I want to use a Bayesian approach to estimate the parameter $p$ of a binomial distribution with the number of Bernoulli experiments $x_i \in \{0,1\}$ being known and fixed to $n$. Choosing my likelihood to be binomial and my prior to be uniform in $[0,1]$, my posterior $f(p|x_1,\dots,x_n)$ becomes a beta distribution.

In the textbook example, the binomial likelihood only takes the sum of positive vs. negative outcomes of $x_i$ into account:

$f(x|p) = p^{\sum x_i} (1-p)^{n-\sum x_i}$

How can I introduce uncertainty about the outcome $x_i$ into my estimate? E.g. assume $x_i$ is determined by a neural network, assigning probabilities $p(x_i=1|input)$ and $p(x_i=0|input) = 1-p(x_i=1|input)$ to each Bernoulli outcome. How can I avoid discarding this potentially important information?

I want to use a Bayesian approach to estimate the parameter $\theta$ of a binomial distribution $\textsf{Binomial}(\theta,n)$ with the number of Bernoulli experiments $x_i \in \{0,1\}$ being known and fixed to $n$.

In the textbook example, the binomial likelihood only takes the sum of positive vs. negative outcomes of $x_i$ into account: \begin{equation} f(x|\theta) = \theta^{\sum x_i} (1-\theta)^{n-\sum x_i} \end{equation}

Together with a uniform prior $f(\theta)=\textsf{Uniform}(0,1)$, the posterior becomes the Beta distribution \begin{equation} f(\theta|s) = \textsf{Beta}(\theta|s+1,n-s+1) . \end{equation} , where $s=\sum x_i$. Unfortunately, $x_i$ is not directly observable. Instead, we are given $\pi_i$ as features to e.g. a neural network, which then provides a probabilistic mapping $f(x_i|\pi_i)$. This probability represents uncertainty about the true, latent value $x_i$.

How can we incorporate this uncertainty about the data into the estimate? E.g. if $f(x_i|\pi_i)=0.51\ \forall\ 0\leq\ i<n$, we would expect the MAP estimate of $\theta$ to be a lot further off from $1$ than if $f(x_i|\pi_i)=0.91$ throughout.

I want to ue a Bayesian approach to estimate the parameter $p$ of a binomial distribution with the number of Bernoulli experiments $x_i \in \{0,1\}$ being known and fixed to $n$. Choosing my likelihood to be binomial and my prior to be uniform in $[0,1]$, my posterior $f(p|x_1,\dots,x_n)$ becomes a beta distribution.

In the textbook example, the binomial likelihood only takes the sum of positive vs. negative outcomes of $x_i$ into account:

$f(x|p) = p^{\sum x_i} (1-p)^{n-\sum x_i}$

How can I introduce uncertainty about the outcome $x_i$ into my estimate? E.g. assume $x_i$ is determined by a neural network, assigning probabilities $p(x_i=1|input)$ and $p(x_i=0|input) = 1-p(x_i=1|input)$ to each Bernoulli outcome. How can I avoid discarding this potentially important information?

I want to use a Bayesian approach to estimate the parameter $p$ of a binomial distribution with the number of Bernoulli experiments $x_i \in \{0,1\}$ being known and fixed to $n$. Choosing my likelihood to be binomial and my prior to be uniform in $[0,1]$, my posterior $f(p|x_1,\dots,x_n)$ becomes a beta distribution.

In the textbook example, the binomial likelihood only takes the sum of positive vs. negative outcomes of $x_i$ into account:

$f(x|p) = p^{\sum x_i} (1-p)^{n-\sum x_i}$

How can I introduce uncertainty about the outcome $x_i$ into my estimate? E.g. assume $x_i$ is determined by a neural network, assigning probabilities $p(x_i=1|input)$ and $p(x_i=0|input) = 1-p(x_i=1|input)$ to each Bernoulli outcome. How can I avoid discarding this potentially important information?