Here are two introductory phrases that do not refer to the nouns or pronouns following them -- which in these cases are the subjects of the sentences.
Subjects are in blue, verbs in red, problem words in purple. The first example is from a letter offering customers a Platinum Visa card:
As a valued Brand Insurance customer, we are pleased to offer you …
The company making the offer has confused itself with the customer to whom the offer is being made. This is a very common syntactical problem. There are two fixes:
Fix #1 -- Change 'we' to 'you', and re-write the remainder of the sentence to suit:
As a valued Brand Insurance customer, you are invited to accept …
Fix #2 -- Move the introductory phrase or clause to follow the noun or pronoun to which it refers. In this location it needs to be as short as possible, so I'll delete the brand name.
We are pleased to offer you, as a valued customer, …
The next example comes from a Forbes online story:
After all, prior to filing its Initial Public Offering (IPO), I argued that Groupon lacks a competitive advantage …
This journalist may very well have argued that Groupon lacked a competitive advantage, but he never filed Groupon's IPO forms. In this case, the best fix is to move the misleading phrase to immediately before 'Groupon', because that's what the phrase is talking about.
After all, I argued that prior to filing its Initial Public Offering (IPO), Groupon lacked a competitive advantage …
Note that I changed 'lacks' to 'lacked', to suit the past-tense verb, 'argued'. Sticklers would also insert a comma after 'that'. I would join them, except for the comma that already lies after 'all'. Two comma in five words seems a bit chunky, but ... it's up to you.
The video tutorial presents all of this with sound, colour and movment.