hey there.
definitely believe in yourself.
i applied to this program and i got in.
i've heard that applications are now closed for this program, but definitely try again next year.
the application process is very holistic; your 'personal statement' defining why you want LSE, why this program (etc.) is extremely critical to your success, as are academic references.
if you have these items, then 'sell yourself' to lse.
they want confidence; not self-doubt.
best of luck.
luigi