Why do some investment firms publish their market predictions?

Why do they publish their market predictions instead of keep them for themselves to maximize their profits

Publicity.

Further:

(assuming they view their predictions as better than random)?

In my opinion, they actually fully know that such predictions are comic, but that it's part of the publicity/investment business complex.

Thus, when dealing with the highest level marketing folks, at the biggest name companies, they're attitude is absolutely no different whether selling beer, Investments or cars. They don't "believe" what they're saying, they just (in each industry as relevant) Emit Stuff in a polite and professional manner, in accordance with the patterns of the industry, and of course you don't overtly lie or break any standards or codes in that industry/jurisdiction.

They do so because it’s free advertising. If they get it wildly wrong, it will be totally forgotten —- it’s not news when a prediction fails to come to pass. Unfortunately, it is often considered news when prediction does come to pass, or even comes close. If it is at 3,850 they will be in the news (again) as being amazingly accurate. If it’s 2900 or 4900, they won’t be called out on it.

Their business is investing other people’s money, if they made perfect prediction that would be a waste of resources. Making non-specific (and of course non-binding) predictions public is just another way to get their name out there and get people to use their services.

Besides the publicity angle, there's also the possibility that a company could use these predictions for more direct profit. Consider the following:

• Buy up lots of stock at $50/share
• Publish that you predict that stock will get to $70/share
• Wait for people following your advice to start pushing the stock price up
• Sell the stock when it gets to $70/share. (or even $60/share)

In your particular case, they're not dealing with a single stock, but there might still be some manipulation -- trying to get people to vote for a specific candidate, without having to directly endorse that candidate.

In the case of this article, there's bias by omission -- to quote the article, highlighting two portions:

A victory for the Republican candidate could push the S&P 500 to as high as 3,900 at year-end under the most optimistic case laid out by Dubravko Lakos-Bujas, the bank’s chief U.S. equity strategist. ... While a number of traders have come to consider a Democratic sweep followed by a prompt fiscal deal among bullish scenarios for the equity market, Lakos-Bujas disagrees, seeing Trump’s victory as the most favorable outcome.

So, by only showing the "most optimistic" of all predictions, they may be cherry picking the data.

Assume we have a some scenario that we want to influence the outcome. Estimates are that Outcome A will result in an S&P increase of 8-12%, normally distributed. But for Outcome B, we have a wider range of predictions, and it's 3-13%, also normally distributed. But we can pick the outlier for B, and publicize that. So instead of comparing the means of 10% vs. 8%, we report on "most optimistic case", and compare 12% vs. 13%.

Leaving aside the question of how accurate these predictions are, since that's already been discussed in other answers: prediction is only half the picture. A large part of investment is figuring out how to use those predictions once you have them ("portfolio analysis").

If you knew exactly what the market was going to do, you'd just pick the investment with the best return and put all your money into that. But market forecasts are probabilistic, not certainties, and they're complicated probabilities because of non-independence - e.g. investing in five different fuel companies is a higher risk strategy than diversifying across five different industries.