Estimate at which point a linear model hits a certain value (including probabilities)

Question

I have a simple 1D set of datapoints with a trend, I want to estimate at which point $X_t$ (i.e., at which point in the future) the model will hit a certain threshold $Y_t$:

I can fit a trendline to estimate the intersection point. But how can I get a probability distribution? I tried to calculate confidence bands and calculate their intersection points, which would give me some information about the "uncertainty" of $X_t$, but it will not give me a continuous probability distribution

I guess it could be done using Bayes Linear Regression, or Bootstrapping? I would be glad to get any advice here and some starting points

Answer 1

The linear regression model in probabilistic terms is

$$ \mu_t = \beta_0 + \beta_1 x_t \\ y_t \sim \mathcal{N}(\mu_t, \sigma^2) $$

so for any $x_t$, you can calculate the probability that $y_t$ is greater than the threshold

$$ \Pr(Y > y_t|x_t) = 1 - \Pr(Y \le y_t|x_t) = 1 - F_{\mu_t,\sigma^2}(y_t) $$

where $F_{\mu_t,\sigma^2}$ is a normal cumulative probability distribution function with mean $\mu_t$ and variance $\sigma^2$.