Brilliant Pokémon have 1.6× the likelihood of appearing, and Shiny Pokémon have 2× the likelihood of appearing
The Pokemon Company themselves indirectly say that you have a higher chance of finding shiny Pokemon by encountering Brilliant Pokemon.
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However, knowing Ty, that's not good enough for them. So here's a direct statement with a couple sources.
Brilliant Pokémon are more likely to be Shiny, depending on the number of Pokémon of that species that the player has caught or defeated. If a Brilliant Pokémon is not Shiny, the game will regenerate its personality value up to 6 additional times (depending on the number of times the player has caught or defeated that species) to try and make it Shiny, resulting in the Pokémon being approximately 7 times more likely than normal to be Shiny. This stacks with the Shiny Charm (which provides 2 bonus rolls), allowing for up to 9 bonus rolls (approximately a 1/456 chance) if the player has the Shiny Charm and caught or defeated at least 500 of that species.
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Hope this answers your question, ty