How do we find probability of a binary event occurring in continuous space?

Question

Let us assume we have a magical 1D line of length 1 cm. And we have an unbiased coin. There are points on the line, where the probability of getting head is high. Example: if the coin lands on 0.5 cm, then it has high probability of getting head.

Initial idea: Discretize the line into intervals (say 10). Then we have 10 bins of 0.1 cm. Randomly pick a bin, flip the coin inside that bin and observe the outcome(head/tail.) We count the number of times it was head in each bin. Assume that the number of flips is limited to 30. After 30 flips, we should be able to say $P(\text{head in }bin_i) = X$

Is there any better way to handle this? What if we want to discretize the line into 100 bins?

Answer 1

What you have here is a binary regression model with a single continuous input variable on the interval $[0,1]$. Your regression model is:

$$Y|X=x \sim \text{Bern}(f(x)) \quad \quad \quad f: [0,1] \rightarrow [0,1],$$

and you are trying to infer the form of the regression function $f$. At present this model is highly general and so you will need to think about what type of binary regression model and inference method you want to use. Usually this would mean considering what kind of functional form or smoothness properties the function $f$ might be expected to have.

You might decide to narrow down the form of this function to some parametric form and then you would have a parametric binary regression model, or you might simply assume some simple smoothness properties and decide to use a nonparametric binary regression model (see e.g., Diaconis and Freedman 1993). Once you decide on a modelling and inference approach, you should sample from your unit interval at appropriate places (determined by experimental design principles in regression)to get a set of data showing the coin-flip outcomes at the sampled points. You would then take that data and use it to estimate the function $f$ using the relevant inference methods for your chosen model.

Answer 2

While this may not be directly answering your question, I can provide an idea using a conditional probability approach, because you want to know the probability of getting head given that you are at some specific position.

As far as I understand, your probability of landing on a bin itself is a random variable, say $Y$. I would consider this random variable to follow a multinoulli (or multinomial distribution), with parameters $p{_i}$ corresponding to the probability of landing on the bin $i$.

Then you do the flipping experiment. Let's represent this experiment by $X$. Given that it is an unbiased coin, this follows a uniform distribution with parameter $p$=0.5.

Then your probability of getting head given that your coin lands on a particular bin is the conditional probability $P(X|Y)$, which equals to $P(X\bigcap Y)/P(Y)$. This is the probability that you want to estimate (you do not estimate probabilities though, you estimate parameters). Your question obviously implies that the probability of getting head and being in the bin $i$ is not independent, e.g. they have a functional relationship.

You can find the distribution of $P(Y)$ by sampling where your coin flips end up at, and then do distribution fitting. If your parameters $p{_i}$ turn out to have a well-defined functional relationship, you may generalize this to an increasing number of intervals. However, even at this point you are unable to know the exact probability distribution of $P(X\bigcap Y)$, and in practice joint probability distributions are hard to define exactly (precisely the reason we develop regression models). If you have an intuition on how this joint probability distribution changes with respect to the result of experiments, you can choose a suitable distribution and estimate its parameters using your observations.

I want to conclude by saying that generally, probabilistic questions like this actually begin with you defining a probability space following your theorization about a phenomenon. Depending on how the probability space is constructed, you may have different solutions. See: https://www.wikiwand.com/en/Bertrand_paradox_(probability)