We used three years of house finch, Haemorhous mexicanus , song recordings spanning four decades in the introduced eastern range to assess how individual level cultural transmission mechanisms drive population level changes in birdsong. First, we developed an agent-based model (available as a new R package called ‘TransmissionBias’) that simulates the cultural transmission of house finch song given different parameters related to transmission biases, or biases in social learning that modify the probability of adoption of particular cultural variants. Next, we used approximate Bayesian computation and machine learning to estimate what parameter values likely generated the temporal changes in diversity in our observed data. We found evidence that strong content bias, likely targeted towards syllable complexity, plays a central role in the cultural evolution of house finch song in the New York metropolitan area. Frequency and demonstrator biases appear to be neutral or absent. Additionally, we estimated that house finch song is transmitted with extremely high fidelity. Future studies can use our simulation framework to better understand how cultural transmission and population declines influence song diversity in wild populations. • We analysed house finch songs from three years spanning four decades in New York. • We compared real data against simulations to make inferences about cultural transmission. • Content bias, likely for syllable complexity, drives cultural evolutionary changes. • Frequency and demonstrator biases were absent and transmission fidelity was high.