How appellate courts should approach conviction appeals is surprisingly unclear. Today the High Court of Australia corrected the Supreme Court of Western Australia on this: Rodi v Western Australia  HCA 44 at , and, also today, the New Zealand Supreme Court granted leave to appeal on whether a conviction appeal after a judge-alone trial requires the appellate court to adopt a review approach or to substitute its view of the evidence for that of the trial judge: Sena v Police  NZSC 92 (leave granted).
[Update: On 6 May 2019 the Supreme Court granted Mr Lundy leave to appeal:  NZSC 45. Leave was limited to the proviso issue. We will have to wait to see whether there was an error made by the Court of Appeal insofar as it may have failed to consider the cumulative effect of the - admittedly, individually weak - inferences that could have supported the defence case. Could they, in combination, have raised a reasonable doubt? On this point we may recall how the reasoning of Ian Binnie in the Bain compensation case was criticised for (allegedly) not considering in combination points that supported the Crown's case. Place your bets. As any Bayesian bookie will tell us, if - for the sake of argument - a reasonable doubt exists when P(G) = 0.94 or less, then to move from a prosecution case that established guilt to P(G) = 0.99 to get to a reasonable doubt, you would have to accept defence evidence which, taken overall, was at least 6 times more consistent with innocence than with guilt. On the fairness side of the coin, broadly speaking, the issue would be whether there was a real risk that the wrongly admitted scientific evidence affected assessment of the probative value of the other evidence in a more than trivial way, by causing the jury to over-estimate the probative value of the evidence against the defendant or to under-estimate the probative value of the evidence that favoured the defendant.]
[Another update: after the Supreme Court delivered its judgment, I discuss the outcome here.]