A family of confidence intervals for a one-dimensional parameter is represented by a confidence distribution. The usual method of calculating confidence intervals from a likelihood function constitutes a one-to-one correspondence between confidence distributions and likelihood functions. This provides a likelihood representation of indirect data summarized by confidence intervals (or subjective judgement represented by a prior distribution) and let us synthesis indirect and direct new data in a combined likelihood function. Such likelihood synthesis avoids some of the problems encountered in Bayesian analysis. In the multi-parameter case, the correspondence between confidence distributions and likelihood functions is less obvious. However, with an an acceptable representation of the likelihood function, inference on a parameter derived from the basic parameter vector can be based on the corresponding marginal confidence distribution. The methodology is illustrated and compared to the Bayesian by an example concerning bowhead whales off Alaska.