An event is a set of outcomes of an experiment in probability. In Bayesian Probability an event is defined as describing the next possible state space using knowledge from the current state.
Events are strictly defined as elements of the sigma algebra, or, equivalently, predicates on the outcome. It's not different in Bayesian probability. It's worth making the definition solely about the mathematical structure, because there are various interpretations of probability, including one used in combinatorics to prove existence of certain objects that has nothing to do with randomness or uncertainty. It's the same math everywhere.
For events describing subsequent states, I think what you're after is the idea of a filtration.
It is more of an observation than a theorem, as it correctly works all the time.
It's a theorem that follows from the axioms of conditional probability. Whether you can find a useful prior is another question.
Entropy is a measure of uncertainty associated with a random variable.
It's a little more general than that. When we define a probability space, we equip it with a sigma algebra (which defines the events). We can also define sub-sigma algebras by taking fewer of the sets in that original one. Sigma algebras themselves form an algebra under the usual set union, intersection, etc., and entropy is a function on that algebra. A random variable X, being a measurable function, has what's called an adapted sigma algebra, that is, the smallest sigma algebra that contains its sets of the form {s : X(x) = k} for all values of k in X's range. We can measure the entropy of that random variable just fine.
A biased sample is not uniform
"Biased" isn't a measure of a sampling procedure, it's a measure of a decision procedure. People speak colloquially of biased samples, but what they really mean is that with the decision procedure they are using plus this sampling method, the expected value of their statistic does not converge to the parameter they are estimating.
The classic example of an unbiased vs a biased procedure is that for variance of a normally distributed random variable. Saying
$$\sigma = \frac{\sum (X - E[X])2}{n}$$
turns out to be a biased estimator. The unbiased estimator is
$$\sigma = \frac{\sum (X - E[X])2}{n-1}$$
On the other hand, the first form has a narrower sampling distribution. Unbiasedness is far from the only criterion for choosing a procedure.
Unbiased Distributions
This is not a thing. Estimators (that is, statistical procedures) may be unbiased. Distributions are what they are.
In general for statistical inference, the only formulation of inference that doesn't turn into mystification and handwaving is decision theory. It's worth couching everything in that.
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A few points:
Events are strictly defined as elements of the sigma algebra, or, equivalently, predicates on the outcome. It's not different in Bayesian probability. It's worth making the definition solely about the mathematical structure, because there are various interpretations of probability, including one used in combinatorics to prove existence of certain objects that has nothing to do with randomness or uncertainty. It's the same math everywhere.
For events describing subsequent states, I think what you're after is the idea of a filtration.
It's a theorem that follows from the axioms of conditional probability. Whether you can find a useful prior is another question.
It's a little more general than that. When we define a probability space, we equip it with a sigma algebra (which defines the events). We can also define sub-sigma algebras by taking fewer of the sets in that original one. Sigma algebras themselves form an algebra under the usual set union, intersection, etc., and entropy is a function on that algebra. A random variable X, being a measurable function, has what's called an adapted sigma algebra, that is, the smallest sigma algebra that contains its sets of the form {s : X(x) = k} for all values of k in X's range. We can measure the entropy of that random variable just fine.
"Biased" isn't a measure of a sampling procedure, it's a measure of a decision procedure. People speak colloquially of biased samples, but what they really mean is that with the decision procedure they are using plus this sampling method, the expected value of their statistic does not converge to the parameter they are estimating.
The classic example of an unbiased vs a biased procedure is that for variance of a normally distributed random variable. Saying
$$\sigma = \frac{\sum (X - E[X])2}{n}$$
turns out to be a biased estimator. The unbiased estimator is
$$\sigma = \frac{\sum (X - E[X])2}{n-1}$$
On the other hand, the first form has a narrower sampling distribution. Unbiasedness is far from the only criterion for choosing a procedure.
This is not a thing. Estimators (that is, statistical procedures) may be unbiased. Distributions are what they are.
In general for statistical inference, the only formulation of inference that doesn't turn into mystification and handwaving is decision theory. It's worth couching everything in that.