Bring back all relevant results when using ngrams with elasticsearch

I indexed my elasticsearch index with ngrams to make it possible to do fuzzy matching and prefix searches quickly. I notice that if I search for documents containing "Bob" in the name field, only results name = Bob return. I would like the response to include documents with name=Bob, but also documents with name = Bobbi, Bobbette, etcetera. The Bob results should have a relatively high score. The other results that don't match exactly, should still appear in the results set, but with lower scores. How can I achieve this with ngrams?

I am using a very small simple index to test. The index contains two documents.

 {
    "_index": "contacts_4",
    "_type": "_doc",
    "_id": "1",
    "_score": 1.0,
    "_source": {
      "full_name": "Bob Smith"
    }
  },
  {
    "_index": "contacts_4",
    "_type": "_doc",
    "_id": "2",
    "_score": 1.0,
    "_source": {
      "full_name": "Bobby Smith"
    }
  }

Solution

Here is a working example (using n-gram tokenizer):

ngram-tokenizer

Mapping

  PUT my_index
  {
  "settings": {
    "analysis": {
      "analyzer": {
        "my_analyzer": {
          "filter": [
            "lowercase"
          ],
          "type": "custom",
          "tokenizer": "my_tokenizer"
        }
      },
      "tokenizer": {
        "my_tokenizer": {
          "token_chars": [
            "letter",
            "digit"
          ],
          "min_gram": "3",
          "type": "ngram",
          "max_gram": "4"
        }
      }
    }
  },
  "mappings": {
    "properties": {
      "full_name": {
        "type": "text",
        "analyzer": "my_analyzer",
        "fields": {
          "keyword": {
            "type": "keyword"
          }
        }
      }
    }
  }
}

Indexing documents

POST my_index/_doc/1
{
  "full_name":"Bob Smith"
}

POST my_index/_doc/2
{
  "full_name":"Bobby Smith"
}

POST my_index/_doc/3
{
  "full_name":"Bobbette Smith"
}

Search Query

GET my_index/_search
{
  "query": {
    "match": {
      "full_name": "Bob"
    }
  }
}

Results

"hits" : [
  {
    "_index" : "my_index",
    "_type" : "_doc",
    "_id" : "1",
    "_score" : 0.1626403,
    "_source" : {
      "full_name" : "Bob Smith"
    }
  },
  {
    "_index" : "my_index",
    "_type" : "_doc",
    "_id" : "2",
    "_score" : 0.13703513,
    "_source" : {
      "full_name" : "Bobby Smith"
    }
  },
  {
    "_index" : "my_index",
    "_type" : "_doc",
    "_id" : "3",
    "_score" : 0.11085624,
    "_source" : {
      "full_name" : "Bobbette Smith"
    }
  }
]

Hope this helps