# Sequential Bayesian updating with binary data (in a case where a beta-bernoulli setup seems inappropriate)

An unknown parameter $$\theta$$ is randomly drawn at time $$t=0$$ according to prior p.d.f. $$\mu_0(\cdot)$$ that has support $$[L,R]\subseteq\mathbb{R}$$.

At each time $$t\in\{1,2,...\}$$ an agent makes an estimate $$a_t\in\mathbb{R}$$ of $$\theta$$ and observes an outcome $$y_t\,|\,(a_t,\theta) \sim\text{Bernoulli}(\sigma(a_t,\theta))$$, where $$\sigma:\mathbb{R}^2\to[0,1]$$ yields the probability of success given $$(a_t,\theta)$$. (Assume $$y_t$$ is independent of all draws $$\{y_\tau\}_{\tau\neq t}$$ and estimates $$\{a_\tau\}_{t\neq t}$$ from other periods.) Let $$\mu_t(\cdot|a_1,y_1,...a_t,y_t)$$ denote the posterior belief (formed using Bayes' rule) after observing the sequence $$\{(\color{red}{a_\tau},\color{blue}{y_\tau})\}_{\tau=1}^t$$ of $$\color{red}{\text{estimates}}$$ and $$\color{blue}{\text{outcomes}}$$.

My (Soft) Question: which $$\mu_0$$ and $$\sigma$$ would make analytically calculating posterior belief $$\mu_t(\cdot|\cdot)$$ tractable? [Edit: An example and a simpler question is found below. Many thanks to user @Tim for the great suggestion!]

Small Request: $$\sigma$$ could indeed be "anything," but ideally I would like something that is monotonically decreasing in how "far'' $$a_t$$ and $$\omega$$ are. (E.g. $$\sigma(a_t,\theta)=\frac{1}{1+(a_t-\theta)^2}).$$ To set ideas, I will now describe a class of functions that seems "reasonable" to me.

Let $$\sigma(a_t,\theta)=h(\theta-a_t)$$ where $$h:R\to[0,1]$$ satisfies

1. $$h(0)=1>\max_{x\in\mathbb{R}\setminus\{0\}}$$ and
2. $$h(x)=h(-x) \text{ and } h'(x)= -h'(-x) \ \forall x\in\mathbb{R}$$.

(i.e. unimodal and symmetric about $$x=0$$, where it achieves a maximum value of $$1$$).

$$\textbf{Example:}$$ Suppose a $$\text{Uniform}([0,1])$$ ($$\equiv \text{Beta}(1,1)$$) prior $$\mu_0(\hat{\theta}):= \mathbb{1}_{[0,1]}(\hat{\theta})$$ and $$\sigma(a_t,\theta):=\frac{1}{1+(a_t-\theta)^2 k}$$, where $$k>0$$ is a known constant. Then, given $$\theta$$ and the first estimate $$a_1$$, the first outcome $$y_1$$ has a $$\text{Bernoulli}\left([{1+(a_t-\theta)^2 k}]^{-1}\right)$$ distribution. After making estimate $$a_1\in\mathbb{R}$$ and observing outcome $$y_1\in\{0,1\}$$, the agent forms posterior belief

$$$$\mu_1(\theta|a_1,y_1) = \begin{cases} \frac{1-[1+(a_1-\theta)^2 k]^{-1}}{1-\int_{0}^{1} [1+(a_1-\hat{\theta})^2 k]^{-1} d\hat{\theta}}\mathbb{1}_{[0,1]}(\theta) &, \text{if } y_1 = 0\\ \frac{[1+(a_1-\theta)^2 k]^{-1}}{\int_{0}^{1} [1+(a_1-\hat{\theta})^2 k]^{-1} d\hat{\theta}}\mathbb{1}_{[0,1]}(\theta) &, \text{if } y_1 = 1 \end{cases}$$$$ where the above follows from Bayes' rule (after some cancellations and simplifications). Focusing on the $$y=1$$ case is sufficient for demonstrating that the resultant posterior belief is not a $$\text{Beta}(\cdot,\cdot)$$ distribution. Simplifying yields $$$$\mu_1(\theta|a_1,y_1) = \frac{\sqrt{k}}{1+(\theta-x)^2} \frac{\mathbb{1}_{[0,1]}(\theta)}{\arctan\left(\frac{\sqrt{k}}{1-(k)(1-a_1)(a_1)}\right)}.$$$$ I don't think I made an error in my derivation. The above is quite complicated, and I am afraid will get more complicated in $$t>1$$.

At the root of these complications is $$\sigma(\cdot,\cdot)$$, so I suppose my question should focus on that.

Simplified Question: assuming a $$\text{Beta}(\alpha,\beta)$$, prior which choice of $$\sigma(\cdot)$$ guarantees that the posterior is also $$\text{Beta}(\cdot,\cdot)$$? (Feel free to assume $$\alpha=1=\beta$$, i.e. $$\text{Uniform}([0,1])$$).

• You made it very abstract, could you give us an actual example of the data you have and the model you have in mind? Why it cant be beta-Bernoulli?
– Tim
Mar 18, 2022 at 22:57
• Hi Tim, thank you so much for taking the time to think about my question. That is a very good suggestion; I provided an example above that hopefully clarifies my problem. Regarding the data: this is actually for a theoretical model, similar to a Bandit problem, so I don't have any actual (real world) data! The data' ({(aₜ,yₜ)}ₜ₌₁,₂,...)' are endogenously produced by the agent in the model (at each t chooses an aₜ, which affects the distribution of yₜ). Mar 19, 2022 at 3:01
• I understand that my question has aspects that may be outside the specialization of many stats stackexchange denizens, but my question really does boil down to one about Bayes' rule and conjugate priors. I tried my best (via terminology, notation, etc.) to present my question in a "familiar" form, but still please do let me know if there is anything I can clarify. Any help I can get on this is deeply appreciated. Mar 19, 2022 at 3:10