# Bayesian Model Averaging in dreamer

Bayesian model averaging is a general mixture distribution, where each mixture component is a different parametric model. Prior weights are placed on each model and the posterior model weights are updated based on how well each model fits the data. Let \(\mu(d)\) represent the mean of the dose response curve at dose \(d\), \(y = \{y_1, \ldots, y_n\}\) be the observed data, and \(m \in \{1, \ldots, M\}\) be an index on the \(M\) parametric models. Then the posterior of the dose response curve, \(\mu(d)\), of the Bayesian model averaging model is

\[\begin{eqnarray*}
p(\mu(d) \mid y) &=& \sum_{m=1}^M p(\mu(d) \mid y, m) p(m \mid y) \\
p(m \mid y) &=& \frac{p(y \mid m) p(m)}{\sum_{m^* } p(y \mid m^* )p(m^*)}
\end{eqnarray*}\]

where \(p(\mu(d) \mid y, m)\) is the posterior mean dose response curve from model \(m\), \(p(m \mid y)\) is the posterior weight of model \(m\), \(p(y \mid m)\) is the marginal likelihood of the data under model \(m\), and \(p(m)\) is the prior weight assigned to model \(m\). In cases where \(p(y \mid m)\) is difficult to compute, Gould (2019) proposed using the observed data’s fit to the posterior predictive distribution as a surrogate in calculating the posterior weights; this is the approach used by dreamer.

dreamer supports a number of models including linear, quadratic, log-linear, log-quadratic, EMAX, exponential, for use as models that can be included in the model averaging approach. In addition, several longitudinal models are also supported (see the dreamer vignette). All of the above models are available for both continuous and binary endpoints.

## Reference

Gould, A. Lawrence. “BMA‐Mod: A Bayesian model averaging strategy for determining dose‐response relationships in the presence of model uncertainty.” *Biometrical Journal* 61.5 (2019): 1141-1159.