How do I implement this example?

Assume z = \{z_t \in \mathbb{R}: t=1,2\} and x=\{x_t \in \mathbb{R}^p: t=1,2..\}. Let D_t=\{(z_1,x_1),(z_2,x_2)...\}.

I want to model z_t given x_t. The normal likelihood is this: z_t | w,\beta,x_t \sim N(x^T_tw,\beta^{-1}). The conjugate prior chosen is this (normal gamma): w,\beta \sim N(w|m_0,\beta^{-1}S_0)Gam(\beta|a_0,b_0). Initial values are m_0=0, S_0 is the identity matrix, a_0=1, and b_0=100.

Let’s suppose I get some new points (z_{t+1},x_{t+1})

Bc we chose a conjugate prior, the posterior is also normal Gamma: w,\beta | D_t,z_{t+1},x_{t+1} \sim N(w|m_{t+1},\beta^{-1}S_{t+1})Gam(\beta|a_{t+1},b_{t+1}) where S_{t+1}, m_{t+1}, and b_{t+1} are defined in an iterative fashion (the exact way they are defined is not necessary for my question).

How would I go about coding up the likelihood and prior? I know this Bayesian process must repeat for every t. Let’s suppose that in this particular t, x_t = [5,2] (so p=2) and z_t is 7. So for obs, zts[-1]=7

xt = [5,2]

with pm.Model() as model:
    # conjugate priors
    wbeta = pm.DensityDist('wbeta', ?some function for Normal Gamma?, shape=2, testval = [1,1])
    w = wbeta[0]
    beta = wbeta[1]
    # likelihood
    obs = pm.Normal("obs", mu=xt*w,sd=np.inv(beta), observed=zts)

What are you trying to do here? Are you trying to set up the likelihood function so you can update each time you have new data?

Yes. And the above example is for one step

You can write down your model as a pymc3 model with the information you have, however, it wont perform update as you wish as pymc3 model cannot figure out the conjugate posterior automatically.

xt = np.array([5, 2])
mu0, S0, a0, b0 = 0, np.eye(2), 1, 100
with pm.Model() as model:
    beta = pm.Gamma('beta', a0, b0)
    w = pm.Normal('w', mu0, S0*(beta**-1))
    obs = pm.Normal('z',, w), beta**-1, observed=7)

Ah that is unfortunate about the conjugate posterior. Thanks