This term always sounds like it means a variable selected at random not a variable with randomness in it. Please use the term ‘stochastic variable’. Edit: or does it mean a variable composed entirely at random without any relation to any other variable?
Edit: I think this post would be much easier to learn from if it was a jupyter notebook with python code intermixed or R markdown. Sometimes the terminology gets away from me and seeing in code what is being said would really help understand what is going on as well as give some training on how to use this knowledge. Edit: there should be a plot illustrating ” which are jointly sampled according to a density p(x,y).” including rugs for the marginal distributions. I could do that if anyone wants. Here is an example describing a different concept.
Unfortunately, this is a well-established mathematical term that’s used universally throughout probability theory. Changing notation to match intuition is not feasible; we must instead change intuition to match notation.
Technically speaking, a random variable is just a measurable function X:Ω→E, where Ω is the underlying sample space and E is some measurable space. Indeed, even if the function X is constant, it is technically speaking still a random variable (in this case it’s called a deterministic random variable).
The problem is that, in practice, people don’t really pay too much attention to precisely what the sample space is, especially if they don’t have some specific reason to care about measure theory. They instead often want to talk about the probability distribution, and the task of figuring out formally and rigorously why this is all well-defined is often omitted. Luckily, the Kolmogorov extension theorem and related results usually allow you to pick a “large enough” sample space that carries all the content you need for your math work.
stochastic variable is certainly less common, but googling it only returns the right thing. it seems like it’d be a valid replacement and I agree it could reduce a common confusion.
“Random variable” is never defined. I though stochastic variable is just a synonym for random variable. I have seen posts where random variable is always written as r.v. and that helps a bit.
From Wikipedia: “In probability theory, the sample space (also called sample description space,[1]possibility space,[2] or outcome space[3]) of an experiment or random trial is the set of all possible outcomes or results of that experiment.
what is a measurable space?
“he function X is constant,” you mean its just one outcome like a die that always lands on one side?
I’m not sure if clarifying this is most useful for the purpose of understanding this post specifically, but for what it’s worth: A measurable space is a set together with a set of subsets that are called “measurable”. Those measurable sets are the sets to which we can then assign probabilities once we have a probability measure P (which in the post we assume to be derived from a density p, see my other comment under your original comment).
“the function X is constant,” you mean its just one outcome like a die that always lands on one side?
I think that’s what the commenter you replied to means, yes. (They don’t seem to be active anymore)
what makes a function measurable?
This is another technicality that might not be too useful to think about for the purpose of this post. A function is measurable if the preimages of all measurable sets are measurable. I.e.: f:X→Z, for two measurable spaces X and Z, is measurable, if f−1(A)⊆X is measurable for all measurable A⊆Z. For practical purposes, you can think of continuous functions or, in the discrete case, just any functions.
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.
This term always sounds like it means a variable selected at random not a variable with randomness in it. Please use the term ‘stochastic variable’. Edit: or does it mean a variable composed entirely at random without any relation to any other variable?
Edit: I think this post would be much easier to learn from if it was a jupyter notebook with python code intermixed or R markdown. Sometimes the terminology gets away from me and seeing in code what is being said would really help understand what is going on as well as give some training on how to use this knowledge. Edit: there should be a plot illustrating ” which are jointly sampled according to a density p(x,y).” including rugs for the marginal distributions. I could do that if anyone wants. Here is an example describing a different concept.
Unfortunately, this is a well-established mathematical term that’s used universally throughout probability theory. Changing notation to match intuition is not feasible; we must instead change intuition to match notation.
Technically speaking, a random variable is just a measurable function X:Ω→E, where Ω is the underlying sample space and E is some measurable space. Indeed, even if the function X is constant, it is technically speaking still a random variable (in this case it’s called a deterministic random variable).
The problem is that, in practice, people don’t really pay too much attention to precisely what the sample space is, especially if they don’t have some specific reason to care about measure theory. They instead often want to talk about the probability distribution, and the task of figuring out formally and rigorously why this is all well-defined is often omitted. Luckily, the Kolmogorov extension theorem and related results usually allow you to pick a “large enough” sample space that carries all the content you need for your math work.
https://en.wikipedia.org/wiki/Random_variable
stochastic variable is certainly less common, but googling it only returns the right thing. it seems like it’d be a valid replacement and I agree it could reduce a common confusion.
“Random variable” is never defined. I though stochastic variable is just a synonym for random variable. I have seen posts where random variable is always written as r.v. and that helps a bit.
From Wikipedia: “In probability theory, the sample space (also called sample description space,[1] possibility space,[2] or outcome space[3]) of an experiment or random trial is the set of all possible outcomes or results of that experiment.
what is a measurable space?
“he function X is constant,” you mean its just one outcome like a die that always lands on one side?
what makes a function measurable?
I’m not sure if clarifying this is most useful for the purpose of understanding this post specifically, but for what it’s worth: A measurable space is a set together with a set of subsets that are called “measurable”. Those measurable sets are the sets to which we can then assign probabilities once we have a probability measure P (which in the post we assume to be derived from a density p, see my other comment under your original comment).
I think that’s what the commenter you replied to means, yes. (They don’t seem to be active anymore)
This is another technicality that might not be too useful to think about for the purpose of this post. A function is measurable if the preimages of all measurable sets are measurable. I.e.: f:X→Z, for two measurable spaces X and Z, is measurable, if f−1(A)⊆X is measurable for all measurable A⊆Z. For practical purposes, you can think of continuous functions or, in the discrete case, just any functions.
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!