you shouldn’t care about) any subset A⊆X. Its probability is given by the integral
In the case that X is discrete and not continuous (e.g., in the case that it is the set of all possible human DNA sequences), one would take a sum instead of an integral:
P(A):=∑x∈Ap(x).The connection to reality is that if we sample x∈X from the random variable X, then its probability of being in the event A is modeled as being precisely P(A). I think with these definitions, it should be possible to read the post again without getting into the technicalities of what a random variable is.
In the end of the article I link to this piece of code of how to do the twin study analysis. I hope that’s somewhat helpful.
FYI Likelihood refers to a function of parameters given the observed data.
L(θ)=P(x∣θ).
Likelihood being larger supports a particular choice of parameter estimate, ergo one may write some hypothesis is likely (in response to the observation of one or more events).
The likelihood of a hypothesis is distinct from the probability of a hypothesis under both bayesianism and frequentism.
Likelihood is not a probability: it does not integrate to unity over the parameter space, and scaling it up to a monotonic transformation does not change its usage or meaning.
I digress, the main point is there is no such thing as the likelihood of an event. Again, Likelihood is a function of the parameter viz. the hypothesis. Every hypothesis has a likelihood (and a probability, presuming you are a bayesian). Every event has a probability, but not a likelihood.
Thanks, I’ve replaced the word “likelihood” by “probability” in the comment above and in the post itself!