# How to update the posterior belief when we are observing a stream of correlated data from a fixed but unknown data source

I want to build a [probabilistic] model that aims to infer the true value of an unknown categorical variable, $$y \in \{1,2,..., K\}$$.

We have a dataset $$(X,y): \mathbb{R}^d\rightarrow \{1,2,..., K\}$$ and we can train a classifier that gives $$d$$-dimensional data, $$X$$, and estimates the output $$y$$.

Now, suppose that $$X$$s are correlated and all coming from a fixed $$y$$. I mean, we are observing $$X^1, X^2,...., X^T,...$$ over time and we know that $$y$$ is fixed for all of them.

For example:

• We receive $$X^1$$ (at time $$t=1$$) and our previously trained classifier produces a guess about $$\hat{y}^1$$.
• Then, we receive $$X^2$$, and we again use the classifier to guess $$\hat{y}^2$$.
• Then, we receive $$X^3$$, and so on.

So, at time $$t=T$$ we have $$\hat{y}^1, \hat{y}^2, ..., \hat{y}^T$$.

Now, the question is: How can I make a model to use these estimations ($$\hat{y}^1, \hat{y}^2, ..., \hat{y}^T$$) and improve my belief about the true $$y$$ over time, considering that:

1. dimension $$d$$ is not small. e.g. $$d >50$$

2. data samples, $$X$$s, are not i.i.d. but all coming from a fixed unknown $$y$$.

3. classifier is not optimal (just trained on some available data) and at each round gives an estimate about the $$\hat{y}^t$$ for the current $$X^t$$.

I have been reading some materials and came across the following but I am not sure which one is better to investigate more into:

• Sequential Hypothesis Testing
• Optimal stopping
• Sequential probability ratio test
• HDI+ROPE decision rule: highest density interval (HDI) region of practical equivalence (ROPE)

Or is there any specific Bayesian framework for it?

## This question has an open bounty worth +50 reputation from moh ending in 23 hours.

The question is widely applicable to a large audience. A detailed canonical answer is required to address all the concerns.

• Hi, what do you mean by "the X's are coming form a fixed unknown y"? Is the true value of X also the same for all X's and the fluctuations are just due to an error-influence? – jottbe Nov 11 at 12:11
• I mean, $X^1, X^2, ...$ are samples that are releasing by a fixed $y$ and the task is to infer what is $y$. For example, we are listening to a person's speech word by word, and the task is to infer the age of the speaker. So, the age is fixed, but unknown.! – moh Nov 11 at 14:32
• Does the model need to generate a probabilistic guess after each X? – Michael Hearn Nov 13 at 19:32
• @MichaelHearn: not necessarily. But it needs to improve its guess over time. – moh 2 days ago
• Do you start with all the classes of Y and have model of what X would be associated with which Y? – Michael Hearn 2 days ago