Response improvement in complex experiments by co-information composite likelihood optimization

Abstract

We propose an adaptive procedure for improving the response outcomes of complex combinatorial experiments. New experiment batches are chosen by minimizing the co-information composite likelihood (COIL) objective function, which is derived by coupling importance sampling and composite likelihood principles. We show convergence of the best experiment within each batch to the globally optimal experiment in finite time, and carry out simulations to assess the convergence behavior as the design space size increases. The procedure is tested as a new enzyme engineering protocol in an experiment with a design space size of order 10⁷.

Appendix: Proofs

1.1 A.1 Proof of Proposition 1

The parameter estimate computed at step t+1 solves the following set of equations over Θ _KSA :

(6.1)

As n→∞, by Assumptions A1–A3, the right hand side of the above equation converges to a quantity proportional to

(6.2)

Note that

$$\mathbb{P}(Y_t \geq\gamma_t, \mathbf{X}_t = \mathbf{x})= \frac {\mathbb{P}(\varphi (\mathbf{x}) + \epsilon\geq \gamma_t)p(\mathbf{x}|\theta_t)}{\rho_t}, $$

where ρ _t =ℙ(Y _t ≥γ _t ). By Assumption A4, there exist θ _t+1∈Θ _KSA such that setting θ=θ _t+1 in (6.2) gives

As n→∞, for the solution of (6.2) evaluated at the optimal point x ^∗, we have

1.2 A.2 Proof of Proposition 2

Let β _t =c _ϵ /ρ _t . Proposition 1 implies

$$p_t (\mathbf{x}^\ast)\geq p_0 \beta_1 \beta_2 \cdots\beta_t = p_0 \prod_{i = 1}^t \beta_t, $$

where p ₀=p ₀(x ^∗). When n is large, Proposition 1 implies

(6.3)

Taking the log in the right hand side of the above expression gives

(6.4)

as T→∞ if . This proves (3.1). Moreover, (6.3) and (6.4) imply the bound in (3.2).