Recent advances in live imaging technology, such as the MS2-GFP system, have enabled the recording of transcriptional data at the single-cell level at ever-greater temporal and spatial resolution. Whereas previously researchers had to rely on static 'snapshots' of developing embryos, such as those provided by Single Molecule Fluorescence in situ Hybridisation (smFISH), it is now possible to record fluorescence microscopy movies of developing embryos in the laboratory. An example of one such live imaging technique is the MS2-GFP system, where gene editing is used to insert a transgene into a gene of interest. When the gene is transcribed, a noisy fluorescent time series is generated which acts as a proxy for transcriptional activity. The dorsal-ventral patterning system in the early Drosophila embryo provides an ideal system for studying transcription using live imaging. In this system, a single input, a member of the Bone Morphogenetic Protein (BMP) family, controls multiple target genes, each of which exhibit transcriptional bursting, where transcripts are produced stochastically in discrete 'bursts' of activity, rather than as a constant, Poissonian process. The aim of analysing these movies is to gain insight into transcriptional regulation in the early embryo, i.e. the relation between the dynamics of mRNA production and cell developmental fate. BMP signalling is of particular interest due to the known involvement of misregulated BMP signalling in developmental defects and cancer. A key problem is how to process and analyse MS2 datasets in order to answer this question. The main output of the thesis is the development of a novel type of Hidden Markov Model (HMM) for extracting kinetic parameters from MS2 movies, with the aim of establishing the relationship between BMP signalling and transcriptional bursting. We first provide an overview of BMP Signalling in Drosophila, followed by a summary of previous theoretical definitions of biological noise and transcriptional bursting in the literature. We then outline the details of the implementation of our algorithm. The algorithm demonstrates a significant improvement in computational efficiency relative to the current state of the art model for MS2 analysis, the Compound State Hidden Markov Model (cpHMM), while allowing for the inference of single-cell transcriptional parameters. Results are shown comparing our algorithm to the original algorithm in terms of computational speed and accuracy, using synthetic and experimental Drosophila data. Finally, we present the in-depth results from using our algorithm to investigate the bursting dynamics of the Drosophila ush and hnt genes. We have been able to establish that regulation of bursting dynamics in this system is achieved through frequency modulation, i.e. by regulating the frequency of bursts, rather than burst duration or amplitude; burst frequency decreases as a function of distance from the embryo midline.