Failure of putative father to claim paternity directly in the two weeks available to him prior to his death from lung cancer is not proof of nonpaternity of illegitimate minor child seeking child's insurance benefits. Secretary has considerable latitude to determine the type and extent of evidence needed to establish paternity so long as such standards are reasonable. Discretion does not include the right to act arbitrarily or without criteria. Court doubts the ability of any ALJ in these circumstances to proceed with the case unaffected by his earlier judgment and suggests it might be appropriate to have a different ALJ "take a completely fresh look" at case. ALJ's conclusion that claimant failed to meet her burden of proof of child's paternity was unreasonable and not supported by substantial evidence.