# Are the following questions relevant in the field of Statistical Inference?

Suppose I am interested in studying the relationship between the age of giraffes in relation to the height and weight of giraffes. Given some measurements on the age, height and weight of giraffes - the most common approach would be to create a regression model age ~ f(height, weight) . For some height and weight measurements, I could predict the age of a giraffe having these measurements.

However, I could also fit a probability distribution function to my data. For instance, I could try to fit a joint multivariate normal probability distribution : P(height, weight, age). Doing so, I think you can answer more questions (relating to statistical inference) compared to the regression model. For example:

1) As in the regression model, we can find out the "most probable age" of a giraffe measuring 17ft and weighing 2000 lbs. This would involve randomly sampling the conditional distribution of the joint multivariate distribution - and then taking the average value of these samples.

2) Unlike the regression model, I can now find out the "most probable age" of a giraffe measuring 17ft and weighing more than 2000 lbs. This would involve adjusting the conditional distribution from 1) and taking the average of random samples (e.g. MCMC) from this conditional distribution.

3) Unlike the regression model, I can also perform conditional inference on the covariates in the probability distribution. For example, given a giraffe is taller than 17ft and between 20-25 years old : what is the most probable weight of this giraffe?

4) And finally, (unlike the regression model) I can also calculate probabilities - for example: What is the probability of observing a giraffe that is younger than 20 years and weighs more than 2000 lbs?

My Question: Can someone please tell me if the points I have listed above can truly be answered using probability distributions (and not answered using regression models) - and if they are in fact "relevant" in statistical research studies? (I imagine that fitting probability distributions to data can be more computationally expensive compared to regression models, and more difficult to explain to non-statisticians.)

Thanks!

Note: If you decide to fit a probability distribution to some data and decide to place priors on the mean, variances and covariances of the model parameters - in some sense, this is analogous to regularization terms in regression models.

• Regression models are intertwined with distribution functions. If I model $Y = f(X)$ I am also modelling the conditional expectation $\mathbb{E}[Y|X]$. Nov 9, 2021 at 18:24
• @ Galen: thank yoy for your reply! this is a great observation! but I don't think that regression models can answer questions about "probabilities"? E.g. What is the probability of observing a giraffe that is younger than 20 years and weighs more than 2000 lbs? Thank you! Nov 9, 2021 at 18:31
• You can always represent the regression model as a model for the conditional distribution of Y given the Xs. That is the point of this book: routledge.com/… Nov 9, 2021 at 18:42
• Thank you! I will look into this book! Nov 9, 2021 at 18:46
• In general: is it easier to do this with probability distribution functions compared to regression models? do probability distribution models have the potential to capture more information and complex patterns in big data compared to regression models? are the 4 points i raised "relevant" in the field of statistical inference? thank you! Nov 9, 2021 at 18:47