In this work we perform an observational data analysis on Einsteinian cubic gravity and $f(P)$ gravity with the objective of constraining the parameter space of the theories. We use the 30 point $z-H(z)$ cosmic chronometer data as the observational tool for our analysis along with the BAO and the CMB peak parameters. The $chi^2$ statistic is used for the fitting analysis and it is minimized to obtain the best fit values for the free model parameters. We have used the Markov chain Monte Carlo algorithm to obtain bounds for the free parameters. To achieve this we used the publicly available textit{CosmoMC} code to put parameter bounds and subsequently generate contour plots for them with different confidence intervals. Besides finding the Hubble parameter $H$ in terms of the redshift $z$ theoretically from our gravity models, we have exercised correlation coefficients and two textit{machine learning} models, namely the linear regression (LR) and artificial neural network (ANN), for the estimation of $H(z)$. For this purpose, we have developed a textit{Python} package for finding the parameter space, performing the subsequent statistical analysis and prediction analysis using machine learning. We compared both of our theoretical and estimated values of $H(z)$ with the observations. It is seen that our theoretical and estimated models from machine learning performed significantly well when compared with the observations.
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