 # Modelling data with a shifted distribution

Data
I have a 2D numpy masked array of runners’ times, where the first index of the array indicates a particular runner and the second index indicates the race (not all runners run in every race, which is why there are masked values).

Modelling
I’d like to model the race times as follows. First, each runner has a ‘quality index’ \mu and each race has a difficulty R, where the \mu have a Dirichlet prior distribution and R is normalised such that it takes a uniform prior on [0,1]. Then, the minimum time a runner could possibly run a race is to be modelled as \lambda\mu R (representing the fact that everyone has a physiological limit), where \lambda is some number, to be constrained by the data, with a uniform prior on [0,1]. Then the distribution of times for all t > \lambda\mu R should be something asymmetrical (e.g. a gamma distribution) such that the expected race time is \mu R, its width is proportional to the expected race time, and there is a long right tail (reflecting the fact that runners can have a ‘bad race’).

Code so far
Mu = pm.Dirichlet(‘mu’, a = np.ones(Nrunner) )
R = pm.Uniform(‘R’, lower=0, upper=1, shape=Nrace)
sigma = pm.HalfNormal(‘sigma’, sd=0.1)
lam = pm.Uniform(‘lam’, lower=0, upper=1)

mu = Mu[:,None] * R[None,:]
shift = lam * mu
mu_g = mu - shift

gamma = pm.Gamma(‘gamma’, mu=mu_g, sd=sigma*mu)
limit = pm.Deterministic(‘limit’, lam * mu)
Y_obs = pm.Deterministic(‘Y_obs’, limit + gamma, observed=t)

Problem
Being new to pymc3, I can’t figure out how to set up the model above properly. I get an error message when the code is runs, since you can’t use the “observed” keyword with Deterministic. Please could someone point me in the right direction for how to phrase this model properly?

Yes, PyMC3 requires that your observed variables have proper likelihoods, and this is not likely to change. I would try using the ShiftedGamma that @aseyboldt defined in this issue: https://github.com/pymc-devs/pymc3/issues/864#issuecomment-285984017

Let me know if you would like more details – I realize implementing a custom distribution right off the bat could be intimidating, but wanted to get a response down for you!

Thanks a lot - I was able to implement that new distribution and it worked as I needed it to!

1 Like