# Bayesian Prior - a distribution conditioned on a set of measure zero? (Definition of a bayesian statistical model)

I am trying to write down the exact definition of a bayesian statistical model in a similar way as the definition for a statistical model.

So far I have the definition:

A statistical model is a pair $$(X,(\mathbb{P}_{\theta})_{\theta\in\Theta})$$, where $$(\mathbb{P}_{\theta})_{\theta\in\Theta}$$ is a family of probability distributions on a measurable space $$(S,\mathcal{A})$$. $$X$$ is a random variable mapping to $$S$$, whose distribution has a free parameter $$\theta \in \Theta$$. This means, we know a function $$\theta \mapsto \rho_{\theta}$$ with $$P_{\theta}(X \in da) = \rho_{\theta}(a)da$$ \begin{align*} \text{with}&&P_{\theta}(X \in da) & := P(X=a) \text{, for }S\text{ discrete and} \\ &&P_{\theta}(X \in da) &:= f(a)da \text{, for density }f(a). \end{align*}

Now I wanted to add that in bayesian models the parameter $$\theta$$ is considered a random variable on its own:

Let $$(S\times\Theta,\mathcal{A}\times\mathcal{B},P)$$, with an appropriate $$\mathcal{B}$$, be a probability space and let $$(X,T)$$ be a random variable mapping to $$(S\times\Theta)$$. For the family $$(\mathbb{P}_\theta)_{\theta\in\Theta}$$ of conditional distributions $$\mathbb{P}_{\theta}(\cdot):=P(\cdot|T=\theta)$$ and the marginal distribution $$\Pi:\mathcal{B}\rightarrow[0,1],\quad \Pi(B) := P(T\in B)= P((X,T)\in S\times B),$$ the statistical model $$(X,\mathbb{P}_{\theta})$$ is called a bayesian statistical model with A-Priori-distribution $$\Pi$$.

My problem/question is, that I know $$\mathbb{P}_{\theta}(\cdot):=P(\cdot|T=\theta)$$ is wrong, because it must be $$P(\cdot|T\in B), \quad B\subset \Theta$$ for it to be a conditional distribution because $$\{T=\theta\}$$ would be of measure zero if $$\Theta$$ is not discrete.

However, I am not sure whether the definition of $$\mathbb{P}_{\theta}(\cdot)$$ then still makes sense as in the end, the 'true parameter' which we want to find is a single value $$\theta_0$$ which produces the observations $$X_1,...,X_n\sim\mathbb{P}_{\theta_0}$$ ? I.o.w. how can I reconcile $$\mathbb{P}_{\theta}\text{ and }P(\cdot|T\in B)\text{ ?}$$

Thank you very much in advance!

• I don't this is a question about Bayesian statistics as much as it is a question about conditional distributions. To make the question more succinct, you could ask, why, in a bivariate normal distribution $p(x,y)$ of $(X,Y)$, can we speak of the conditional distribution $p(y|X=x)$, considering that $X=x$ is an event with probability zero? Jul 13, 2021 at 13:11
• Perhaps stats.stackexchange.com/questions/230545/… addresses the issue? For more like this, use this site search.
– whuber
Jul 13, 2021 at 13:48
• @whuber Thank you very much! I didn't know yet that conditioning on an event implicitly means conditioning on the sigma algebra generated by it, nor did I even know about conditioning on sigma algebras, so I will definitely look into it!
– klm
Jul 13, 2021 at 14:10