We investigate star forming scaling relations using Bayesian inference on a comprehensive data sample of low- (z<0.1) and high-redshift (1<z<5) star forming regions. This full data set spans a wide range of host galaxy stellar mass ($M_{*} sim10^6-10^{11} M_{odot}$) and clump star formation rates (SFR $ sim10^{-5}-10^2 M_odot yr^{-1}$). We fit the power-law relationship between the size (r$_{Halpha}$) and luminosity (L$_{Halpha}$) of the star forming clumps using the Bayesian statistical modeling tool Stan that makes use of Markov Chain Monte Carlo (MCMC) sampling techniques. Trends in the scaling relationship are explored for the full sample and subsets based on redshift and selection effects between samples. In our investigation we find no evidence of redshift evolution of the size-luminosity scaling relationship, nor a difference in slope between lensed and unlensed data. There is evidence of a break in the scaling relationship between high and low star formation rate surface density ($Sigma_{SFR}$) clumps. The size-luminosity power law fit results are L$_{Halpha}sim$ r$_{Halpha}^{2.8}$ and L$_{Halpha}sim$ r$_{Halpha}^{1.7}$ for low and high $Sigma_{SFR}$ clumps, respectively. We present a model where star forming clumps form at locations of gravitational instability and produce an ionized region represented by the Str{o}mgren radius. A radius smaller than the scale height of the disk results in a scaling relationship of $L propto r^3$ (high $Sigma_{SFR}$ clumps), and a scaling of $L propto r^2$ (low $Sigma_{SFR}$ clumps) if the radius is larger than the disk scale height.
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