Gemara

Kesuvos.85b: A man deposited seven pearls wrapped in a garment by R. Miyasha. R. Miyasha died, without leaving any instructions. Levi claimed that he deposited them.

R. Ami: (We believe Levi.) Firstly, R. Miyasha is not estimated to be rich. Secondly, Leah has an identifying Siman.

This applies only if Levi was not frequent in the house. If he was, perhaps another man deposited them, and Levi saw them there.

Rav Nachman and R. Aba ruled identically in similar cases of deposits of a gold cup and a silk garment.

Bava Metzi'a 24a: A silver cup was stolen from Mar Zutra Chasida's host. He noticed a Talmid drying his hands on someone else's garment. He reasoned that if the Talmid does not mind using another's property, he is the thief. He coerced the Talmid; the Talmid confessed.

Bava Basra 58a: A woman chastised her daughter: you should be secretive about Zenus! Your (alleged) father has no idea that only one of my 10 children is from him!

Her husband overheard this; he commanded that all his property be given to one son. No one knew which son he meant.

R. Bena'ah (to the sons): You should all bang on his grave until he reveals to you which son he meant.

All the sons did so, except for one. R. Bena'ah awarded him all the property.

Rishonim

Rif and Rosh (Kesuvos 44a and 9:8): The Gemara brings the cases in which R. Ami, Rav Nachman and R. Aba believed the depositor to show that no one argues with this Halachah.

Rosh (ibid.): We claim for orphans any claim that their father could have made, but only if it is common. Therefore, we do not claim that perhaps their father found a Metzi'ah. Both of the reasons are necessary. It does not suffice to give Simanim, for then someone regular in a rich person's house could tell Simanim of things to someone who is not regular there.

Hagahos Ashri: If the Mes was rich, we say that he bought the item. However, if one gives a Siman that could not be known if it was not a deposit, e.g. that it is wrapped in a turban, we return it even if the Mes was rich. The orphans need not show their possessions to let us see whether one has the Siman given, even if witnesses saw it deposited. We discuss when someone happened to see the item by them. The orphans must admit (if the item is by them).

Rosh (ibid.): R. Moshe of Narvona infers that this is only if the father did not specify and the orphans are unsure. If they say that their father said that it is his, they swear and do not pay. R. Yonah says that if the claimant was regularly in the house his claim is Batel and the orphans may use or sell the item. When there is no Vadai claim, we do not say that if the Mes who was not rich, every valuable thing left over must be left until Eliyahu. Perhaps he found a Metzi'ah!

The Rif and Rosh (Bava Metzi'a 13a and 2:5) bring the episode in which Mar Zutra Chasida coerced a Talmid (Rosh - with sticks) to confess.

Rambam (Hilchos Sanhedrin 24:1): Moshe died without giving instructions. Levi claims that he had a deposit by Moshe and gives superb Simanim on it. Levi did not regularly enter Moshe's house. If the judge knows that Moshe was not rich enough have such an object and he believes that it was not Moshe's, he takes it from the heirs and gives it to Levi. The same applies to all similar cases. It depends on what the judge believes to be the truth. The Torah requires two witnesses for a judge to rule according to them even if he does not know whether or not the matter is true.

Rambam (2): This is letter of the law. After the spread of Batei Din with judges of improper deeds or insufficient Chachmah... we do not take from heirs without a clear proof. We do not rely on assessment of the Mes or the one claiming.

Beis Yosef (CM 15 DH Kosav ha'Rambam): The Rif and Rosh cite a Gaon who requires a clear proof nowadays to disqualify someone from swearing. They do not say so about taking a deposit from heirs. The Tur cites the Rambam, but omits that nowadays we require a clear proof to take from heirs. The Rambam holds that just like nowadays we do not rely on estimation for other judgments, the same applies to a deposit. In Hilchos She'alah u'Pikadon (6:4) and Hilchos Nachalos (11:1) the Rambam says that we give to the depositor when the conditions are fulfilled. This is letter of the law. He relies on what he wrote here that nowadays we require proof.

Poskim

Shulchan Aruch (CM 15:4): A judge may withdraw from a case if he sees that the claimant is a liar, but not if the defendant is a liar, lest the liar profit. He must investigate to annul his scheme. An expert judge who is unique in his generation can obligate him to pay if he clearly assesses that he is liable.

Beis Yosef (DH ha'Dayan, citing Rosh 107:6): If every judge would refuse to judge a liar, the liar would be exempt. If a judge evaluates that a party is not answering questions to avoid obligating himself, the judge can obligate him as if he answered. An expert judge may rely on a clear estimation, even though he cannot clearly exposed the lie. We find that R. Bena'ah did so. He learned from Shlomo. When each of two women claimed that the live baby was hers, Shlomo ruled that the true mother is the one who had compassion on the baby. In many cases Chachamim rule according to estimation.

Rema: In this way a judge can demand a guarantor before granting time to a party (to find proofs). The judge acts according to what he sees. He can require an oath when the Halachah does not, in order to clarify the truth.

Shulchan Aruch (5): A judge can judge monetary cases according to what he is firmly convinced to be the truth, even if there is not a clear proof. After the spread of judges lacking proper deeds or Chachmah ... we do not take from heirs without a clear proof. We do not rely on assessment of the Mes or the one claiming.

SMA (16): This is like the Rambam. I do not know why the Rema did not mention that the Rif, Rosh and Tur hold that even nowadays we take from heirs through estimation

Rema: If it seems to Beis Din that one party stole from the other and they cannot make him pay through Din, they can decree on others not to do business with him. If the party is a woman, they can decree that no one marry her or engage in Shiduchim for her until she returns what is not hers.

Beis Yosef (DH Kosav Maharik): Maharik (14:2,3) says that a judge may do what seems improper if he sees that a party is lying or not cooperating. He must act according to the truth, not according to the claims if they are false.

Shulchan Aruch (297:1): Levi told Moshe's orphans that he had a deposit by Moshe and gave superb Simanim on it. If the judge knows that Moshe was not rich enough to have such an object, he gives it to Levi. This is if Levi did not regularly enter Moshe's house. After the spread of Batei Din with judges of improper deeds or insufficient Chachmah, we do not take from heirs based on a judge's estimation.