To test an algorithm under development, I need such kind of a data set as follows. Denote \theta the parameter and y the data vector. The data elements of y, y_i, i=1,2,... , arrive one by one. The posterior p(\theta | y_1,y_2,...y_i) is expected to change along with the arrival of new data elements. For example, the posterior p(\theta | y_1,y_2,...y_i) has one peaky mode, then when y_{i+1} arrives, the posterior p(\theta | y_1,y_2,...y_i,y_{i+1}) becomes to have two or more modes. A real data case is much more preferable, while a way to simulate such a data set is also advisable.
One easy idea that comes in mind would be to have a series of oscillation peaks, say for instance deriving from a Fourier transform of some pulsating signal observed in the time-domain, distributed over a precise range of frequency. Thus, your dataset could be something like amplitude of the signal versus frequency.
If you assume your parameter \theta to be the frequency position of the oscillation peaks, then you would have a posterior distribution having as many modes as the number of peaks contained in the given frequency range.
For simulating data bins cumulating over time (your y vector), you could simply add data bins that extend your frequency range from a previously calculated full range of frequency signal. In this case, you can add, say, as many oscillation peaks as you like, hence add modes to your final posterior as the frequency range increases.
The example dataset can be very simple to be simulated and I know this can suit for what you intend to test. I hope this can be of any help.
Dear Bin,
unfortunately I have no experience with datasets changing while you analize them since I deal with astrophysical phenomena for which we require very long, but length-fixed, observations.
I think that in any case the example of time-series would be a realistic one that you can take into account. A possibility would be that of analyzing time-series, hence a dataset that you collect as the time evolves, in order to test your model on some evolving phenomenon on very short time scales (say few hours or even minutes), so that you can practically observe the signal you collect changing in time.
Thanks Enrico for your kind reply.
I also had an experience in analyzing astrophysical data by Bayesian methods. We have a paper published, titled "Bayesian Methods for Analysis and Adaptive Scheduling of Exoplanet Observations". You can just contact me if you're interested in a possible collaboration between us.
Thank you Bin for proposing that, I shall be happy to see the paper you suggested. I will keep you in mind for a possible collaboration in the future. You can also check a paper of mine entitled "A Bayesian approach to scaling relations for amplitudes of solar-like oscillations in Kepler stars" on MNRAS where I deal with Bayesian methods.
Regards.