Hi,

This answer is not specific to Bayesian optimization, though it will also work there.

A common trick is to leave the p_i's unconstrained. Later, when you need the probability distribution, you can create the transformed variables p^\prime_i = \frac{\exp{p_i}}{\sum_i \exp{p_i}}. The p^\prime values then define a valid discrete probability distribution. (This is also called the "softmax function".)

Here is a simple example in python with numpy:

import numpy as np

p = np.array([-1, 5, 3, 2])

p_primes = np.exp(p) / np.sum(np.exp(p))

You could also capture covariance structure by placing something like a multivariate normal prior over the p_i's, though you would then need to optimize, etc., over the parameters of the covariance matrix.

Have a good day,

Brandon

P.S. Sorry, it seems that latex and source code are not formatted very well here.

Ajit and Brandon thanks for answering.

I think I will try to use the polar coordinates from the first orthant of a unit 4-sphere to represent my problem. Then I have 4 angels a1-a4 and I get 5 coordinates x1--x5 and setting p_i=(x_i)^2 gives the constraint sum p_i =1.