The presumption of innocence refers to two things: the burden of proof and the standard of proof. It means that the burden of proof is on the prosecution, and the standard of proof is beyond reasonable doubt. The presumption of innocence applies at a criminal trial when the ultimate issue is whether the defendant is guilty.
The presumption of innocence is not the same thing as the right to a fair trial, although in some generalised conceptions of fair trial the presumption of innocence is regarded as a requirement for trial fairness. But it need not be, and in any event the presumption of innocence has exceptions.
The most obvious exception to the presumption of innocence is that if the defendant is relying on the defence of insanity, he must prove that on the balance of probabilities he was insane at the relevant time. This exception is justified on the basis that he will know more about his state of mind than the prosecution, so he should have the burden, and, as it seems unfair to require him to prove insanity to the standard of beyond reasonable doubt, the lesser standard of the balance of probabilities applies.
The law has not regarded the need to raise a reasonable doubt on the issue as a standard of proof. There are only two standards of proof: beyond reasonable doubt, and the balance of probabilities. See, for example Z v Dental Complaints Assessment Committee (blogged here 25 July 2008) and the Index to this site, and the entries under the heading "Standard of proof", for exceptions to this generalisation.
Another illustration of an exception to the presumption of innocence is where, on proof that he had a particular quantity of illicit drug in his possession, the defendant must prove on the balance of probabilities that he did not have an intention to supply it to another person, if he is to avoid conviction for possession of the drug for supply. This exception is more controversial than the insanity one, because the prosecution should have access to evidence of the defendant's dealings, and should be able to prove circumstances indicating an intention to supply the drug. That is not particularly different from requiring the prosecution to prove the mental elements of possession of the drug. So this exception to the presumption of innocence may be thought to lack sufficient justification in policy terms and may be a breach of the right to be presumed innocent. See Hansen v R  NZSC 7 (blogged 20 February 2007). In the UK the raising of a reasonable doubt has been held to be "proving" absence of a mental element: R v DPP ex p. Kebilene  AC 326, so that amounts to recognition of a third standard of proof.
Sometimes a reverse onus of proof is applied to the issue of the magnitude of a confiscation order in proceeds of crime applications. This situation existed in Grayson and Barnham v United Kingdom  ECHR 877 (23 September 2008). The focus here was on whether a provision requiring the applicants to show on the balance of probabilities that they did not have sufficient assets to meet a confiscation order, the amount of which corresponded to the court's assessment of their benefits from their respective offending, was in breach of the fair trial requirement in Art 6§1 of the Convention. This was important, not only because of the amount of money involved, but also because in default of payment prison sentences which were quite hefty were imposed.
The Strasbourg court has long regarded the presumption of innocence as part of the general notion of a fair hearing: Saunders v. United Kingdom, judgment of 17 December 1996, Reports of Judgments and Decisions 1996-VI, § 68. Also, proceedings of this nature are treated as analogous to sentencing: Phillips v. United Kingdom (no. 41087/98, §§ 35 and 39, ECHR 2001-VII.
Well, in the present case it was the applicants (the original accuseds) who had the means of proving their present inability to pay. The Court quoted Lord Steyn, with whom the other Law Lords agreed, in R v Rezvi  UKHL 1, who approved the following reasons given by the Court of Appeal supporting the exception to the presumption of innocence in this context:
"(a) It is only after the necessary convictions that any question of confiscation arises. This is of significance, because the trial which results in the conviction or convictions will be one where the usual burden and standard of proof rests upon the prosecution. In addition, a defendant who is convicted of the necessary offence or offences can be taken to be aware that if he committed the offences of which he has been convicted, he would not only be liable to imprisonment or another sentence, but he would also be liable to confiscation proceedings.
(b) The prosecution has the responsibility for initiating the confiscation proceedings unless the court regards them as inappropriate ...
(c) There is also the responsibility placed upon the court not to make a confiscation order when there is a serious risk of injustice. As already indicated, this will involve the court, before it makes a confiscation order, standing back and deciding whether there is a risk of injustice. If the court decides there is, then the confiscation order will not be made.
(d) There is the role of this court on appeal to ensure there is no unfairness."
Those backstop matters – the appropriateness of the proceedings, the avoidance of a serious risk of injustice, and the appellate court's role of ensuring there is no unfairness – were reflected in the ECtHR's conclusion (49):
"The Court agrees with the judgments of the Court of Appeal in the instant cases ... that it was not incompatible with the notion of a fair hearing in criminal proceedings to place the onus on each applicant to give a credible account of his current financial situation. In each case, having been proved to have been involved in extensive and lucrative drug dealing over a period of years, it was not unreasonable to expect the applicants to explain what had happened to all the money shown by the prosecution to have been in their possession, any more than it was unreasonable at the first stage of the procedure to expect them to show the legitimacy of the source of such money or assets. Such matters fell within the applicants' particular knowledge and the burden on each of them would not have been difficult to meet if their accounts of their financial affairs had been true."
There was therefore no violation of the right to a fair trial arising from the reverse onus. The applicants had argued that they were in the unfair position of having to prove a negative, and (for one of the applicants) that the assumptions made by the lower court when it assessed the benefit obtained from offending continued to operate as assumptions when the court decided the ability to pay. The point seems to be that not only did the applicant have to prove inability on the balance of probabilities, but also he had to do so in the face of an additional assumption against him. This was not explored by the Court.
The argument raises two concerns: did the additional assumption against the defendant, assuming it existed, mean that unfairness occurred in the sense that the standard of proof he had to satisy was raised by requiring him to adduce more cogent evidence than he otherwise would have had to adduce; or, was there unfairness arising from a bias by the court against the defendant, which again required him to adduce more cogent evidence that should have been the case? European law asserts that the standard, the balance of probabilities, does not change with the seriousness of the consequences of the decision (see blog discussing Z, above). However, here there was nothing to indicate that the courts in the case of each of the applicants had applied the wrong standard or that they were biased.