This paper presents Bayesian techniques for conservative claims about software reliability, particularly when evidence suggests the software's executions are not statistically independent. We formalise informal notions of "doubting" that the executions are independent, and incorporate such doubts into reliability assessments. We develop techniques that reveal the extent to which independence assumptions can undermine conservatism in assessments, and identify conditions under which this impact is not significant. These techniques - novel extensions of conservative Bayesian inference (CBI) approaches - give conservative confidence bounds on the software's failure probability per execution. With illustrations in two application areas - nuclear power-plant safety and autonomous vehicle (AV) safety - our analyses reveals: 1) the confidence an assessor should possess before subjecting a system to operational testing. Otherwise, such testing is futile - favourable operational testing evidence will eventually decrease one's confidence in the system being sufficiently reliable; 2) the independence assumption supports conservative claims sometimes; 3) in some scenarios, observing a system operate without failure gives less confidence in the system than if some failures had been observed; 4) building confidence in a system is very sensitive to failures - each additional failure means significantly more operational testing is required, in order to support a reliability claim.

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