In answering this question, I'm assuming you are doing a CBC study within Lighthouse Studio, and using the HB estimation routine within Lighthouse Studio.
Under Analysis + Analysis Manager, select the HB drop-down. Then, click the "gear icon" to access the settings for the HB estimation. Along the left-side of the screen, select Advanced Output Options. Then, check "Successive estimates of alpha (CSV)".
Then, Run your HB estimation. When it's done, go to your project's folder (where you created the project on your hard drive), within the "Analysis Runs" folder. Browse to the folder that has the name of your HB run. You will find a file that includes ..._alpha.CSV in the name.
This file contains multiple rows, where each row represents the alpha estimate at each iteration of the MCMC algorithm. Typically, you ignore the first 5K or 10K iterations (because we believe we haven't yet obtained convergence). But, you can take an average across the remaining rows (alpha draws) to come up with a point estimate of the alpha to summarize preferences for the population. Or, you can analyze all those later draws to represent the distributional uncertainty around the population means.