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Most of the time when I read Japanese sentences, I have no confidence whether
the speaker or another person is the subject. Let's see one example as
follows.

> 主人は、料理にかけては、誰にも引けを取らないと信じている。

I think the speaker is a woman (housewife). But what I don't understand is who
the believer is? Here I have two interpretations as follows:

  * A: The speaker is the believer 

> When it comes to cooking, I believe that my husband is better than anyone
> else.

  * B: The speaker's husband is the believer

> When it comes to cooking, my husband believes that he is better than anyone
> else.

Is there a method to determine the correct subject in this kind of sentence?

# Edit

Rather than posting a new question that seems to be related to this question,
let me ask an additional question here. If you think it should be separated, I
will do it.

Considering [broccoli forest's
comment](https://japanese.stackexchange.com/questions/35859/who-is-the-
subject-of-this-sentence#comment70574_35861) as follows,

> It's perfectly natural if you parse it 「主人は誰にも引けを取らない」と信じている。 where
> 「主人は誰にも引けを取らない」 is what the writer thinks.

it leads me to the additional question "Does each of the following make the
writer the subject?"

  * Adding 「・・・」as @broccoli forest did.

> **「** 主人は、料理にかけては、誰にも引けを取らない **」** と信じている。

  * Adding こと.

> 主人は、料理にかけては、誰にも引けを取らない **こと** と信じている。

  * Adding ということ.

> 主人は、料理にかけては、誰にも引けを取らない **ということ** と信じている。
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grammar
Who is the subject of this sentence?
The general answer is to use a mixture of **grammatical, contextual, and
cultural cues**.

In this case, it seems like the subject should be the husband.

**Grammatically:** If you remove 「、料理にかけては、」which just refines the context to
just cooking, you get 「主人は誰にも引けを取らないと信じている。」which maybe makes it a little
easier to see that 主人 is the only potential explicit subject. The speaker
could be an implicit subject, but then the author has the option to use が
instead, e.g., 「料理にかけては、主人が誰にも引けを取らないと信じている」, in which case it becomes quite
likely the wife is the subject. The original version could still have the wife
as the subject, but the fact that the author chose not to go with this kind of
option instead might give us a minor nudge in the direction of thinking the
husband is the subject.

**Cultural Cues:** 誰にも引けを取らない is quite the confident statement. Assuming the
wife was directly saying she believes this, either she is bragging in an
unusually direct way, in which case she might as well drop と信じている entirely, or
she is likely to hedge heavily. So I interpret the presence of と信じている as being
a further hint that it is the husband's belief, not the wife's.

**Contextual:** The thread of thought of the speaker would likely allow us
decide which way makes more sense, but unfortunately no context is presented
in the original question.

* * *

Regarding the post-edit additional question:  
The first of the three is more clearly the wife, since the husband gets
"erased" from the "highlighted context" by the closing quote to some degree.

For the other two, I don't think the additions change which subject is implied
from a strict grammatical point of view, as the subject of 信じている is still left
implicit, and we are still forced to decide if the initial 主人は provides that
subject, or if it is omitted entirely (implying the wife).

I can't comment on whether the additions shift things in a
usage/cultural/nuance way, as both sentences look a little odd to me.

* * *

[Disclaimer: I'm not a 100% confident about all aspects of my interpretation,
so if an advanced Japanese speaker presents a contradicting answer, I would go
with theirs instead of mine.]
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answer
The given quote, "主人は、料理にかけては、誰にも引けを取らないと信じている" is incomplete and confusing
even to native Japanese speakers.

It can be taken either way as you say;

  * My husband believes he is a better cook than anyone else, or

  * I believe my husband is a better cook than anyone else.

But I guess the quoted sentense suggests that the speaker's husband believes
himself being an excelent cook," unless the speaker says
"主人は、料理にかけては誰にも引けを取らないと、 **私は** 信じている."

As Japanese language omits the subject very often unlike English language, you
need to be trained to speculate and judge what the subject is from the
context. I don't think there is an established rule or playbook to identify
it.
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